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filter=lfs diff=lfs merge=lfs -text diff --git a/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/2301.04187v1.pdf.txt b/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/2301.04187v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d479cd05ccaf5cc70e5e777dbbbb90dddb4c261e --- /dev/null +++ b/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/2301.04187v1.pdf.txt @@ -0,0 +1,757 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Gravitational wave source populations: Disentangling an AGN component +V. Gayathri,1, 2 Daniel Wysocki,2 Y. Yang,1 R. O’Shaughnessy,3 Z. Haiman,4 H. Tagawa,4 and I. Bartos1 +1Department of Physics, University of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA +2Leonard E. Parker Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, +USA∗ +3Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA +4Department of Astronomy, Columbia University, 550 W. 120th St., New York, NY, 10027, USA +ABSTRACT +The astrophysical origin of the over 90 compact binary mergers discovered by the LIGO and Virgo +gravitational wave observatories is an open question. While the unusual mass and spin of some of the +discovered objects constrain progenitor scenarios, the observed mergers are consistent with multiple +interpretations. A promising approach to solve this question is to consider the observed distributions of +binary properties and compare them to expectations from different origin scenarios. Here we describe a +new hierarchical population analysis framework to assess the relative contribution of different formation +channels simultaneously. +For this study we considered binary formation in AGN disks along with +phenomenological models, but the same framework can be extended to other models. We find that +high-mass and high-mass-ratio binaries appear more likely to have an AGN origin compared to the +same origin as lower-mass events. Future observations of high-mass black hole mergers could further +disentangle the AGN component from other channels. +1. INTRODUCTION +Understanding the origin of binary black hole merg- +ers is the first step in utilizing black hole mergers to +probe a range of astrophysical processes. +The LIGO +(Aasi et al. 2015) and Virgo (Acernese et al. 2015) grav- +itational wave observatories have discovered about 90 +binary mergers so far (Abbott et al. 2021a), provid- +ing important information on the astrophysical popu- +lation of mergers. Binary black hole systems can form +through various channels, including isolated stellar bi- +naries (Portegies Zwart & Yungelson 1998; Belczynski +et al. 2002; Marchant et al. 2016; de Mink & Mandel +2016) or triples (Antonini et al. 2014; Kimpson et al. +2016; Veske et al. 2020), dynamical interactions in star +clusters (Sigurdsson & Hernquist 1993; Portegies Zwart +& McMillan 2000), primordial black holes formed in the +early universe (Carr & Hawking 1974), and in the ac- +cretion disks of active galactic nuclei (AGNs; McKernan +et al. 2012; Bartos et al. 2017; Stone et al. 2017; Tagawa +et al. 2020b; McKernan et al. 2020, 2022). +The increased number of binary black hole observa- +tions allows for a more detailed investigation of the pop- +ulation’s mass and spin distributions. +While individ- +ual events may provide anecdotal suggestions hinting at +∗ gayathri.v@ligo.org +one formation channel or another, only an interpreta- +tion of the full census can enable one to disentangle the +potential contributions from multiple formation scenar- +ios. Several studies have previously explored how to dis- +entangle multiple channels, largely relying on compar- +ison to phenomenologically-motivated estimates of the +detailed outcomes of full formation scenarios (Doctor +et al. 2020; Gerosa & Fishbach 2021; Gayathri et al. +2021, 2020; Yang et al. 2020b; Tagawa et al. 2021; Kim- +ball et al. 2021). +Some studies also shown a mixture +of channels is strongly preferred over any single channel +dominating the detected population (Zevin et al. 2021). +The latest population analysis carried out by LIGO- +Virgo-KAGRA, GWTC3 (Abbott et al. 2021a,b), iden- +tified several population features that may be indica- +tive of the binaries’ origin. First, there appears to be +a peak in the binary black hole mass spectrum around +30-40 M⊙ compared to a more simple power-law type +population (Tiwari & Fairhurst 2021; Talbot & Thrane +2018; Edelman et al. 2022; Sadiq et al. 2022; Fishbach & +Holz 2017). At the same time a few observed black holes +had unusual properties, such as masses in the so-called +upper mass gap (≳ 50 M⊙), highly unequal masses in +the binary, high spin and precessing mergers, which are +rare in stellar evolution and might be indicative of al- +ternative formation scenarios. +arXiv:2301.04187v1 [gr-qc] 10 Jan 2023 + +2 +Here we introduce a flexible approach to compare +the predictions of detailed formation models with +observations while simultaneously accounting for the +potentially-confounding contributions from a flexible +phenomenologically-parameterized model for compact +binary formation. +Our paper is organised as follows. In section 2 we in- +troduce binary formation in AGN disks, phenomenolog- +ical descriptions for formation in non-AGN sources, and +our flexible parametric population inference method. In +section 3, we talk about the analysis’s key findings. In +section 4, we summarize our findings and comment on +future directions. +2. METHODS +2.1. Binary mergers in AGN disks +We construct a one-parameter model for BBH forma- +tion and merger within an AGN disk, parameterized +by the maximum mass mmax of the natal BH distri- +bution. Specifically, we adopt a seed BH mass distri- +bution which follows the Salpeter mass function with +index 2.35, dN/dm ∝ m−2.35 with given mmax. +For +neutron stars (NS) we assume a normal distribution +m/M⊙ ∼ N(1.49, 0.19). +The BHs and NSs are as- +sumed to orbit a supermassive black hole in an AGN, +migrating into the disk and inward from their natal lo- +cations. Close to the AGN, these objects undergo multi- +ple encounters, facilitating binary formation and merger. +Other AGN parameters are fiducial values that are ex- +pected to be typical, while there are large uncertainties. +The range of possible values in the AGN models param- +eter space is discussed in (McKernan et al. 2018). +Following (Bartos et al. 2017), we adopted a geometri- +cally thin, optically thick, radioactively efficient, steady- +state accretion disk expected in AGNs. We used a vis- +cosity parameter α = 0.1, radioactive efficiency ϵ = 0.1, +fiducial supermassive BH mass M• = 106 M⊙ and ac- +cretrion rate 0.1 ˙MEdd, where +˙MEdd is the Eddington +accretion rate. +Using Yang et al. (2020) and Tagawa +et al. (2020b,a), we have computed the expected mass +and spin distributions of binary mergers in AGNs. +Figure +1 +shows +the +binary +black +hole +merger +intrinsic +parameter +distribution +for +the +AGN +model with different initial mass limits (mmax += +[15M⊙, 35M⊙, 50M⊙, 75M⊙]). +As the natal BH mass +upper limit mmax increases, more massive binary com- +ponents and total masses are allowed. At the same time, +for high mmax, asymmetric systems are more frequent. +By contrast, we have observed that the spin distribu- +tion properties are largely independent of our choice for +mmax limit. +Figure 1. +Parameter distributions for binary black holes +formed in our AGN disk formation models; line color indi- +cates the maximum natal BH mass. As expected, as the max- +imum mass increases, total (M) mass upper limits increase. +Additionally, a higher BH maximum natal mass mmax in- +creases the relative frequency of asymmetric mergers, partic- +ularly highly asymmetric mergers with q < 1/10. +2.2. Phenomenology of AGN and non-AGN sources +To allow for binary black holes which have a non- +AGN origin, we follow previous work and introduce a +few-parameter mixture model family. As illustrated in +Figure 2, for the non-AGN component, we allow binary +black holes to arise from a mixture of a power law and +Gaussian components, as detailed in Table 11 of Abbott +et al. (2020). +In the power law component, the pri- +mary is drawn from a pure power law mass distribution +(with some unknown mmax,pl < 50M⊙ and unknown +primary power law index); the mass ratio is drawn from +another power law; and the spins are drawn from an + +10- +mmax = 15 +mmax = 35 +10-2, +mmax = 50 +mmax = 75 +10-3 +PDF +10-4 +10-5 +10-6 +50 +100 +150 +200 +250 +300 +Taotal Mass (Mo)100 +PDF +10-1 +mmax= 15 +mmax=35 +mmax= 50 +mmax = 75 +0.2 +0.4 +0.6 +0.8 +1.0 +Mass Ratio3 +unknown Beta distribution. +In the Gaussian compo- +nent, the primary and secondary are drawn from two +independent Gaussian distributions with unknown mean +and variance, with both means confined a priori to be +near 30 − 40M⊙ to be consistent with expectations for +PISN supernovae. +By including a second component +this phenomenological model can allow for non-power- +law features and also allow for spin distributions that +vary with mass. Each component k has some undeter- +mined overall rate Rk. The top panel of Figure 2 shows +the general non-AGN model. +Our overall model is therefore a mixture model, pa- +rameterized by the unknown (continuous) AGN merger +rate and its (discrete) maximum mass mmax, along with +all parameters of the non-AGN mixture model. +The +overall merger rate density dN/dV dXdt can therefore +be expressed as a sum +dN +dV dXdt = Ragnpagn(X|mmax)+Rgpg(X|Λg)+Rplppl(X|Λpl) +where X are binary parameters and where pq, Λq are the +model distributions and parameters for the qth compo- +nent (AGN, Gaussian, and power-law respectively). +To systematically assess how well the distinctive fea- +tures in AGN formation scenarios can be disentangled +from this large model family, we will perform a se- +quence of calculations with increasing model complexity, +as shown in the panels of Figure 2. Specifically, we con- +sider without any non-AGN component; the power-law +and Gaussian (PL+G) model only; the power-law and +AGN model (PL+AGN), without any Gaussian compo- +nent; and finally the most general model with all three +components. +2.3. Population inference +We describe and demonstrate a flexible parametric +method to infer the event rate as a function of compact +binary parameters, accounting for Poisson error and se- +lection biases. In (Abbott et al. 2021c), analysed the +Multi-Spin model which is the joint mass-spin model for +binary black holes. Independent analyses have shown +that there is a feature in the BBH mass spectrum around +30-40 M⊙, which is modeled as a Gaussian peak on top +of a power law continuum. It is an empirical way of mod- +eling extra features, but here we tried to understand it +feature with AGN models. +In this section we review the population inference +Wysocki et al. (2019) used for multi-formation chan- +nel contributions. Binaries with intrinsic parameters x +would merge at a rate dN/dVc dtdx = R p(x), where N +is the number of detections, Vc is the comoving volume, +R is the space-time-independent rate of binary coales- +cence per unit comoving volume and p(x) is the probabil- +ity of x from detected binaries. The binaries intrinsic pa- +rameters includes mass mi and spins Si, where i = 1, 2. +The likelihood of the astrophysical BBH population at a +Figure 2. Graphical representations of the BBH population +analysis in m1 − m2−spin parameter. +Top panel for the +PL+G, the middle panel for the PL+AGN and the bottom +panel for the PL+G+AGN model. +given merger rate R (Loredo 2004; Mandel et al. 2019; +Thrane & Talbot 2019) and given binary intrinsic pa- +rameters X ≡ (m1, m2, χ1, χ2), where χi = Si/m2 +i , +given the data for N detections D = (d1, ..., dN). This +the likelihood is given by +L(R, X) ≡ p(D|R, X), +(1) +L(R, X) ∝ e−µ(R,X) +N +� +n=1 +� +dx ℓn(x) R p(x, X), +(2) +where µ(R, X) is the expected number of detections un- +der a given population parametrization X. Using Bayes’ + +m2(Mo) +spin +BBH +spin +PL +mi(Mo)m2(Mo) +BBH +spin +PL+AGN +m1 +1(Mo)m2(Mo) +spin +G +BBH +spin +PL +AGN +mi(Mo)4 +theorem one may obtain a posterior distribution on R +and X after assuming some prior p(R, X). For compu- +tational efficiency and to enable direct comparison with +discrete formation models, we use the Gaussian likeli- +hood approximation technique introduced in (Delfavero +et al. 2022) to characterize each BBH observation’s like- +lihood ℓn(x). +Using this formalism, we estimate what fraction of +binary black holes are generated via the AGN channel +using detected gravitational wave merger information. +To do this study we have upgraded current parametric +methods with a mixture model feature. Here we have a +freedom to do the analysis with number of models which +has different astrophysical binary distributions. +3. ANALYSIS +As we discussed before, we perform population infer- +ence analysis with mixture model feature and detected +confident detection binary black hole detection. For this +study we have considered different the astrophysical bi- +nary distributions from AGN ( with different mmax), +power-law, Gaussian peak and its combinations (see Fig- +ure 2). +3.1. Astrophysical merger rate +We have estimated the astrophysical black hole merger +rate for a given astrophysical model with a given +number of GW detection. +For this study, we have +used the parameter estimation samples from obtained +by LIGO-Virgo-KAGRA Collaboration (Abbott et al. +2021d,e, 2019, 2021f) and it available in the Gravita- +tional Wave Open Science Center (https: //www.gw- +openscience.org) with the mixed model. Here we follow +the same selection criteria as (Abbott et al. (2021b)), +we have considered events with a false alarm rate of +< 0.25yr−1. +Table +1 +shows +the +merger +rates +inferred +for +joint PL only, +G only, +AGN only (with different +mmax), PL+AGN model (with different mmax) and +PL+G+AGN models (with different mmax). We have +estimated the merger rate results derived using each in- +dividual model component as well as combined. We have +observed that the inferred merger rate from the AGN- +only models with different choices for mmax produces +largely consistent results peaking near 50 Gpc−3yr−1. +For the smallest maximum BH natal mass mmax = +15M⊙, the inferred single-component AGN merger rate +peaks around 80 Gpc−3yr−1. For all other models, they +peak around the same R. Similarly, we have inferred +the merger rate distribution for the single-component +Gaussian, and power-law components. As expected, the +merger rate from the power-law component dominates +overall, as it incorporates and describes many frequent +mergers of the lowest-mass binary black holes. In the +case of PL+AGN, the inferred AGN and PL compo- +nents have a well-determined merger rate, with median +AGN merger rate ≃ 8/, Gpc−3yr−1 and PL merger rate +≃ 30/, Gpc−3yr−1. Changing the maximum natal BH +mass mmax has a very mild impact on the inferred AGN +merger rate, and almost no effect on the inferred PL +merger rate. +In +the +case +of +PL+G+AGN +analyses, +the +in- +ferred AGN, G and PL components have a well- +determined merger rate, with median AGN merger rate +≃ 7/, Gpc−3yr−1, G merger rate ≃ 4/, Gpc−3yr−1 and +PL merger rate ≃ 30/, Gpc−3yr−1. +As we have seen +in the PL+AGN study, we have not seen any major +effect on PL or G merger rate when we change the +AGN model. Our inferred merger rates deduced from +the multi-component model are consistent with infer- +ences performed using single-component models alone, +suggesting that inference isolates only the contribution +from each component. +3.2. Inferred merger rate versus mass +To better appreciate how well our inference directly +projects out the relative contribution from each com- +ponent, Figure 3.2 and 3.2 shows our inferred merger +rate versus mass for PL+AGN and PL+G+AGN mod- +els analyses respectively. In each plot we have shown +each model component in mass space. +As we expected the low mass region is highly con- +tributed by the PL model compared to other models. +We have observed a peak in PL model distribution +around 7M⊙ for both PL+AGN as well as PL+G+AGN +analyses. The peak is prominent for the PL+G+AGN +study compared to PL+AGN. The high mass region is +represented by only AGN, as we expected to see. Note +that, the AGN model not only contributes in high mass +region it also contributes full mass space as shown in 3.2 +and 3.2. +3.3. Inferred merger rate versus mass ratio +Similarly here we show the inferred merger rate ver- +sus mass ratio for PL+AGN and PL+G+AGN models +analyses. +Figure 3.2 and 3.2 shows our inferred merger rate ver- +sus mass ratio for PL+AGN and PL+G+AGN mod- +els analyses respectively. In each plot we have shown +each model component in mass ratio space. As we ex- +pected the low mass ratio region contributed by AGN +model for PL+AGN analysis and AGN & G models for +PL+G+AGN analysis. For high q, the dominate contri- +bution from PL model, that is consistence with detected +events. + +5 +Analysis +models +mmax = 15 +mmax = 35 +mmax = 50 +mmax = 75 +PL only +71.9+19.9 +−20.4 +- +- +- +- +G only +19.1+2.5 +−2.3 +- +- +- +- +AGN only +84.7+19.5 +−18.5 +49.8+6.1 +−5.5 +52.9+7.8 +−6.1 +53.2+7.7 +−6.2 +PL+AGN +PL +29.3+9.9 +−7.2 +28.8+11.3 +−6.8 +25.7+7.6 +−6.3 +26.8+9.5 +−6.2 +AGN +8.7+6.3 +−4.5 +8.3+8.3 +−3.7 +12.7+6.5 +−4.1 +11.9+4.6 +−3.2 +PL+AGN+G +PL +21.3+7.3 +−5.2 +23.5+6.6 +−5.5 +21.6+6.7 +−5.0 +23.3+7.1 +−5.0 +AGN +6.7+5.6 +−4.0 +6.2+5.0 +−3.5 +12.7+5.0 +−4.9 +12.9+4.3 +−4.3 +G +10.2+2.2 +−2.3 +9.3+2.6 +−3.7 +5.9+2.9 +−1.9 +5.6+2.2 +−1.5 +Table 1. The astrophysical rates from PL+AGN and PL+AGN+G models. Each row corresponds to each analysis and each +column corresponds to different AGN models with different initial mass limit. +Figure 3. +The inferred merger rate versus mass for +PL+G+AGN and PL+G+ AGN model analyses. The solid, +dashed and dotted lines for AGN, G and PL models compo- +nents. +3.4. Power-law model parameters +As we discussed before, the contribution of a power- +law model to the overall merger rate does not change +substantially if we include or omit other model com- +Figure 4. The inferred merger rate versus mass ratio for +PL+AGN and PL+G+ AGN model analyses. +The solid, +dashed and dotted lines for AGN, G and PL models compo- +nents. +ponents like AGN or G. Among the models we con- +sider, this quasi-universality is expected: the PL model +most effectively reproduces the merger rate versus mass +for the lowest-mass and most frequently merging binary + +AGN mmax = 15 +100 +AGN mmax = 35 +AGN mmax = 50 +AGN mmax = 70 +10-1 +R*p(m1) +10-2 +10-3 +10-4 +101 +102 +mi(Mo)101 +AGNmm=15 +AGN mm = 35 +AGN mm = 50 +100 +AGN mm = 70 +10-1 +(Tw)d +* +10-2 +R +10-3 +10-4 +101 +102 +mi(Mo)AGN mmax = 15 +AGN mmax = 35 +102 +AGN mmax = 50 +AGN mmax = 70 +101 +R* p(q) +100 +10-1 +10-1 +100 +bAGNmm=15 +AGN mm = 35 +102 +AGN mm = 50 +AGN mm = 70 +101 +R*p(q) +100 +10-1 +10-1 +100 +b6 +black holes. +While the overall merger rate from this +component is stable to our choice of the mixture, the +model parameters recovered for PL depend strongly on +which other confounding contributions are also present, +as suggested by Figure 3.2 and 3.2. +While the PL +mass ratio distribution does not depend strongly on in- +cluding or omitting AGN or G, the power law slope +α and minimum mass mmin do change substantially. +The estimated α median value with 65% credible inter- +vals from different analysis as 1.6+0.2 +−0.2, 6.7+3.1 +−2.4,8.4+2.3 +−3.4, +and 1.8+0.5 +−0.5 for PL-only, PL+G, PL+AGN (mmin=50) +and PL+G+AGN (mmin=50) respectively. Similarly, +mmin estimation are 2.5+0.3 +−0.3, 8.4+0.2 +−0.3, 8.5+0.2 +−0.4, and +6.7+1.6 +−1.3 for PL-only, PL+G, PL+AGN (mmin=50) and +PL+G+AGN (mmin=50) respectively. +For example, +the α estimation suggests that while a pure power-law +model favours mmin close to the lower limit our priors al- +low, incorporating other components causes the power- +law component’s minimum mass to favour larger masses. +With the pertinent mass range for the power-law chang- +ing substantially via different mmin, unsurprisingly. The +α estimation has a wide range of inferred power law ex- +ponents, as the PL may dominate only an extremely +narrow range of masses; see Figure 3.2. +4. CONCLUSION +In this paper, we have directly compared a one- +parameter model for AGN binary black hole formation +with the reconstructed sample of binary black holes +identified via gravitational wave observations. To decon- +volve the AGN component from binaries with different +origin, we allow for BBH formation in both AGN and +phenomenological channels. We consistently find a sig- +nificant contribution to the merger rate from the AGN +component (≃ O(5/Gpc−3yr−1). +Our inferred AGN +contribution follows by our prior belief on the maximum +mass of BBH formed from other channels, which we pre- +sume is less than 50M⊙ due to pair-instability impacts +on stellar evolution and death. +As in previous studies (Yang et al. 2020a,b; Gayathri +et al. 2020, 2021; Vajpeyi et al. 2022), our models for +AGN BBH formation predict a wide range of BBH mass +ratios and frequent significant spins. At present, because +the distinctive signatures of AGN formation are prefer- +entially imparted only to the most massive BBH, the +extant BBH sample does not yet contain enough events +to provide overwhelming evidence in favour of an AGN +component, consistent with prior work (Vajpeyi et al. +2022) subsequent observations could support or rule out +this channel. +Acknowledgements We gratefully acknowledge the +support of LIGO and Virgo for the provision of com- +putational resources. G.V. and D.W. acknowledge the +support of the National Science Foundation under grant +PHY-2207728. +I.B. acknowledges the support of the +National Science Foundation under grants #1911796, +#2110060 and #2207661 and of the Alfred P. Sloan +Foundation. +This research has made use of data, +software and/or web tools obtained from the Gravita- +tional Wave Open Science Center (https: //www.gw- +openscience.org), a service of LIGO Laboratory, the +LIGO Scientific Collaboration and the Virgo Collabora- +tion. 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L., et al. 2021, ApJ, +910, 152, doi: 10.3847/1538-4357/abe40e + diff --git a/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/load_file.txt b/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fba64f55fbc530c18b446fbd8108c376e027cb86 --- /dev/null +++ b/1dE2T4oBgHgl3EQf5AjB/content/tmp_files/load_file.txt @@ -0,0 +1,708 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf,len=707 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX631 Gravitational wave source populations: Disentangling an AGN component V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Gayathri,1, 2 Daniel Wysocki,2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Yang,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' O’Shaughnessy,3 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Haiman,4 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Tagawa,4 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Bartos1 1Department of Physics, University of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA 2Leonard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Parker Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA∗ 3Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA 4Department of Astronomy, Columbia University, 550 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 120th St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=', New York, NY, 10027, USA ABSTRACT The astrophysical origin of the over 90 compact binary mergers discovered by the LIGO and Virgo gravitational wave observatories is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' While the unusual mass and spin of some of the discovered objects constrain progenitor scenarios, the observed mergers are consistent with multiple interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' A promising approach to solve this question is to consider the observed distributions of binary properties and compare them to expectations from different origin scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Here we describe a new hierarchical population analysis framework to assess the relative contribution of different formation channels simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For this study we considered binary formation in AGN disks along with phenomenological models, but the same framework can be extended to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We find that high-mass and high-mass-ratio binaries appear more likely to have an AGN origin compared to the same origin as lower-mass events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Future observations of high-mass black hole mergers could further disentangle the AGN component from other channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' INTRODUCTION Understanding the origin of binary black hole merg- ers is the first step in utilizing black hole mergers to probe a range of astrophysical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The LIGO (Aasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2015) and Virgo (Acernese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2015) grav- itational wave observatories have discovered about 90 binary mergers so far (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021a), provid- ing important information on the astrophysical popu- lation of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Binary black hole systems can form through various channels, including isolated stellar bi- naries (Portegies Zwart & Yungelson 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' de Mink & Mandel 2016) or triples (Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Kimpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Veske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020), dynamical interactions in star clusters (Sigurdsson & Hernquist 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Portegies Zwart & McMillan 2000), primordial black holes formed in the early universe (Carr & Hawking 1974), and in the ac- cretion disks of active galactic nuclei (AGNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' McKernan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Bartos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Tagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' McKernan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The increased number of binary black hole observa- tions allows for a more detailed investigation of the pop- ulation’s mass and spin distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' While individ- ual events may provide anecdotal suggestions hinting at ∗ gayathri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='v@ligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='org one formation channel or another, only an interpreta- tion of the full census can enable one to disentangle the potential contributions from multiple formation scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Several studies have previously explored how to dis- entangle multiple channels, largely relying on compar- ison to phenomenologically-motivated estimates of the detailed outcomes of full formation scenarios (Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Gerosa & Fishbach 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Gayathri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Tagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Kim- ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Some studies also shown a mixture of channels is strongly preferred over any single channel dominating the detected population (Zevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The latest population analysis carried out by LIGO- Virgo-KAGRA, GWTC3 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021a,b), iden- tified several population features that may be indica- tive of the binaries’ origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' First, there appears to be a peak in the binary black hole mass spectrum around 30-40 M⊙ compared to a more simple power-law type population (Tiwari & Fairhurst 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Talbot & Thrane 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Sadiq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Fishbach & Holz 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' At the same time a few observed black holes had unusual properties, such as masses in the so-called upper mass gap (≳ 50 M⊙), highly unequal masses in the binary, high spin and precessing mergers, which are rare in stellar evolution and might be indicative of al- ternative formation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='04187v1 [gr-qc] 10 Jan 2023 2 Here we introduce a flexible approach to compare the predictions of detailed formation models with observations while simultaneously accounting for the potentially-confounding contributions from a flexible phenomenologically-parameterized model for compact binary formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Our paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In section 2 we in- troduce binary formation in AGN disks, phenomenolog- ical descriptions for formation in non-AGN sources, and our flexible parametric population inference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In section 3, we talk about the analysis’s key findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In section 4, we summarize our findings and comment on future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Binary mergers in AGN disks We construct a one-parameter model for BBH forma- tion and merger within an AGN disk, parameterized by the maximum mass mmax of the natal BH distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Specifically, we adopt a seed BH mass distri- bution which follows the Salpeter mass function with index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='35, dN/dm ∝ m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='35 with given mmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For neutron stars (NS) we assume a normal distribution m/M⊙ ∼ N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The BHs and NSs are as- sumed to orbit a supermassive black hole in an AGN, migrating into the disk and inward from their natal lo- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Close to the AGN, these objects undergo multi- ple encounters, facilitating binary formation and merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Other AGN parameters are fiducial values that are ex- pected to be typical, while there are large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The range of possible values in the AGN models param- eter space is discussed in (McKernan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Following (Bartos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2017), we adopted a geometri- cally thin, optically thick, radioactively efficient, steady- state accretion disk expected in AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We used a vis- cosity parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1, radioactive efficiency ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1, fiducial supermassive BH mass M• = 106 M⊙ and ac- cretrion rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 ˙MEdd, where ˙MEdd is the Eddington accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Using Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' (2020) and Tagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' (2020b,a), we have computed the expected mass and spin distributions of binary mergers in AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Figure 1 shows the binary black hole merger intrinsic parameter distribution for the AGN model with different initial mass limits (mmax = [15M⊙, 35M⊙, 50M⊙, 75M⊙]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As the natal BH mass upper limit mmax increases, more massive binary com- ponents and total masses are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' At the same time, for high mmax, asymmetric systems are more frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' By contrast, we have observed that the spin distribu- tion properties are largely independent of our choice for mmax limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Parameter distributions for binary black holes formed in our AGN disk formation models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' line color indi- cates the maximum natal BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As expected, as the max- imum mass increases, total (M) mass upper limits increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Additionally, a higher BH maximum natal mass mmax in- creases the relative frequency of asymmetric mergers, partic- ularly highly asymmetric mergers with q < 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Phenomenology of AGN and non-AGN sources To allow for binary black holes which have a non- AGN origin, we follow previous work and introduce a few-parameter mixture model family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As illustrated in Figure 2, for the non-AGN component, we allow binary black holes to arise from a mixture of a power law and Gaussian components, as detailed in Table 11 of Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In the power law component, the pri- mary is drawn from a pure power law mass distribution (with some unknown mmax,pl < 50M⊙ and unknown primary power law index);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' the mass ratio is drawn from another power law;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' and the spins are drawn from an 10- mmax = 15 mmax = 35 10-2, mmax = 50 mmax = 75 10-3 PDF 10-4 10-5 10-6 50 100 150 200 250 300 Taotal Mass (Mo)100 PDF 10-1 mmax= 15 mmax=35 mmax= 50 mmax = 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 Mass Ratio3 unknown Beta distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In the Gaussian compo- nent, the primary and secondary are drawn from two independent Gaussian distributions with unknown mean and variance, with both means confined a priori to be near 30 − 40M⊙ to be consistent with expectations for PISN supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' By including a second component this phenomenological model can allow for non-power- law features and also allow for spin distributions that vary with mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Each component k has some undeter- mined overall rate Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The top panel of Figure 2 shows the general non-AGN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Our overall model is therefore a mixture model, pa- rameterized by the unknown (continuous) AGN merger rate and its (discrete) maximum mass mmax, along with all parameters of the non-AGN mixture model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The overall merger rate density dN/dV dXdt can therefore be expressed as a sum dN dV dXdt = Ragnpagn(X|mmax)+Rgpg(X|Λg)+Rplppl(X|Λpl) where X are binary parameters and where pq, Λq are the model distributions and parameters for the qth compo- nent (AGN, Gaussian, and power-law respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' To systematically assess how well the distinctive fea- tures in AGN formation scenarios can be disentangled from this large model family, we will perform a se- quence of calculations with increasing model complexity, as shown in the panels of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Specifically, we con- sider without any non-AGN component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' the power-law and Gaussian (PL+G) model only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' the power-law and AGN model (PL+AGN), without any Gaussian compo- nent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' and finally the most general model with all three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Population inference We describe and demonstrate a flexible parametric method to infer the event rate as a function of compact binary parameters, accounting for Poisson error and se- lection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021c), analysed the Multi-Spin model which is the joint mass-spin model for binary black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Independent analyses have shown that there is a feature in the BBH mass spectrum around 30-40 M⊙, which is modeled as a Gaussian peak on top of a power law continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' It is an empirical way of mod- eling extra features, but here we tried to understand it feature with AGN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In this section we review the population inference Wysocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' (2019) used for multi-formation chan- nel contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Binaries with intrinsic parameters x would merge at a rate dN/dVc dtdx = R p(x), where N is the number of detections, Vc is the comoving volume, R is the space-time-independent rate of binary coales- cence per unit comoving volume and p(x) is the probabil- ity of x from detected binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The binaries intrinsic pa- rameters includes mass mi and spins Si, where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The likelihood of the astrophysical BBH population at a Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Graphical representations of the BBH population analysis in m1 − m2−spin parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Top panel for the PL+G, the middle panel for the PL+AGN and the bottom panel for the PL+G+AGN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' given merger rate R (Loredo 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Thrane & Talbot 2019) and given binary intrinsic pa- rameters X ≡ (m1, m2, χ1, χ2), where χi = Si/m2 i , given the data for N detections D = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=', dN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' This the likelihood is given by L(R, X) ≡ p(D|R, X), (1) L(R, X) ∝ e−µ(R,X) N � n=1 � dx ℓn(x) R p(x, X), (2) where µ(R, X) is the expected number of detections un- der a given population parametrization X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Using Bayes’ m2(Mo) spin BBH spin PL mi(Mo)m2(Mo) BBH spin PL+AGN m1 1(Mo)m2(Mo) spin G BBH spin PL +AGN mi(Mo)4 theorem one may obtain a posterior distribution on R and X after assuming some prior p(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For compu- tational efficiency and to enable direct comparison with discrete formation models, we use the Gaussian likeli- hood approximation technique introduced in (Delfavero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2022) to characterize each BBH observation’s like- lihood ℓn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Using this formalism, we estimate what fraction of binary black holes are generated via the AGN channel using detected gravitational wave merger information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' To do this study we have upgraded current parametric methods with a mixture model feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Here we have a freedom to do the analysis with number of models which has different astrophysical binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' ANALYSIS As we discussed before, we perform population infer- ence analysis with mixture model feature and detected confident detection binary black hole detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For this study we have considered different the astrophysical bi- nary distributions from AGN ( with different mmax), power-law, Gaussian peak and its combinations (see Fig- ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Astrophysical merger rate We have estimated the astrophysical black hole merger rate for a given astrophysical model with a given number of GW detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For this study, we have used the parameter estimation samples from obtained by LIGO-Virgo-KAGRA Collaboration (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2021d,e, 2019, 2021f) and it available in the Gravita- tional Wave Open Science Center (https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='gw- openscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='org) with the mixed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Here we follow the same selection criteria as (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' (2021b)), we have considered events with a false alarm rate of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='25yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Table 1 shows the merger rates inferred for joint PL only, G only, AGN only (with different mmax), PL+AGN model (with different mmax) and PL+G+AGN models (with different mmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We have estimated the merger rate results derived using each in- dividual model component as well as combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We have observed that the inferred merger rate from the AGN- only models with different choices for mmax produces largely consistent results peaking near 50 Gpc−3yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For the smallest maximum BH natal mass mmax = 15M⊙, the inferred single-component AGN merger rate peaks around 80 Gpc−3yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For all other models, they peak around the same R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Similarly, we have inferred the merger rate distribution for the single-component Gaussian, and power-law components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As expected, the merger rate from the power-law component dominates overall, as it incorporates and describes many frequent mergers of the lowest-mass binary black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In the case of PL+AGN, the inferred AGN and PL compo- nents have a well-determined merger rate, with median AGN merger rate ≃ 8/, Gpc−3yr−1 and PL merger rate ≃ 30/, Gpc−3yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Changing the maximum natal BH mass mmax has a very mild impact on the inferred AGN merger rate, and almost no effect on the inferred PL merger rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In the case of PL+G+AGN analyses, the in- ferred AGN, G and PL components have a well- determined merger rate, with median AGN merger rate ≃ 7/, Gpc−3yr−1, G merger rate ≃ 4/, Gpc−3yr−1 and PL merger rate ≃ 30/, Gpc−3yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As we have seen in the PL+AGN study, we have not seen any major effect on PL or G merger rate when we change the AGN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Our inferred merger rates deduced from the multi-component model are consistent with infer- ences performed using single-component models alone, suggesting that inference isolates only the contribution from each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Inferred merger rate versus mass To better appreciate how well our inference directly projects out the relative contribution from each com- ponent, Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 shows our inferred merger rate versus mass for PL+AGN and PL+G+AGN mod- els analyses respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In each plot we have shown each model component in mass space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As we expected the low mass region is highly con- tributed by the PL model compared to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We have observed a peak in PL model distribution around 7M⊙ for both PL+AGN as well as PL+G+AGN analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The peak is prominent for the PL+G+AGN study compared to PL+AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The high mass region is represented by only AGN, as we expected to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Note that, the AGN model not only contributes in high mass region it also contributes full mass space as shown in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Inferred merger rate versus mass ratio Similarly here we show the inferred merger rate ver- sus mass ratio for PL+AGN and PL+G+AGN models analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 shows our inferred merger rate ver- sus mass ratio for PL+AGN and PL+G+AGN mod- els analyses respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' In each plot we have shown each model component in mass ratio space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As we ex- pected the low mass ratio region contributed by AGN model for PL+AGN analysis and AGN & G models for PL+G+AGN analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For high q, the dominate contri- bution from PL model, that is consistence with detected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 5 Analysis models mmax = 15 mmax = 35 mmax = 50 mmax = 75 PL only 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9+19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9 −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4 G only 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 AGN only 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 PL+AGN PL 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 AGN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 PL+AGN+G PL 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 AGN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='0 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 G 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The astrophysical rates from PL+AGN and PL+AGN+G models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Each row corresponds to each analysis and each column corresponds to different AGN models with different initial mass limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The inferred merger rate versus mass for PL+G+AGN and PL+G+ AGN model analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The solid, dashed and dotted lines for AGN, G and PL models compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Power-law model parameters As we discussed before, the contribution of a power- law model to the overall merger rate does not change substantially if we include or omit other model com- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The inferred merger rate versus mass ratio for PL+AGN and PL+G+ AGN model analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The solid, dashed and dotted lines for AGN, G and PL models compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' ponents like AGN or G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Among the models we con- sider,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' this quasi-universality is expected: the PL model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='most effectively reproduces the merger rate versus mass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='for the lowest-mass and most frequently merging binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='R*p(m1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='mi(Mo)101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGNmm=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='(Tw)d 10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='mi(Mo)AGN mmax = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mmax = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='R* p(q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='bAGNmm=15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='AGN mm = 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='R*p(q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' While the overall merger rate from this component is stable to our choice of the mixture, the model parameters recovered for PL depend strongly on which other confounding contributions are also present, as suggested by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' While the PL mass ratio distribution does not depend strongly on in- cluding or omitting AGN or G, the power law slope α and minimum mass mmin do change substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The estimated α median value with 65% credible inter- vals from different analysis as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5 for PL-only, PL+G, PL+AGN (mmin=50) and PL+G+AGN (mmin=50) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Similarly, mmin estimation are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='4, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='3 for PL-only, PL+G, PL+AGN (mmin=50) and PL+G+AGN (mmin=50) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' For example, the α estimation suggests that while a pure power-law model favours mmin close to the lower limit our priors al- low, incorporating other components causes the power- law component’s minimum mass to favour larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' With the pertinent mass range for the power-law chang- ing substantially via different mmin, unsurprisingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' The α estimation has a wide range of inferred power law ex- ponents, as the PL may dominate only an extremely narrow range of masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' CONCLUSION In this paper, we have directly compared a one- parameter model for AGN binary black hole formation with the reconstructed sample of binary black holes identified via gravitational wave observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' To decon- volve the AGN component from binaries with different origin, we allow for BBH formation in both AGN and phenomenological channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' We consistently find a sig- nificant contribution to the merger rate from the AGN component (≃ O(5/Gpc−3yr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Our inferred AGN contribution follows by our prior belief on the maximum mass of BBH formed from other channels, which we pre- sume is less than 50M⊙ due to pair-instability impacts on stellar evolution and death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' As in previous studies (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Gayathri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Vajpeyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2022), our models for AGN BBH formation predict a wide range of BBH mass ratios and frequent significant spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' At present, because the distinctive signatures of AGN formation are prefer- entially imparted only to the most massive BBH, the extant BBH sample does not yet contain enough events to provide overwhelming evidence in favour of an AGN component, consistent with prior work (Vajpeyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2022) subsequent observations could support or rule out this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Acknowledgements We gratefully acknowledge the support of LIGO and Virgo for the provision of com- putational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' acknowledge the support of the National Science Foundation under grant PHY-2207728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' acknowledges the support of the National Science Foundation under grants #1911796, #2110060 and #2207661 and of the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Sloan Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' This research has made use of data, software and/or web tools obtained from the Gravita- tional Wave Open Science Center (https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='gw- openscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collabora- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' LIGO is funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' National Science Foun- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian in- stitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' This material is based upon work supported by NSF’s LIGO Laboratory, which is a major facility fully funded by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' REFERENCES Aasi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} +page_content=' 2015, Class.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQf5AjB/content/2301.04187v1.pdf'} diff --git a/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/2301.13604v1.pdf.txt b/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/2301.13604v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b50fa6ff458759aed7a4ec6b9bfdb77bd2c5a7e2 --- /dev/null +++ b/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/2301.13604v1.pdf.txt @@ -0,0 +1,2465 @@ +Nonlinearities in Macroeconomic Tail Risk through +the Lens of Big Data Quantile Regressions +Jan Pr¨user +TU Dortmund +Department of Statistics +Florian Huber1 +University of Salzburg +Department of Economics +Abstract. +Modeling and predicting extreme movements in GDP is notoriously difficult +and the selection of appropriate covariates and/or possible forms of nonlinearities are key in +obtaining precise forecasts. +In this paper, our focus is on using large datasets in quantile +regression models to forecast the conditional distribution of US GDP growth. +To capture +possible non-linearities we include several nonlinear specifications. The resulting models will +be huge dimensional and we thus rely on a set of shrinkage priors. +Since Markov Chain +Monte Carlo estimation becomes slow in these dimensions, we rely on fast variational Bayes +approximations to the posterior distribution of the coefficients and the latent states. +We +find that our proposed set of models produces precise forecasts. These gains are especially +pronounced in the tails. Using Gaussian processes to approximate the nonlinear component +of the model further improves the good performance in the tails. +JEL: C11, C32, C53 +KEYWORDS: Growth at risk, quantile regression, global-local priors, non-linear models, +large datasets. +1We would like to thank the editor, Mike McCracken, three anonymous referees, and participants at the +International Symposium on Forecasting 2022 at the University of Oxford and at the Statistische Woche in M¨unster +for helpful comments. Jan Pr¨user gratefully acknowledges the support of the German Research Foundation (DFG, +468814087). Please address correspondence to: Jan Pr¨user. Department of Statistics, TU Dortmund. Address: +CDI Building, Room 122, 44221 Dortmund, Germany. Email: prueser@statistik.tu-dortmund.de. +arXiv:2301.13604v1 [econ.EM] 31 Jan 2023 + +1 +Introduction +Modeling and predicting the conditional distribution of output growth has attracted considerable +academic attention in recent years. Starting at least with the influential paper by Adrian et al. +(2019), focus has shifted towards analyzing whether there exist asymmetries between a predictor +(in their case financial conditions) and output growth across different quantiles of the empirical +distribution. Several other papers (Adrian et al., 2018; Ferrara et al., 2019; Gonz´alez-Rivera +et al., 2019; Delle Monache et al., 2020; Plagborg-Møller et al., 2020; Reichlin et al., 2020; +Figueres and Jaroci´nski, 2020; Adams et al., 2021; Mitchell et al., 2022) have started to focus on +modeling full predictive distributions using different approaches and information sets. However, +most of these contributions have been confined to models which exploit small datasets and, at +least conditional on the quantile analyzed, assume linear relations between GDP growth and the +predictors.2 +Times of economic stress such as the global financial crisis (GFC) or the Covid-19 pandemic +have highlighted that exploiting information contained in many time series and allowing for +nonlinearities improves predictive performance in turbulent periods (see, e.g., Huber et al., +2023). Since economic dynamics change in volatile economic regimes, models that control for +structural breaks allow for different effects of economic shocks over time or imply nonlinear +relations between GDP growth and its predictors often excel in forecasting applications (see +D’Agostino et al., 2013; Carriero et al., 2016; Adrian et al., 2021; Clark et al., 2022b; Pfarrhofer, +2022; Huber et al., 2023). Moreover, another important empirical regularity is that the set of +predictors might change over time. This is because variables which are seemingly unimportant +in normal periods (such as financial conditions) play an important role in recessions and yield +important information on future behavior of output growth. +This discussion highlights that the effect of predictors on output growth depends on the +quantile under consideration and thus appears to be state dependent and modeling the tran- +sition might call for nonlinear econometric models. The key challenge, however, is to identify +the different determinants of GDP growth across quantiles while taking possible nonlinearities +into account. In this paper, we aim to solve these issues by proposing a Bayesian quantile re- +gression (QR) which can be applied to huge information sets, and which is capable of capturing +nonlinearities of unknown form. Our model is a standard QR model that consists of two parts. +2A recent exception is Kohns and Szendrei (2021) who estimate large-scale quantile regressions and then apply +ex-post sparsification to sharpen predictive inference. +2 + +The first assumes a linear relationship between the covariates and quantile-specific GDP growth +whereas the second component assumes an unknown and possibly highly nonlinear relationship +between the two. The precise form of nonlinearities is captured through three specifications. +One is parametric and based on including polynomials up to a certain order, whereas the re- +maining two are nonparametric. Among these nonparametric specifications we include B-splines +(see Shin et al., 2020) and Gaussian processes (see Williams and Rasmussen, 2006). Both have +been shown to work well when it comes to function estimation and forecasting. +The combination of a linear and nonlinear term implies that the dimension of the parame- +ter space increases substantially. Since all these models can be cast in terms of a linear regression +conditional on appropriately transformed covariates, we can use regularization techniques to de- +cide on whether more flexibility is necessary and which variables should enter the model. We +achieve this through several popular shrinkage priors that have excellent empirical properties in +large dimensions and are relatively easy to implement. These shrinkage priors enable us to select +promising subsets of predictors and the degree of nonlinearities for each quantile separately. +Posterior inference using Markov Chain Monte Carlo (MCMC) techniques in these dimen- +sions proves to be an issue because we have to estimate a large-scale regression model for all +quantiles of interest. This procedure needs to be repeated a large number of times if we wish to +carry out an out-of-sample forecasting exercise. To reduce the computational burden enormously +we estimate the QRs using Variational Bayes (VB).3 This estimation strategy approximates the +exact full conditional posterior distributions with simpler approximating distributions. These +approximating densities are obtained by minimizing the Kullback-Leibler (KL) distance be- +tween some known density q and the exact posterior distribution p. Hence, integration in huge +dimensions is replaced by a simpler optimization routine. Our approach is fast and allows for +computing all results of our forecasting exercise without the use of high performance computing +environments. +We apply our techniques to the large dimensional FRED-QD dataset (McCracken and Ng, +2016) and focus on single and multi-step-ahead forecasting of US GDP growth over a hold-out +period ranging from 1991Q2 to 2021Q3. The different nonlinear models we consider are high +dimensional and feature up to around 1,000 coefficients per equation. +The empirical results can be summarized as follows. Using huge information sets and +nonlinear models in combination with priors that introduce substantial shrinkage pays off for tail +3For an introduction, see Blei et al. (2017) and an algorithm for QRs is provided in Bufrei (2019). +3 + +forecasts. In both tails, forecast improvements relative to the small-scale QR model developed in +Adrian et al. (2019) are sizable. When we focus on the center of the distribution the differences +become smaller. Once we allow for nonlinearities we find modest improvements in predictive +accuracy. Comparing the different nonlinear specifications reveals that Gaussian processes offer +the largest improvements vis-´a-vis the linear QR. This indicates that a successful tail forecasting +model should be able to extract important information from huge datasets, while controlling for +possibly nonlinear relations. When we focus on the key properties of the proposed priors we +observe that priors that imply a dense model (characterized by many small coefficients) yield +good tail forecasts. +The paper is structured as follows. +The next section introduces the general QR and +the scale-location mixture representation to cast the model in terms of a standard generalized +additive regression with auxiliary latent variables. We then focus on the different priors used, +provide additional details on the nonlinear components of the models, briefly discuss VB, outline +how we estimate the posterior distributions of the parameters and latent quantities, and illustrate +the computational properties of our approach. Section 3 discusses our empirical findings. The +final section summarizes and concludes the paper. +An Online Appendix includes additional +technical details, empirical results and more precise information on the used dataset. +2 +Bayesian analysis of general QRs +2.1 +The likelihood function +In this paper, our goal is to model the dependence between the qth quantile of GDP growth +yt and a panel of K predictors in {xt}T +t=1 with K being huge. The covariates include a wide +range of macroeconomic and financial indicators. Possible nonlinearites between yt and xt are +captured through a function gq(xt), with gq : RK → R. +The fact that K is large and the +inclusion of nonlinear functions of xt implies that the number of parameters is large relative to +the number of observations T. +Our workhorse model is the QR developed in Koenker and Bassett (1978). As opposed +to the standard QR, our model decomposes the qth quantile function Qq(yt) in a linear and +nonlinear part and a non-standard error distribution: +(1) +yt = x′ +tβq + gq(xt) + εt, +4 + +where βq is a K−dimensional vector of quantile-specific regression coefficients and εt is a shock +term with density fq such that the qth quantile equals zero: +� 0 +−∞ +fq(εt)dεt = q. +Conditional on the quantile, this model resembles a generalized additive model (GAM), see +Hastie and Tibshirani (1987). +We approximate gq(xt) using nonlinear transformations of xt: +(2) +gq(xt) ≈ +M +� +m=1 +γqmzm(xt) = z′ +tγq +with γq = (γq1, . . . , γqM)′, zt = (z1(xt), . . . , zM(xt))′ and zm(xt) denotes a basis function that +depends on xt with γqm denoting the corresponding basis coefficient. This basis function de- +pends on the specific approximation model used to infer the nonlinear effects and our additive +representation nests models commonly used in the machine learning literature (such as Gaussian +processes, splines, neural networks but also more traditional specifications such as time-varying +parameter models). We will discuss the precise specification of zm (and thus zt) in more detail +in Sub-section 2.3. Here it suffices to note that depending on the specification, M could be very +large. For instance, in the Gaussian process case, M = T and thus the number of regression +coefficients would be K + T. +If fq remains unspecified, estimation of βq and γq is achieved by solving the following +optimization problem: +arg max +{βq,γq} += +T +� +t=1 +ρq(yt − x′ +tβq − z′ +tγq), +with ρq(l) = l[q − I(l < 0)] denoting the loss function. This optimization problem is straight- +forward to solve but, if K + M is large, regularization is necessary. This motivates a Bayesian +approach to estimation and inference. +From a Bayesian perspective, carrying out posterior inference requires the specification of +a likelihood and suitable priors. Following Yu and Moyeed (2001) we assume that the shocks εt +follow an asymmetric Laplace distribution (ALD) with density: +fq(εt) = q(1 − q) exp (−ρq(εt)). +The key thing to notice is that the qth quantile equals zero and the parameter q controls the +5 + +skewness of the distribution. Kozumi and Kobayashi (2011) show that one can introduce auxil- +iary latent quantities to render the model with ALD distributed shocks conditionally Gaussian. +This is achieved by exploiting a scale-location mixture representation (West, 1987): +εqt = θqνqt + τq√σqνqtut, +θq = 1 − 2q +q(1 − q), +τ 2 +q = +2 +q(1 − q), +νqt ∼ E +� 1 +σq +� +, +ut ∼ N(0, 1), +where E +� +1 +σq +� +denotes the exponential distribution and σq is a scaling parameter. Hence, con- +ditional on knowing νq = (νq1, . . . , νqT )′, θq, τq, σq and appropriately selecting gq, the model is a +linear regression model with response ˆyt = yt −θqνqt and Gaussian shocks that are conditionally +heteroskedastic. This conditional likelihood will form the basis of our estimation strategy. +To complete the model specification we assume that +1 +σq ∼ G(c0, d0), where c0 is the shape +and d0 the rate parameter of the Gamma distribution which we set both to zero in order to +obtain a flat prior. The choice of the prior distribution on βq and γq is essential for our high +dimensional QRs. We discuss different suitable choices in the next section. +2.2 +Priors for the quantile regression coefficients +For the large datasets we consider in this paper, M + K ≫ T and thus suitable shrinkage priors +are necessary to obtain precise inference. Kohns and Szendrei (2021) and Mitchell et al. (ming) +use flexible shrinkage priors in large-scale QRs and show that these work well for tail forecasting. +We build on their findings by considering a range of different priors on βq and γq. All these +priors belong to the class of so-called global-local shrinkage priors (Polson and Scott, 2010) and +have the following general form: +βq|ψβ +q1, . . . , ψβ +qK, λβ +q ∼ +K +� +j=1 +N(0, ψβ +qjλβ +q ), +ψβ +qj ∼ u, +λβ +q ∼ π, +γq|ψγ +q1, . . . , ψγ +qM, λγ +q ∼ +M +� +j=1 +N(0, ψγ +qjλγ +q), +ψγ +qj ∼ u, +λγ +q ∼ π, +with λs +q (s ∈ {β, γ}) denoting a quantile-specific global shrinkage parameter and ψs +qj are local +scaling parameters that allow for non-zero coefficients in the presence of strong global shrinkage +(i.e., with λs +q close to zero). The functions u and π refer to mixing densities which, if suitably +chosen, translate into different shrinkage priors. In this paper, all the priors we consider can be +cast into this form but differ in the way the mixing densities u and π are chosen. Since these +6 + +priors are well known, we briefly discuss them in the main text and relegate additional technical +details to the Online Appendix. +We focus on five shrinkage priors that have been shown to work well in a wide variety of +forecasting applications (see, e.g., Huber and Feldkircher, 2019; Cross et al., 2020; Chan, 2021; +Pr¨user, 2022). The first prior we consider is the Ridge prior. The Ridge prior is a special case +of a global-local prior with local parameters set equal to 1 and a global shrinkage parameter +which follows an inverse Gamma distribution. Formally, this implies setting ψs +qj = 1 for all q, j +and λs +q ∼ G−1(e0, e1). The hyperparameters e0 and e1 control the tightness of the prior. We +set these equal to e0 = e1 = 0. This prior shrinks all coefficients uniformly towards zero and +provides little flexibility to allow for idiosyncratic (i.e., variable-specific) deviations from the +overall shrinkage pattern. +This issue is solved by estimating the local shrinkage parameters. The Horseshoe (HS, +see, Carvalho et al., 2010), our second prior, does this. This prior sets u and π to a half-Cauchy +distribution: +� +ψs +qj ∼ C+(0, 1) and �λsq ∼ C+(0, 1). +The HS possesses excellent posterior +contraction properties (see, e.g., Ghosh et al., 2016; Armagan et al., 2013; van der Pas et al., +2014). Moreover, it does not rely on any additional tuning parameters. +Another popular global-local shrinkage prior is the Normal-Gamma (NG) prior of Griffin +and Brown (2010). This prior assumes that u and π are Gamma densities. More formally, +ψs +qj ∼ G(ϑ, λs +qϑ/2) and λs +q ∼ G(c0, d0), with ϑ being a hyperparameter that controls the tail +behavior of the prior, and c0 and d0 are hyperparameters that determine the overall degree of +shrinkage. We set c0 = d0 = 0 and ϑ = 0.1. This choice implies heavy global shrinkage on the +coefficients but also implies fat tails of the marginal prior of the coefficients after integrating +out the local scaling parameters. The Bayesian LASSO is obtained as a special case of the NG +prior with ϑ = 1. +Finally, the Dirichlet-Laplace prior (Bhattacharya et al., 2015) assumes that the local scal- +ing parameter ψs +qj is a product of a Dirichlet-distributed random variate φs +qj ∼ Dir(α, . . . , α) and +a parameter ˜ +ψs +qj ∼ E(1/2) that follows an exponential distribution. Hence, the Dirichlet-Laplace +prior sets ψs +qj = (φs +qj)2 ˜ +ψs +qj. On the global scaling parameters we use a Gamma distribution +� +λβ +q ∼ G(Kα, 1/2) and +� +λγ +q ∼ G(Mα, 1/2). We set α = 1 +K for the linear part and α = +1 +M for +the non-linear part. +7 + +2.3 +Capturing nonlinearities in high dimensional QRs +In extreme periods such as the GFC or the Covid-19 pandemic, nonlinearities in macroeconomic +data become prevalent. We control for this by having a nonlinear part in our QR. As stated in +(2), we capture possible nonlinearities in xt through nonlinear transformations zm(xt). +The first and simplest nonlinear specification maps xt into the space of polynomials. Bai +and Ng (2008) capture nonlinearities in macro data through polynomials and by relying on +factor-based predictive regressions. We follow this approach and define the corresponding basis +function as follows: +zt = ((x2 +t )′, (x3 +t )′, . . . , (xN +t )′)′. +Deciding on the order of the polynomial N is a model selection issue and suitable shrinkage +priors can be adopted. In our empirical work, we focus on the cubic case. This specification +will overweight large movements in xt and should thus be suitable for quickly capturing sharp +downturns in the business cycle. In this case, the number of coefficients triples since M = 3K. +The resulting nonlinear model is called Polynomial-QR. +Adding cubic terms allows us to capture nonlinearities in a relatively restricted manner. +Since the precise form of nonlinearities is typically unknown, the remaining two specifications +we consider are nonparametric and only require relatively mild prior assumptions on the form +of nonlinear interactions. The first of these two is the B-Spline (see, e.g., De Boor, 2001, for a +review). B-Splines have a proven track record in machine learning and computer science (Shin +et al., 2020). +For the B-spline, we assume that each element in xt exerts a (possibly) nonlinear effect +on yt that might differ across covariates. This implies that gq(xt) equals: +gq(xt) ≈ +K +� +k=1 +Φk(x•,j)γq,k. +Here, we let Φk denote a T ×r matrix of B-spline basis functions that depend on the jth covariate +in X = (x′ +1, . . . , x′ +T )′, x•,j and r is the number of knots. In this case, the number of nonlinear +coefficients is M = rK. In our empirical work we place the knots at the following quantiles of +x•,j: {0, 0.05, 0.1, 0.25, 0.50, 0.75, 0.90, 0.95, 1}, implying that r = 9 and thus M = 9K. We +will henceforth call this model Spline-QR. +The last specification we consider is the Gaussian process (GP) regression. GP regression +8 + +is a nonparametric estimation method that places a GP prior on the function gq(xt): +gq(xt) ∼ GP(µq(xt), K(xt, xt)). +The mean function µq(xt) is, without loss of generality, set equal to zero and K(xt, xt) is a +kernel function that encodes the relationship between xt and xt for t, t = 1, . . . , T. It is worth +noting that our additive specification implies that if the mean function is set equal to zero, the +model is centered on a standard QR. +Since xt is observed in discrete time steps, the GP prior implies a Gaussian prior on +gq = (gq(x1), . . . , gq(xT ))′: +gq ∼ N(0T , K(w)), +where K(w) is a T × T-dimensional matrix with (t, t)th element K(xt, xt). w = (w1, w2)′ is +a set of hyperparameters that determine the properties of the kernel (and thus the estimated +function). +The GP regression is fully specified if we determine the kernel function K. In this paper, +we use the Gaussian (or squared exponential) kernel: +K(xt, xt) = w1 × exp +� +−w2 +2 ||xt − xt||2� +. +The hyperparameters w are set according to the median heuristic proposed in Arin et al. (2017). +What we discuss above is the function-space view of the GP regression. An alternative +way of expressing the GP is the so-called weight-space view. The weight-space view is obtained +by integrating out gq, yielding the following regression representation: +y = Xβq + Zγq + ε, +with y denoting the stacked dependent variables, Z is the lower Cholesky factor of K and +γq ∼ N(0, IT ). Notice that gq = Zγq. Hence, the Cholesky factor of the kernel matrix provides +the basis functions, and the parameters can be readily estimated. In this case, the number +of nonlinear coefficients is M = T. Since we use a shrinkage prior on γq, the corresponding +implied kernel is given by ZBγ +q Z′. The M × M matrix Bγ +q is a prior covariance matrix with +Bγ +q = λγ +q × diag(ψγ +q1, . . . , ψγ +qM). Approximating gq using GPs leads to the GP-QR specification. +This completes our choice of nonlinear techniques used in the big data QR. Alternative +9 + +choices (such as allowing for time-varying parameters, neural networks or Bayesian additive +regression trees) can be straightforwardly introduced in this general framework. +2.4 +A brief introduction to variational Bayes +The high dimensionality of the state space calls for alternative techniques to carry out posterior +inference. We opt for using variational approximations to the joint posterior density. In this +section, we provide a discussion on how VB works in general. For an excellent in-depth introduc- +tion, see Blei et al. (2017). In machine learning, variational techniques have been commonly used +to estimate complex models such as deep neural networks (see, e.g., Polson and Sokolov, 2017). +In econometrics, recent papers use VB in huge dimensional multivariate time series models such +as VARs (Gefang et al., 2022; Chan and Yu, 2020) or state space models to speed up estimation +(Koop and Korobilis, 2023). In a recent paper, Korobilis and Schr¨oder (2022), propose a QR +factor model and estimate it using VB techniques. +To simplify the exposition, we fix the prior variances. The appendix provides information +on how we estimate the prior variances (and associated hyperparameters) using VB. Let ξq = +(βq, γq, σq, νq) denote a generic vector which stores all unknowns of the model, with νq = +(νq1, . . . , νqT ) denoting the latent components. +Our aim is to approximate the joint posterior distribution p(ξq|y) using an analytically +tractable approximating distribution q(ξq). This variational approximation is found by mini- +mizing the Kullback-Leibler (KL) distance between p and q. One can show that minimization +of the KL distance is equivalent to maximizing the evidence lower bound (ELBO) defined as: +(3) +ELBO = Eq(ξq) (log p(ξq, y)) − Eq(ξq) (log q(ξq)) , +with Eq(ξq) denoting the expectation with respect to q(ξq). This implies that finding the approx- +imating density q replaces the integration problem (which is typically solved through MCMC +sampling) with an optimization problem (which is fast and thus scales well into high dimensions). +A common and analytically tractable choice of approximating densities assumes that q(ξq) +is factorized as follows: +q(ξq) = +S +� +s=1 +qs(ξqs), +where ξqs denotes a partition of ξq. A particular example (which we use in this paper) would +specify ξq1 = (β′ +q, γ′ +q)′, ξq2 = σq and ξq3 = νq. +10 + +This class is called the mean field variational approximation and assumes that the different +blocks ξqs are uncorrelated.4 Notice that all our priors on ξq can be written as: +p(ξq) = +S +� +s=1 +p(ξqs), +and using the fact that: +Eq(ξq)(log p(ξq, y)) = Eq(ξq)(log p(y|ξq)) + +S +� +s=1 +Eq(ξq)(log p(ξqs)), +the ELBO can be stated as: +ELBO = Eq(ξq)(log p(y|ξq)) + +S +� +s=1 +Eq(ξq)(log p(ξqs)) − +S +� +s=1 +Eq(ξq)(log q(ξqs)). +Wand et al. (2011) prove that under the variational family the optimal approximating densities +are closely related to the full conditional posterior distributions: +q∗ +s(ξq) = exp +� +Eq(ξq)(log p(ξqs|y, ξq,−s) +� +, +where ξq,−s is the vector ξq with the sth component excluded. Hence, if p(ξqs|y, ξq,−s) is known +(which is the case for the QR regression based on the auxiliary representation discussed in the +previous subsection), the elements in ξqs can be updated iteratively (by conditioning on the +expected values of ξq,−s) until the squared difference of the ELBO or of all elements of ξqs is +smaller than some small ϵ between two subsequent iterations. +2.5 +Approximate Bayesian inference in general QRs +In this section we briefly state the three approximating densities (q∗ +s(ξ)) used to estimate the +parameters and latent quantities in the QR regression. We provide derivations for the three +approximating densities of the three parameter groups: ˜βq = (β′ +q, γ′ +q)′, σq and νq in the Online +Appendix. +We start by discussing the approximating densities for the regression and basis coefficients. +4Frazier et al. (2022) state that mean field VB approximations might perform poorly in models with a large +number of latent variables. However, they also note that the resulting model forecasts could still perform well in +practice. +11 + +A Gaussian distribution approximates the posterior of ˜βq: +p( ˜βq|•) ≈ N +� +E( ˜βq), ˆΣκq +� +, +with variance and mean given by, respectively: +ˆΣ ˜βq = +� T +� +t=1 +ftf ′ +t +τ 2q +E +� 1 +νqt +� +E +� 1 +σq +� ++ B−1 +0q +�−1 +, +E( ˜βq) = ˆΣ ˜βq +� +���E +� 1 +σq +� +T +� +t=1 +E +� 1 +νqt +� ft +� +yt − θq +� +E +� +1 +νqt +��−1� +τ 2q +� +��� . +ft = (x′ +t, z′ +t)′ and B−1 +0q = diag(Bβ +q , Bγ +q )−1 is a prior precision matrix with Bβ +q = λβ +q ×diag(ψβ +q1, . . . , ψβ +qK) +and Bγ +q = λγ +q × diag(ψγ +q1, . . . , ψγ +qK). The approximating densities used to estimate the prior hy- +perparameters are provided in Section 1 of the Online Appendix. +The latent variable νqt follows a generalized inverse Gaussian (GIG) distribution: GIG(r, A, B)5 +with +p(νqt|•) ≈ GIG +� +� +� +� +� +1 +2, 2E +� 1 +σq +� ++ θ2 +q +τ 2q +E +� 1 +σq +� +� +�� +� +Aq +, +E +� +1 +σq +� +τ 2q +�� +yt − f ′ +tE( ˜βq) +�2 ++ f ′ +t ˆΣ ˜βqft +� +� +�� +� +Bq +� +� +� +� +� . +The moments of νqt are given by +E +� +νj +qt +� += +�� +Bq +� +Aq +�j K1/2+j +�� +AqBq +� +K1/2 +�� +AqBq +� , +where Kx denotes the modified Bessel function of the second kind. +Finally, we approximate +p +� 1 +σq +|• +� +≈ G(cq1, dq1) +5We use the following parametrization of the GIG distribution: log (GIG(x)) ∝ (0.5 − 1) log(x) − +� +Ax + 1 +2 +B +x +� +. +12 + +with +cq1 = c0 + 1.5T, +dq1 = d0 + +T +� +t=1 +E(νqt) + +1 +2τ 2q +T +� +t=1 +(E +� 1 +νqt +� � +yt − f ′ +tE( ˜βq) +�2 ++ 2θq(f ′ +tE( ˜βq) − yt) ++ E(νqt)θ2 +q + E +� 1 +νqt +� +f ′ +t ˆΣ ˜βqft), +and E +� +1 +σq +� += cq1 +dq1 . +2.6 +Comparing computation times between VB and MCMC +These steps, in combination with the updating steps for the priors detailed in the Online Ap- +pendix, form the basis of our VB algorithm. As stated in the introduction, the key advantage of +using VB instead of more precise MCMC-based techniques is computational efficiency. Before +we turn to our empirical work, we illustrate this point using synthetic data. +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Number of Variables +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +Time in Minutes +Runtime: VB vs MCMC +VB +MCMC +Figure 1: Comparison of computation times against the number of covariates M + K +To illustrate the computational merits of employing VB-based approximations, Fig. 1 +shows the estimation times for different values of M + K using our VB-based QR (for a specific +quantile) and the QR estimated through the Gibbs sampler. The MCMC algorithm is repeated +10, 000 times. The figure shows that the computational burden increases lightly in the number +of covariates for VB. When we focus on MCMC estimation, the computational requirements +increase sharply in the number of covariates. Especially in our empirical work, where K + M +is often above 1, 000, VB proves to be a fast alternative to MCMC-based quantile regressions. +It is also worth stressing that if the number of quantiles to estimate is large (and no parallel +computing facilities are available), MCMC-based estimation becomes excessively slow. +13 + +3 +Forecasting output growth using huge dimensional QRs +In this section, we present our forecasting results. The next sub-section provides information on +the dataset and the forecasting setup. We then proceed by discussing the results from QRs that +exclude the nonlinear part in Sub-section 3.2. The question whether nonlinearities are important +is investigated in Sub-section 3.3, and Sub-section 3.4 deals with how forecast accuracy changes +over time. Sub-section 3.5 discusses the determinants of the different tail forecasts and differences +in the shrinkage properties across priors. +3.1 +Data overview and forecasting setup +We use the quarterly version of the McCracken and Ng (2016) dataset. The data set covers in- +formation about the real economy (output, labor, consumption, orders and inventories), money, +prices and financial markets (interest rates, exchange rates, stock market indexes). All series +are seasonally adjusted and transformed to be approximately stationary. The set of variables +included in xt and their transformation codes are described in Table 1 of the Online Appendix. +All models we consider also include the first lag of GDP growth.6 Forecasts are carried out using +direct forecasting by appropriately lagging the elements in xt. +Our sample runs from 1971Q1 to 2021Q3 and we use the period 1991Q2 to 2021Q3 as +our hold-out period. +The forecasting design is recursive. +This implies that we estimate all +our models on an initial training sample with data until 1991Q1 and produce one-quarter- +and four-quarters-ahead predictive distributions for 1991Q2 and 1992Q1, respectively. After +obtaining these, we add the next observation (1991Q2) and recompute the models to obtain the +corresponding predictive densities for 1991Q3 and 1992Q2. This procedure is repeated until we +reach the end of the hold-out period. +As a measure of overall forecasting accuracy we focus on the continuous ranked probability +score (CRPS). The CRPS is a measure of density forecasting accuracy and generalizes the mean +absolute error (MAE) to take into account how well a given model predicts higher order moments +of a target variable. +The CRPS measures overall density fit. Considering overall CRPSs possibly masks rele- +vant idiosyncrasies of model performance across quantiles. If a decision maker is interested in +downside risks to GDP growth, she might value a model more that does well at the critical 5 or +10 percentiles as opposed to the remaining regions of the predictive distribution. To shed light +6We find that including more lags of GDP growth only has small effects on the empirical results. +14 + +on asymmetries across different predictive quantiles, we focus on the quantile score (QS): +QSqt = (yt − Qqt)(q − 1{yt≤Qqt}), +where Qqt is the forecast of the qth quantile of yt and 1{yt≤Qqt} denotes the indicator function +that equals one if yt is below the forecast for the qth quantile. +The QS can also be used to construct quantile-weighted (qw) CRPS scores (Gneiting and +Ranjan, 2011). These qw-CRPSs can be specified to put more weight on certain regions of the +predictive distribution. In general, the qw-CRPS is computed as: +qw-CRPS = +2 +J − 1 +J−1 +� +j=1 +ω(ζj)QSsjt, +with ζj = j/J, J − 1 = 19 denoting the number of quantiles we use to set up the qw-CRPS and +sj selects the jth element from the set of quantiles we consider. This set ranges from 0.05 to +0.95 with a step size of 0.05 and thus, s1 = 0.05, s2 = 0.10, . . . , s19 = 0.95. +We use two weighting functions ω(ζj) that focus on different regions of the predictive +density. These schemes are motivated in Gneiting and Ranjan (2011). The first (CRPS-left) +puts more weight on the left tail (i.e. downside risks) and is specified as ω(ζj) = (1 − ζj)2, +while the second (CRPS-tails) puts more weight on both tails as opposed to the center of the +distribution: ω(ζj) = (2ζj − 1)2. +Notice that if we use equal weights we obtain a discrete +approximation to the CRPS. +3.2 +Results based on linear QRs +We start discussing the QRs that set g(xt) = 0 for all t. Here, our goal is to show that including +more information pays off relative to the model proposed in Adrian et al. (2019). Hence, we +benchmark the QR models to the model which only includes lagged GDP growth and the NFCI. +This model is henceforth called the ABG model and estimated in the same way as in the original +paper. +Table 1 shows average (over time) qw-CRPSs relative to the ABG model. Numbers smaller +than one suggest that a given model outperforms the ABG benchmark whereas numbers exceed- +ing unity indicate that the model produces less precise density forecasts. +The table reveals a great deal of heterogeneity with respect to different priors. Popular GL +priors such as the HS, the NG or the DL lead to forecasts that are often slightly worse than the +15 + +Table 1: CRPS for linear models +One-quarter-ahead +Four-quarters-ahead +Model +CRPS +CRPS-tails +CRPS-left +CRPS +CRPS-tails +CRPS-left +HS +1.06 +1.06 +1.05 +0.99 +0.95 +1.02 +RIDGE +0.88 +0.84 +0.83 +0.87 +0.86 +0.87 +NG +1.01 +0.98 +0.98 +1.01 +0.96 +1.03 +LASSO +0.91 +0.89 +0.91 +0.87 +0.85 +0.88 +DL +1.10 +1.09 +1.08 +1.09 +1.06 +1.14 +Notes: We highlight in light gray (dark gray) rejection of equal forecasting accuracy against the +benchmark model at significance level 10% (5%) using the test in Diebold and Mariano (1995) with +adjustments proposed by Harvey et al. (1997). Results are shown relative to the AGB model and +are based on the full sample. +ones obtained from the benchmark. However, priors such as the Ridge or the LASSO (which is +particular known for over-shrinking significant signals (see, e.g., Griffin and Brown, 2010)) yield +forecasts that are better than the benchmark forecasts for both forecast horizons and across the +different variants of the CRPS. Our findings corroborate recent results in Carriero et al. (2022) +who show that large QRs with shrinkage improve upon the ABG benchmark. +This is especially pronounced in the case of the Ridge prior. In this case, the accuracy +gains vis-´a-vis the ABG benchmark reach 17 percent and, in most cases, accuracy differences +are statistically significant according to the Diebold and Mariano (1995) test. +Turning to the different forecast horizons reveals that specifications that do well in terms +of short-term forecasting also produce precise longer-term predictions. For the LASSO-based +model, four-quarters-ahead accuracy gains are slightly more pronounced whereas for the Ridge +we do not find discernible differences across both forecast horizons. +Next, we drill deeper into the quantile-specific forecasting performance by considering QSs +for q ranging from q ∈ {0.05, 0.1, 0.25, 0.5, 0.75, 0.95, 0.99}. These, for one-step-ahead forecasts, +are shown in Fig. +2 and Fig. +3 provides the four-steps-ahead results. +Before starting our +discussion it is worth stressing that many of these differences are statistically significant with +respect to the DM test. The corresponding results are provided in the Online Appendix (see +Fig. 15 and 16). +Similar to the findings based on the CRPSs, there is a great deal of heterogeneity across +priors. Both the LASSO and the Ridge prior improve upon the ABG benchmark for all quantiles +by relatively large margins. These gains appear to be more pronounced in the tails, reaching +over 20 percent in terms of the QSs. When focusing on the center of the distribution (i.e., the +median forecast), the gains are much smaller. In general, the other priors perform considerably +16 + +5% +10% +25% +50% +75% +90% +95% +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +HS +RIDGE +NG +LASSO +DL +One-quarter-ahead quantile scores +Figure 2: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out +period. +5% +10% +25% +50% +75% +90% +95% +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +HS +RIDGE +NG +LASSO +DL +One-year-ahead quantile scores +Figure 3: Four-quarters-ahead quantile scores for different values of q, averaged over the hold- +out period. +worse. The only exception turns out to be the NG prior which, displays an excellent performance +in the left tail, while being still outperformed by the LASSO and the Ridge prior. +Considering four-quarters-ahead tail forecasts yield a similar but less pronounced picture. +17 + +For higher-order forecasts, priors that did well at the one-quarter-ahead horizon (LASSO and +Ridge) also yield precise tail forecasts. One remarkable difference from short-term forecasts is +that higher order median forecasts appear to be much more precise than the ones obtained from +the ABG benchmark specification. +This brief discussion gives rise to a simple recommendation for practitioners. If interest is +on producing precise tail forecasts (irrespective of the forecast horizon) it pays off to use large +QRs coupled with either a LASSO or Ridge-type prior. Since the Ridge prior is much simpler +(i.e., it only features a single hyperparameter) and the empirical performance is very similar to +the LASSO, our focus from now on will be on comparing the Ridge-based QR with a range of +non-linear specifications. +3.3 +Allowing for nonlinearities in large scale QRs +In the previous sub-section we have shown that using big QRs leads to tail forecasts that are +superior to the ones of the benchmark ABG specification. Conditional on the quantile, these +models are linear in the parameters. However, recent literature (see, e.g., Clark et al., 2022b) +suggests that nonlinearities become more important in the tails. Hence, we now address this +question within our approximate framework. +Table 2 shows relative CRPSs for the different nonlinear models. As opposed to Table 1, +all results are now benchmarked against the QR with the Ridge prior. This allows us to directly +measure the performance gains from introducing nonlinearities relative to setting gq(xt) = 0. +Notice that the absence of gray shaded cells in the table indicates that the DM test does not +point towards significant differences in forecast accuracy between the linear and the different +nonlinear QRs. +Despite this, a few interesting insights emerge from the table. First, many numbers in the +table are close to unity and differences are not statistically significant from the best performing +linear QR.7 This indicates that once we include many predictors, additionally controlling for +nonlinearities of different forms only yields small positive (and sometimes negative) gains in +terms of tail forecasting accuracy. Second, the first finding strongly depends on the approxi- +mation techniques chosen. Among all three specifications, using GPs is superior to using either +polynomials or B-Splines to approximate the unknown function gq. Second, and focusing on +7For the GP-QR specifications with ridge prior we obtain p-values between 0.1 and 0.2 using the test in +Diebold and Mariano (1995) with adjustments proposed by Harvey et al. (1997). +18 + +Table 2: CRPSs for nonlinear models +One-quarter-ahead +Four-quarters-ahead +Model +CRPS +CRPS-tails +CRPS-left +CRPS +CRPS-tails +CRPS-left +Polynomials +HS +1.02 +1.00 +1.08 +0.96 +0.97 +1.00 +RIDGE +0.98 +0.94 +1.01 +0.96 +0.97 +1.00 +NG +1.03 +0.99 +1.07 +0.95 +0.96 +1.00 +LASSO +1.05 +1.04 +1.08 +1.02 +1.00 +0.99 +DL +1.07 +1.04 +1.15 +1.22 +1.20 +1.29 +B-Splines +HS +1.13 +1.16 +1.18 +1.10 +1.14 +1.05 +RIDGE +1.08 +1.08 +1.08 +1.09 +1.13 +1.04 +NG +1.15 +1.17 +1.20 +1.13 +1.17 +1.07 +LASSO +1.07 +1.06 +1.09 +1.02 +1.01 +0.99 +DL +0.98 +1.00 +1.01 +1.04 +1.08 +1.02 +Gaussian Processes +HS +0.96 +0.94 +0.97 +1.05 +1.02 +1.10 +RIDGE +0.97 +0.95 +0.98 +0.96 +0.95 +0.97 +NG +0.98 +0.95 +0.98 +1.06 +1.02 +1.08 +LASSO +1.04 +1.04 +1.07 +1.01 +0.99 +1.00 +DL +1.02 +0.97 +1.00 +1.22 +1.20 +1.29 +Results are shown relative to the linear QR with a Ridge prior and are based on the +full sample. +GP-QR specifications, the specific prior chosen matters appreciably. Whereas the results for the +conditionally linear models clearly suggest that the LASSO and Ridge priors are producing the +most precise density forecasts. The results for the nonlinear models tell a slightly different story. +We observe that the Ridge does well again but, for one-quarter-ahead tail forecasts, is outper- +formed by the HS. The LASSO, by contrast, is the weakest specification. Since the LASSO is +known to overshrink significant signals (see, e.g., Griffin and Brown, 2010), it could be that it +misses out important information arising from the GP-based basis functions. Third, and finally, +if we consider four-quarters-ahead predictions the QR coupled with a GP and a Ridge prior +becomes the single best performing model again. +To again gain a better understanding on which quantiles of the predictive distribution +drive the CRPSs, Figs. 4 and 5 are similar to Figs. 2 and 3 and show the QSs for different +quantiles. These are normalized to the linear QR with ridge prior so that numbers smaller than +one indicate that nonlinearities improve predictive accuracy for a given quantile and numbers +exceeding one imply that nonlinearities decrease forecasting accuracy. +In general, both figures tell a consistent story: nonlinearities help in the right tail across +19 + +both forecast horizons, for all three nonlinear specifications, and for most priors considered. The +only exception to this pattern are four-quarters-ahead right tail forecasts of GDP growth when +B-Splines are used. When there are gains, they are often sizable. For instance, in the case of +the QR-GP model we observe accuracy improvements up to 25 percent relative to the linear QR +model. +5% +10% +25% +50% +75% +90% +95% +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +Polynomials +5% +10% +25% +50% +75% +90% +95% +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +B-Splines +5% +10% +25% +50% +75% +90% +95% +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +Gaussian process +HS +RIDGE +NG +LASSO +DL +One-quarter-ahead quantile scores +Figure 4: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out +period and normalized to the QR with a Ridge prior. +When we focus on the left tail, accuracy premia often turn negative. In some cases (such as +for GP models with Ridge, NG and HS priors) there are accuracy gains for predicting downside +risks but these gains are only rather small (reaching five percent in the case of the QR-GP +regression with a Ridge prior). +3.4 +Heterogeneity of forecast accuracy over time +Up to this point, our analysis focused on averages over time. In the next step we will focus on +how forecasting performance changes over the hold-out period. To shed light on the importance +of nonlinearities over time, we again compare the different nonlinear specifications to the linear +QR regression with a Ridge prior. Figs. 5 and 6 show the cumulative CRPSs relative to the +linear benchmark QR for one-quarter and four-quarters-ahead forecasts. +20 + +5% +10% +25% +50% +75% +90% +95% +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +Polynomials +5% +10% +25% +50% +75% +90% +95% +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +B-Splines +5% +10% +25% +50% +75% +90% +95% +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +Gaussian process +HS +RIDGE +NG +LASSO +DL +One-year-ahead quantile scores +Figure 5: Four-quarters-ahead quantile scores for different values of q, averaged over the hold- +out period and normalized to the QR with a Ridge prior. +We start by focusing on the one-quarter-ahead forecasts first. For this specification, the +density accuracy performance is heterogenous over time. In the first part of the sample, models +using either polynomials or Gaussian processes coupled with a DL prior yield CRPSs that are +superior to the linear benchmark. However, these accuracy gains vanish during the GFC. When +we put more weight on tail forecasting accuracy (and consider GP-QRs), the gains disappear as +early as during the 2001 recession that followed the 9/11 terrorist attacks and the burst of the +dot-com bubble. +In the pandemic, we observe a sharp increase in predictive accuracy for several priors (most +notably the Ridge and NG priors). This pattern is more pronounced for the weighted variants of +the CRPSs. Considering the other nonlinear model specifications gives rise to similar insights. +Spline-based approximations to gq generally perform poorly up until the pandemic. During the +pandemic, even this specification improves sharply against the linear benchmark specification. +This pattern is particularly pronounced for the GP-QRs. +Considering the performance of the models and priors that did well on average (GP- +QRs with Ridge and the HS) reveals that most of these gains are actually driven by superior +performance during the pandemic. +21 + +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +CRPS +Polynomials +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +B-Splines +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +1.05 +1.1 +Gaussian process +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +CRPS-tails +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +0.85 +0.9 +0.95 +1 +1.05 +1.1 +1.15 +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +CRPS-left +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +1 +1.5 +2 +2.5 +3 +3.5 +1992 +1994 +1996 +1998 +2000 +2002 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +2020 +0.85 +0.9 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +HS +RIDGE +NG +LASSO +DL +CRPS over time One-Quarter-ahead +Figure 6: Cumulative one-quarter-ahead CRPS relative to the linear QR with the Ridge prior +over the hold-out period. +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +CRPS +Polynomials +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2 +B-Splines +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.5 +1 +1.5 +2 +2.5 +3 +Gaussian process +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +CRPS-tails +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +CRPS-left +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +HS +RIDGE +NG +LASSO +DL +CRPS over time One-Year-ahead +Figure 7: Cumulative four-quarter-ahead CRPS relative to linear QR with a Ridge prior over +the hold-out period. +22 + +Turning to four-quarters-ahead forecasts provides little new insights. Models using the +DL prior do not excel in the first part of the hold-out period and are generally outperformed by +the linear QR. However, accuracy improvements during the GFC and the pandemic are quite +pronounced for splines and the GP-QRs. +To sum up this discussion, our results indicate that forecast performance is heterogenous +over time. Different models such as the Polynomial-QR and the GP-QR with a DL prior out- +perform in the early part of the hold-out period. This performance premium vanishes during +the first two recessions observed in the sample. By contrast, other models such as QR-GP with +either the NG or the LASSO do not gain much in tranquil periods but excel during recessions. +3.5 +Properties and determinants of the quantile forecasts +The previous sub-sections have outlined that QRs and QRs with nonlinear components perform +well in terms of tail forecasting. In this sub-section, our goal is to investigate which variables +determine the quantile forecasts and in what respect successful shrinkage priors differ from their +less successful counterparts. +The presence of nonlinearities complicates our investigation since it is not clear on how +to measure the effect of xt on a given quantile of yt in the presence of nonlinearities. As a +simple solution, we follow Clark et al. (2022a) and approximate the nonlinear, quantile-specific +model using a linear posterior summary (see Woody et al., 2021). Specifically, we estimate the +following regression model: +Qq,t = x′ +t ˆαq + ˆεt, +ˆεt ∼ N(0, σ2 +ˆεq). +On the linearized coefficients we use a Horseshoe prior and on the error variances an inverse +Gamma prior. To achieve interpretability and decouple shrinkage and selection (see Hahn and +Carvalho, 2015), we then apply the SAVS estimator proposed in Ray and Bhattacharya (2018) +to the posterior mean of ˆαq.8 This will yield a sparse variant of ˆαq, that is easy to interpret +and can be understood as the best linear approximation to the corresponding quantile forecast +arising from the nonlinear model. For brevity, we focus on one-step-ahead forecasts (i.e. xt +includes a single lag of all variables). Results for four-quarters-ahead are included in the Online +Appendix. +8Huber et al. (2021) and Hauzenberger et al. (2021) apply SAVS to multivariate time series models and show +that it works well for forecasting. +23 + +HS +RIDGE +NG +LASSO +DL +-0.3 +-0.2 +-0.1 +0 +Linear +10% +M1REAL +HS +RIDGE +NG +LASSO +DL +-0.4 +-0.2 +0 +0.2 +0.4 +50% +PCESVx +PAYEMS +M1REAL +PCESVx +M1REAL +PCESVx +M1REAL +HS +RIDGE +NG +LASSO +DL +-0.5 +0 +0.5 +90% +PCESVx +M1REAL +M1REAL +PCESVx +USSERV +M1REAL +USSERV +M1REAL +CLAIMS +PCESVx +USSERV +M1REAL +HS +RIDGE +NG +LASSO +DL +-0.4 +-0.2 +0 +0.2 +0.4 +Polynomials +TCU +CUMFNS +PAYEMS +CUMFNS +M1REAL +CUMFNS +PAYEMS +PRFIx +HS +RIDGE +NG +LASSO +DL +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +UNRATE +M1REAL +PCESVx +PRFIx +UEMP5T +CLAIMS +HS +RIDGE +NG +LASSO +DL +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +PCESVx +M1REAL +TLBCBB +TLBBBD +PCESVx +HWIx +M1REAL +TLBCBB +TLBBBD +UNRATE +M1REAL +TLBBBD +PCECC9 +UNRATE +CLAIMS +TNWMVB +HS +RIDGE +NG +LASSO +DL +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +B-Splines +CUMFNS +USPRIV +CUMFNS +USCONS +M1REAL +CUMFNS +USPRIV +SRVPRD +FPIx +M1REAL +UMCSEN +HS +RIDGE +NG +LASSO +DL +-0.6 +-0.4 +-0.2 +0 +UEMP15 +UEMP15 +UEMP15 +M1REAL +M1REAL +HS +RIDGE +NG +LASSO +DL +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +UEMP15 +HWIx +HWIURA +TLBCBB +TLBBBD +UEMP15 +HWIx +HWIURA +TLBCBB +TLBBBD +UEMP15 +HWIx +HWIURA +TLBCBB +TLBBBD +USSERV +M1REAL +PCECC9 +UEMP15 +M1REAL +TLBCBB +TLBBBD +HS +RIDGE +NG +LASSO +DL +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +Gaussian process +PRFIx +PRFIx +M1REAL +SRVPRD +PRFIx +LNS120 +HS +RIDGE +NG +LASSO +DL +-0.5 +0 +0.5 +M1REAL +M1REAL +M1REAL +M1REAL +PCESVx +M1REAL +HS +RIDGE +NG +LASSO +DL +-0.5 +0 +0.5 +1 +PCESVx +USSERV +UEMP15 +UEMP15 +M1REAL +PCESVx +UEMP15 +USSERV +M1REAL +CLAIMS +PCESVx +M1REAL +Figure 8: One-quarter-ahead linearized posterior summaries across quantiles +Figure 8 shows the results of this exercise across nonlinear specifications and priors. Start- +ing with the left tail forecasts and linear models suggests that most quantile forecasts are not +related to elements in xt in a robust manner. There are only two exceptions. The first relates +to the NG prior. In this case, real money growth (M1real) survives the sparsification step and +the relationship indicates that declines in money growth imply an increase in tail risks (i.e. a +decline in GDP growth in the ten percent quantile). The other exception relates to the DL +prior. In this case, employment growth in education and health services (USEHS) remains. If +we focus on nonlinear models other variables appear to be correlated with forecasts of tail risks. +Among the different priors, we find some variables which show up repeatedly. Among these +are all nonfarm employees (PAYEMS), money growth, capacity utilization in manufacturing +(CUMFNS) and private fixed investment (both residential and non-residential). Most of these +variables are forward looking in nature and thus consistent with our intuition that economic +agents form expectations about the state of the economy in the future and thus change their +investment decisions accordingly. Notice that the relationship between private fixed investment +is particularly pronounced for GP-QRs under the HS, NG and the DL prior. Another pattern +worth mentioning is that the LASSO-based forecasts are generated from sparse models across +both linear and nonlinear specifications. +Once we focus on the center of the distribution we find that forecasts from linear models +24 + +are driven by one or two variables. Most prominently, specifications that do well in terms of point +forecasts (such as the Ridge and Lasso) yield point forecasts that display a strong relationship +with (lagged) money growth. In case we adopt a nonlinear specification, some differences arise +across specifications. +For polynomials, median forecasts under all priors except the DL are +related to very few predictors, with money growth and short-run unemployment showing up +for the NG and LASSO models. The DL prior implies a more dense model. This could be a +possible reason for the rather weak performance of this specification. When we turn to spline- +based models we again find a similar pattern. Money growth shows up in the case of the NG and +LASSO and short-run unemployment predicts median output growth if we adopt a HS, Ridge +or NG prior. Models that capture nonlinearities through GPs, our best performing nonlinear +specifications, give rise to a very consistent pattern across priors. In all cases, lagged money +growth appears to be a robust predictor of GDP growth. And it impacts GDP growth forecasts +negatively. +Finally, when our focus is on right-tail forecasts, all models become much more dense. +Variables that have been showing up in the case of left-tail and point forecasts again show +up (most notably money growth and short-term unemployment). Additional variables such as +initial unemployment claims or prices remain in the sparse model as well. But there is no clear +pattern across models except for the fact that money growth also remains in the set of robust +predictors even if much shrinkage is introduced. +The analysis based on linearized coefficients provides information on which variables are +predictive for output growth forecasts across quantiles. However, the analysis in Sub-sections 3.2 +and 2.3 suggests that differences in forecast performance are driven by the prior. To understand +which properties of a given prior exert a positive effect on predictive accuracy, we now focus +on the shrinkage hyperparameters of the different priors. Comparing the amount of shrinkage +introduced through the different priors is not straightforward. Here, our measure of choice is +based on using the re-scaled log determinant of the prior covariance matrices as a measure of +overall shrinkage for each respective prior. Since all prior covariance matrices are diagonal this +simply amounts to summing over the log of the diagonal elements of B0q and then normalizing +through by the number of diagonal elements. +This constitutes a rough measure of overall +shrinkage and we can compute it for each quarter in the hold-out period. Again, we will focus +on shrinkage introduced in one-quarter-ahead predictive regressions. The four-quarters-ahead +results are qualitatively similar and included in the Online Appendix. +25 + +Log-determinants of the prior covariance matrices over the hold-out period are depicted +in Fig. 9. The figure includes (if applicable) solid lines which refer to the amount of shrinkage +introduced on the linear coefficients and dashed lines which refer to the log-determinants of the +prior covariances that relate to the shrinkage factors on the basis coefficients of the different +nonlinear models. +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-8 +-7.8 +-7.6 +-7.4 +-7.2 +-7 +-6.8 +-6.6 +-6.4 +-6.2 +10% +Linear +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9.5 +-9 +-8.5 +-8 +-7.5 +-7 +-6.5 +-6 +-5.5 +-5 +Polynomials +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-10 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +B-Splines +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9.5 +-9 +-8.5 +-8 +-7.5 +-7 +-6.5 +-6 +-5.5 +-5 +Gaussian process +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-8 +-7.5 +-7 +-6.5 +-6 +-5.5 +50% +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-10 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-8.2 +-8 +-7.8 +-7.6 +-7.4 +-7.2 +-7 +-6.8 +-6.6 +-6.4 +-6.2 +90% +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9.5 +-9 +-8.5 +-8 +-7.5 +-7 +-6.5 +-6 +-5.5 +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-10 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +1992 +1995 +1997 +2000 +2002 +2005 +2007 +2010 +2012 +2015 +2017 +2020 +-9.5 +-9 +-8.5 +-8 +-7.5 +-7 +-6.5 +-6 +HS +RIDGE +NG +LASSO +DL +HS-nonlinear +RIDGE-nonlinear +NG-nonlinear +LASSO-nonlinear +DL-nonlinear +Figure 9: Overall shrinkage in one-step-ahead predictive QRs +From this figure, a few interesting insights emerge. First, the different priors introduce +different degrees of shrinkage. Overall, two priors stand out in terms of the amount of shrinkage +they introduce. +The first one is the DL. This is rather surprising given the fact that this +prior performs worst in the forecasting horse race but also leads to posterior summaries which +feature several non-zero coefficients. Our conjecture is that this prior forces the vast majority of +coefficients to effectively zero but several coefficients remain sizable and the corresponding set +of variables is still too large and overfitting issues arise. The prior that introduces the largest +amount of shrinkage is the LASSO. In this case, almost all coefficients are very small. These +observations are corroborated by boxplots, included in the Online Appendix (see Figs. 3 to 6 in +the Online Appendix), which show the scaling parameters over three sub-samples. Our results +imply that models which feature a large number of shrunk coefficients provide better forecasts +than models which feature many coefficients that are effectively zero and some coefficients that +are non-zero and sizable. This is consistent with findings in Giannone et al. (2021) who provide +26 + +empirical evidence that macroeconomic data is rather dense as opposed to sparse. Notice that +the fact that dense models produce accurate tail forecasts is not inconsistent with our analysis +based on linearized posterior summaries. This is because the linearized model under a shrinkage +and sparsification approach strikes a balance between achieving a good model fit while keeping +the model as simple as possible. Hence, if the covariates in the panel co-move, shrinkage and +sparsification techniques will select one of these variables. +Second, in almost all cases the amount of shrinkage introduced on the nonlinear part of +the different models is much larger than the degree of shrinkage on linear coefficients. This holds +for most priors, nonlinear methods and over all time periods. One exception is the Spline-QR +specification with a DL prior and when the right tail is considered. Interestingly, this specific +combination of much stronger shrinkage on the linear part of the model and less shrinkage on +the nonlinear part leads to good forecasts in the right tail (see Fig. 4). +Third, and finally, there is (with some notable exceptions) relatively little time-variation +in the amount of shrinkage over the hold-out period. The only exception are the GP-QRs. In +this case, the amount of shrinkage decreases appreciably from 2013 onward. +4 +Concluding remarks +In this paper, we have shown that combining QRs with nonlinear specifications and large datasets +leads to precise quantile forecasts of GDP growth. +Since the resulting models are high di- +mensional, we consider several popular shrinkage priors to regularize estimates. MCMC-based +estimation of these huge dimensional models is slow. Hence, we speed up computation by us- +ing VB approximation methods that approximate the joint posterior distribution using simpler +approximating densities. +The empirical results indicate that our methods work remarkably well when the CRPS is +taken under consideration. When we put more weight on the tail forecasting performance, we +find that most of the overall gains are driven by a strong performance in both the left and right +tail while the performance in the center of the distribution is close to the predictive accuracy of +the simple quantile regression proposed in Adrian et al. (2019). These results, however, differ +across priors and nonlinear specifications. In principle, it can be said that models featuring +simple shrinkage priors, such as the LASSO or Ridge, in combination with GPs to capture +nonlinearities of arbitrary form yield the most precise forecasts. +27 + +References +Adams, P. A., Adrian, T., Boyarchenko, N., and Giannone, D. (2021). Forecasting macroeco- +nomic risks. International Journal of Forecasting, 37(3):1173–1191. +Adrian, T., Boyarchenko, N., and Giannone, D. (2019). Vulnerable growth. American Economic +Review, 109(4):1263–89. +Adrian, T., Boyarchenko, N., and Giannone, D. (2021). 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MIT press Cambridge, MA. +Woody, S., Carvalho, C. M., and Murray, J. S. (2021). Model interpretation through lower- +dimensional posterior summarization. +Journal of Computational and Graphical Statistics, +30(1):144–161. +Yu, K. and Moyeed, R. A. (2001). Bayesian quantile regression. Statistics & Probability Letters, +54(4):437–447. +31 + diff --git a/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/load_file.txt b/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b34b7b3d47378373d26e19f12a522b0d3fc0bd38 --- /dev/null +++ b/2dFRT4oBgHgl3EQfnDfl/content/tmp_files/load_file.txt @@ -0,0 +1,1362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf,len=1361 +page_content='Nonlinearities in Macroeconomic Tail Risk through the Lens of Big Data Quantile Regressions Jan Pr¨user TU Dortmund Department of Statistics Florian Huber1 University of Salzburg Department of Economics Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Modeling and predicting extreme movements in GDP is notoriously difficult and the selection of appropriate covariates and/or possible forms of nonlinearities are key in obtaining precise forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this paper, our focus is on using large datasets in quantile regression models to forecast the conditional distribution of US GDP growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To capture possible non-linearities we include several nonlinear specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The resulting models will be huge dimensional and we thus rely on a set of shrinkage priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since Markov Chain Monte Carlo estimation becomes slow in these dimensions, we rely on fast variational Bayes approximations to the posterior distribution of the coefficients and the latent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We find that our proposed set of models produces precise forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These gains are especially pronounced in the tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Using Gaussian processes to approximate the nonlinear component of the model further improves the good performance in the tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' JEL: C11, C32, C53 KEYWORDS: Growth at risk, quantile regression, global-local priors, non-linear models, large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 1We would like to thank the editor, Mike McCracken, three anonymous referees, and participants at the International Symposium on Forecasting 2022 at the University of Oxford and at the Statistische Woche in M¨unster for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Jan Pr¨user gratefully acknowledges the support of the German Research Foundation (DFG, 468814087).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Please address correspondence to: Jan Pr¨user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Department of Statistics, TU Dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Address: CDI Building, Room 122, 44221 Dortmund, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Email: prueser@statistik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='13604v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='EM] 31 Jan 2023 1 Introduction Modeling and predicting the conditional distribution of output growth has attracted considerable academic attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Starting at least with the influential paper by Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2019), focus has shifted towards analyzing whether there exist asymmetries between a predictor (in their case financial conditions) and output growth across different quantiles of the empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Several other papers (Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Ferrara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Gonz´alez-Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Delle Monache et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Plagborg-Møller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Reichlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Figueres and Jaroci´nski, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2022) have started to focus on modeling full predictive distributions using different approaches and information sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, most of these contributions have been confined to models which exploit small datasets and, at least conditional on the quantile analyzed, assume linear relations between GDP growth and the predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 Times of economic stress such as the global financial crisis (GFC) or the Covid-19 pandemic have highlighted that exploiting information contained in many time series and allowing for nonlinearities improves predictive performance in turbulent periods (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since economic dynamics change in volatile economic regimes, models that control for structural breaks allow for different effects of economic shocks over time or imply nonlinear relations between GDP growth and its predictors often excel in forecasting applications (see D’Agostino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Carriero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Pfarrhofer, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Moreover, another important empirical regularity is that the set of predictors might change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is because variables which are seemingly unimportant in normal periods (such as financial conditions) play an important role in recessions and yield important information on future behavior of output growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This discussion highlights that the effect of predictors on output growth depends on the quantile under consideration and thus appears to be state dependent and modeling the tran- sition might call for nonlinear econometric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The key challenge, however, is to identify the different determinants of GDP growth across quantiles while taking possible nonlinearities into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this paper, we aim to solve these issues by proposing a Bayesian quantile re- gression (QR) which can be applied to huge information sets, and which is capable of capturing nonlinearities of unknown form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our model is a standard QR model that consists of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2A recent exception is Kohns and Szendrei (2021) who estimate large-scale quantile regressions and then apply ex-post sparsification to sharpen predictive inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2 The first assumes a linear relationship between the covariates and quantile-specific GDP growth whereas the second component assumes an unknown and possibly highly nonlinear relationship between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The precise form of nonlinearities is captured through three specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' One is parametric and based on including polynomials up to a certain order, whereas the re- maining two are nonparametric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Among these nonparametric specifications we include B-splines (see Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020) and Gaussian processes (see Williams and Rasmussen, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Both have been shown to work well when it comes to function estimation and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The combination of a linear and nonlinear term implies that the dimension of the parame- ter space increases substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since all these models can be cast in terms of a linear regression conditional on appropriately transformed covariates, we can use regularization techniques to de- cide on whether more flexibility is necessary and which variables should enter the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We achieve this through several popular shrinkage priors that have excellent empirical properties in large dimensions and are relatively easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These shrinkage priors enable us to select promising subsets of predictors and the degree of nonlinearities for each quantile separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Posterior inference using Markov Chain Monte Carlo (MCMC) techniques in these dimen- sions proves to be an issue because we have to estimate a large-scale regression model for all quantiles of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This procedure needs to be repeated a large number of times if we wish to carry out an out-of-sample forecasting exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To reduce the computational burden enormously we estimate the QRs using Variational Bayes (VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 This estimation strategy approximates the exact full conditional posterior distributions with simpler approximating distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These approximating densities are obtained by minimizing the Kullback-Leibler (KL) distance be- tween some known density q and the exact posterior distribution p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, integration in huge dimensions is replaced by a simpler optimization routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our approach is fast and allows for computing all results of our forecasting exercise without the use of high performance computing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We apply our techniques to the large dimensional FRED-QD dataset (McCracken and Ng, 2016) and focus on single and multi-step-ahead forecasting of US GDP growth over a hold-out period ranging from 1991Q2 to 2021Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The different nonlinear models we consider are high dimensional and feature up to around 1,000 coefficients per equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The empirical results can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Using huge information sets and nonlinear models in combination with priors that introduce substantial shrinkage pays off for tail 3For an introduction, see Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2017) and an algorithm for QRs is provided in Bufrei (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3 forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In both tails, forecast improvements relative to the small-scale QR model developed in Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2019) are sizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we focus on the center of the distribution the differences become smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Once we allow for nonlinearities we find modest improvements in predictive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Comparing the different nonlinear specifications reveals that Gaussian processes offer the largest improvements vis-´a-vis the linear QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This indicates that a successful tail forecasting model should be able to extract important information from huge datasets, while controlling for possibly nonlinear relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we focus on the key properties of the proposed priors we observe that priors that imply a dense model (characterized by many small coefficients) yield good tail forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The next section introduces the general QR and the scale-location mixture representation to cast the model in terms of a standard generalized additive regression with auxiliary latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We then focus on the different priors used, provide additional details on the nonlinear components of the models, briefly discuss VB, outline how we estimate the posterior distributions of the parameters and latent quantities, and illustrate the computational properties of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Section 3 discusses our empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The final section summarizes and concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' An Online Appendix includes additional technical details, empirical results and more precise information on the used dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2 Bayesian analysis of general QRs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 The likelihood function In this paper, our goal is to model the dependence between the qth quantile of GDP growth yt and a panel of K predictors in {xt}T t=1 with K being huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The covariates include a wide range of macroeconomic and financial indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Possible nonlinearites between yt and xt are captured through a function gq(xt), with gq : RK → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The fact that K is large and the inclusion of nonlinear functions of xt implies that the number of parameters is large relative to the number of observations T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our workhorse model is the QR developed in Koenker and Bassett (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As opposed to the standard QR, our model decomposes the qth quantile function Qq(yt) in a linear and nonlinear part and a non-standard error distribution: (1) yt = x′ tβq + gq(xt) + εt, 4 where βq is a K−dimensional vector of quantile-specific regression coefficients and εt is a shock term with density fq such that the qth quantile equals zero: � 0 −∞ fq(εt)dεt = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Conditional on the quantile, this model resembles a generalized additive model (GAM), see Hastie and Tibshirani (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We approximate gq(xt) using nonlinear transformations of xt: (2) gq(xt) ≈ M � m=1 γqmzm(xt) = z′ tγq with γq = (γq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , γqM)′, zt = (z1(xt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , zM(xt))′ and zm(xt) denotes a basis function that depends on xt with γqm denoting the corresponding basis coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This basis function de- pends on the specific approximation model used to infer the nonlinear effects and our additive representation nests models commonly used in the machine learning literature (such as Gaussian processes, splines, neural networks but also more traditional specifications such as time-varying parameter models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We will discuss the precise specification of zm (and thus zt) in more detail in Sub-section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Here it suffices to note that depending on the specification, M could be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For instance, in the Gaussian process case, M = T and thus the number of regression coefficients would be K + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' If fq remains unspecified, estimation of βq and γq is achieved by solving the following optimization problem: arg max {βq,γq} = T � t=1 ρq(yt − x′ tβq − z′ tγq), with ρq(l) = l[q − I(l < 0)] denoting the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This optimization problem is straight- forward to solve but, if K + M is large, regularization is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This motivates a Bayesian approach to estimation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' From a Bayesian perspective, carrying out posterior inference requires the specification of a likelihood and suitable priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Following Yu and Moyeed (2001) we assume that the shocks εt follow an asymmetric Laplace distribution (ALD) with density: fq(εt) = q(1 − q) exp (−ρq(εt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The key thing to notice is that the qth quantile equals zero and the parameter q controls the 5 skewness of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Kozumi and Kobayashi (2011) show that one can introduce auxil- iary latent quantities to render the model with ALD distributed shocks conditionally Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is achieved by exploiting a scale-location mixture representation (West, 1987): εqt = θqνqt + τq√σqνqtut, θq = 1 − 2q q(1 − q), τ 2 q = 2 q(1 − q), νqt ∼ E � 1 σq � , ut ∼ N(0, 1), where E � 1 σq � denotes the exponential distribution and σq is a scaling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, con- ditional on knowing νq = (νq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , νqT )′, θq, τq, σq and appropriately selecting gq, the model is a linear regression model with response ˆyt = yt −θqνqt and Gaussian shocks that are conditionally heteroskedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This conditional likelihood will form the basis of our estimation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To complete the model specification we assume that 1 σq ∼ G(c0, d0), where c0 is the shape and d0 the rate parameter of the Gamma distribution which we set both to zero in order to obtain a flat prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The choice of the prior distribution on βq and γq is essential for our high dimensional QRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We discuss different suitable choices in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 Priors for the quantile regression coefficients For the large datasets we consider in this paper, M + K ≫ T and thus suitable shrinkage priors are necessary to obtain precise inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Kohns and Szendrei (2021) and Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (ming) use flexible shrinkage priors in large-scale QRs and show that these work well for tail forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We build on their findings by considering a range of different priors on βq and γq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' All these priors belong to the class of so-called global-local shrinkage priors (Polson and Scott, 2010) and have the following general form: βq|ψβ q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , ψβ qK, λβ q ∼ K � j=1 N(0, ψβ qjλβ q ), ψβ qj ∼ u, λβ q ∼ π, γq|ψγ q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , ψγ qM, λγ q ∼ M � j=1 N(0, ψγ qjλγ q), ψγ qj ∼ u, λγ q ∼ π, with λs q (s ∈ {β, γ}) denoting a quantile-specific global shrinkage parameter and ψs qj are local scaling parameters that allow for non-zero coefficients in the presence of strong global shrinkage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', with λs q close to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The functions u and π refer to mixing densities which, if suitably chosen, translate into different shrinkage priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this paper, all the priors we consider can be cast into this form but differ in the way the mixing densities u and π are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since these 6 priors are well known, we briefly discuss them in the main text and relegate additional technical details to the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We focus on five shrinkage priors that have been shown to work well in a wide variety of forecasting applications (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Huber and Feldkircher, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Cross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Chan, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Pr¨user, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first prior we consider is the Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The Ridge prior is a special case of a global-local prior with local parameters set equal to 1 and a global shrinkage parameter which follows an inverse Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Formally, this implies setting ψs qj = 1 for all q, j and λs q ∼ G−1(e0, e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The hyperparameters e0 and e1 control the tightness of the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We set these equal to e0 = e1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This prior shrinks all coefficients uniformly towards zero and provides little flexibility to allow for idiosyncratic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', variable-specific) deviations from the overall shrinkage pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This issue is solved by estimating the local shrinkage parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The Horseshoe (HS, see, Carvalho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2010), our second prior, does this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This prior sets u and π to a half-Cauchy distribution: � ψs qj ∼ C+(0, 1) and �λsq ∼ C+(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The HS possesses excellent posterior contraction properties (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Armagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' van der Pas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Moreover, it does not rely on any additional tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Another popular global-local shrinkage prior is the Normal-Gamma (NG) prior of Griffin and Brown (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This prior assumes that u and π are Gamma densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' More formally, ψs qj ∼ G(ϑ, λs qϑ/2) and λs q ∼ G(c0, d0), with ϑ being a hyperparameter that controls the tail behavior of the prior, and c0 and d0 are hyperparameters that determine the overall degree of shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We set c0 = d0 = 0 and ϑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This choice implies heavy global shrinkage on the coefficients but also implies fat tails of the marginal prior of the coefficients after integrating out the local scaling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The Bayesian LASSO is obtained as a special case of the NG prior with ϑ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Finally, the Dirichlet-Laplace prior (Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2015) assumes that the local scal- ing parameter ψs qj is a product of a Dirichlet-distributed random variate φs qj ∼ Dir(α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , α) and a parameter ˜ ψs qj ∼ E(1/2) that follows an exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, the Dirichlet-Laplace prior sets ψs qj = (φs qj)2 ˜ ψs qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' On the global scaling parameters we use a Gamma distribution � λβ q ∼ G(Kα, 1/2) and � λγ q ∼ G(Mα, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We set α = 1 K for the linear part and α = 1 M for the non-linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 Capturing nonlinearities in high dimensional QRs In extreme periods such as the GFC or the Covid-19 pandemic, nonlinearities in macroeconomic data become prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We control for this by having a nonlinear part in our QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As stated in (2), we capture possible nonlinearities in xt through nonlinear transformations zm(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first and simplest nonlinear specification maps xt into the space of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Bai and Ng (2008) capture nonlinearities in macro data through polynomials and by relying on factor-based predictive regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We follow this approach and define the corresponding basis function as follows: zt = ((x2 t )′, (x3 t )′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , (xN t )′)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Deciding on the order of the polynomial N is a model selection issue and suitable shrinkage priors can be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In our empirical work, we focus on the cubic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This specification will overweight large movements in xt and should thus be suitable for quickly capturing sharp downturns in the business cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, the number of coefficients triples since M = 3K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The resulting nonlinear model is called Polynomial-QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Adding cubic terms allows us to capture nonlinearities in a relatively restricted manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since the precise form of nonlinearities is typically unknown, the remaining two specifications we consider are nonparametric and only require relatively mild prior assumptions on the form of nonlinear interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first of these two is the B-Spline (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', De Boor, 2001, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' B-Splines have a proven track record in machine learning and computer science (Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For the B-spline, we assume that each element in xt exerts a (possibly) nonlinear effect on yt that might differ across covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This implies that gq(xt) equals: gq(xt) ≈ K � k=1 Φk(x•,j)γq,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Here, we let Φk denote a T ×r matrix of B-spline basis functions that depend on the jth covariate in X = (x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , x′ T )′, x•,j and r is the number of knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, the number of nonlinear coefficients is M = rK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In our empirical work we place the knots at the following quantiles of x•,j: {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95, 1}, implying that r = 9 and thus M = 9K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We will henceforth call this model Spline-QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The last specification we consider is the Gaussian process (GP) regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' GP regression 8 is a nonparametric estimation method that places a GP prior on the function gq(xt): gq(xt) ∼ GP(µq(xt), K(xt, xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The mean function µq(xt) is, without loss of generality, set equal to zero and K(xt, xt) is a kernel function that encodes the relationship between xt and xt for t, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' It is worth noting that our additive specification implies that if the mean function is set equal to zero, the model is centered on a standard QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since xt is observed in discrete time steps, the GP prior implies a Gaussian prior on gq = (gq(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , gq(xT ))′: gq ∼ N(0T , K(w)), where K(w) is a T × T-dimensional matrix with (t, t)th element K(xt, xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' w = (w1, w2)′ is a set of hyperparameters that determine the properties of the kernel (and thus the estimated function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The GP regression is fully specified if we determine the kernel function K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this paper, we use the Gaussian (or squared exponential) kernel: K(xt, xt) = w1 × exp � −w2 2 ||xt − xt||2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The hyperparameters w are set according to the median heuristic proposed in Arin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' What we discuss above is the function-space view of the GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' An alternative way of expressing the GP is the so-called weight-space view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The weight-space view is obtained by integrating out gq, yielding the following regression representation: y = Xβq + Zγq + ε, with y denoting the stacked dependent variables, Z is the lower Cholesky factor of K and γq ∼ N(0, IT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Notice that gq = Zγq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, the Cholesky factor of the kernel matrix provides the basis functions, and the parameters can be readily estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, the number of nonlinear coefficients is M = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since we use a shrinkage prior on γq, the corresponding implied kernel is given by ZBγ q Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The M × M matrix Bγ q is a prior covariance matrix with Bγ q = λγ q × diag(ψγ q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , ψγ qM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Approximating gq using GPs leads to the GP-QR specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This completes our choice of nonlinear techniques used in the big data QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Alternative 9 choices (such as allowing for time-varying parameters, neural networks or Bayesian additive regression trees) can be straightforwardly introduced in this general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 A brief introduction to variational Bayes The high dimensionality of the state space calls for alternative techniques to carry out posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We opt for using variational approximations to the joint posterior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this section, we provide a discussion on how VB works in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For an excellent in-depth introduc- tion, see Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In machine learning, variational techniques have been commonly used to estimate complex models such as deep neural networks (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Polson and Sokolov, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In econometrics, recent papers use VB in huge dimensional multivariate time series models such as VARs (Gefang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Chan and Yu, 2020) or state space models to speed up estimation (Koop and Korobilis, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In a recent paper, Korobilis and Schr¨oder (2022), propose a QR factor model and estimate it using VB techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To simplify the exposition, we fix the prior variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The appendix provides information on how we estimate the prior variances (and associated hyperparameters) using VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Let ξq = (βq, γq, σq, νq) denote a generic vector which stores all unknowns of the model, with νq = (νq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , νqT ) denoting the latent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our aim is to approximate the joint posterior distribution p(ξq|y) using an analytically tractable approximating distribution q(ξq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This variational approximation is found by mini- mizing the Kullback-Leibler (KL) distance between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' One can show that minimization of the KL distance is equivalent to maximizing the evidence lower bound (ELBO) defined as: (3) ELBO = Eq(ξq) (log p(ξq, y)) − Eq(ξq) (log q(ξq)) , with Eq(ξq) denoting the expectation with respect to q(ξq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This implies that finding the approx- imating density q replaces the integration problem (which is typically solved through MCMC sampling) with an optimization problem (which is fast and thus scales well into high dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' A common and analytically tractable choice of approximating densities assumes that q(ξq) is factorized as follows: q(ξq) = S � s=1 qs(ξqs), where ξqs denotes a partition of ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' A particular example (which we use in this paper) would specify ξq1 = (β′ q, γ′ q)′, ξq2 = σq and ξq3 = νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 10 This class is called the mean field variational approximation and assumes that the different blocks ξqs are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 Notice that all our priors on ξq can be written as: p(ξq) = S � s=1 p(ξqs), and using the fact that: Eq(ξq)(log p(ξq, y)) = Eq(ξq)(log p(y|ξq)) + S � s=1 Eq(ξq)(log p(ξqs)), the ELBO can be stated as: ELBO = Eq(ξq)(log p(y|ξq)) + S � s=1 Eq(ξq)(log p(ξqs)) − S � s=1 Eq(ξq)(log q(ξqs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Wand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2011) prove that under the variational family the optimal approximating densities are closely related to the full conditional posterior distributions: q∗ s(ξq) = exp � Eq(ξq)(log p(ξqs|y, ξq,−s) � , where ξq,−s is the vector ξq with the sth component excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, if p(ξqs|y, ξq,−s) is known (which is the case for the QR regression based on the auxiliary representation discussed in the previous subsection), the elements in ξqs can be updated iteratively (by conditioning on the expected values of ξq,−s) until the squared difference of the ELBO or of all elements of ξqs is smaller than some small ϵ between two subsequent iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 Approximate Bayesian inference in general QRs In this section we briefly state the three approximating densities (q∗ s(ξ)) used to estimate the parameters and latent quantities in the QR regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We provide derivations for the three approximating densities of the three parameter groups: ˜βq = (β′ q, γ′ q)′, σq and νq in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We start by discussing the approximating densities for the regression and basis coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 4Frazier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2022) state that mean field VB approximations might perform poorly in models with a large number of latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, they also note that the resulting model forecasts could still perform well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 11 A Gaussian distribution approximates the posterior of ˜βq: p( ˜βq|•) ≈ N � E( ˜βq), ˆΣκq � , with variance and mean given by, respectively: ˆΣ ˜βq = � T � t=1 ftf ′ t τ 2q E � 1 νqt � E � 1 σq � + B−1 0q �−1 , E( ˜βq) = ˆΣ ˜βq � ���E � 1 σq � T � t=1 E � 1 νqt � ft � yt − θq � E � 1 νqt ��−1� τ 2q � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' ft = (x′ t, z′ t)′ and B−1 0q = diag(Bβ q , Bγ q )−1 is a prior precision matrix with Bβ q = λβ q ×diag(ψβ q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , ψβ qK) and Bγ q = λγ q × diag(ψγ q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , ψγ qK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The approximating densities used to estimate the prior hy- perparameters are provided in Section 1 of the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The latent variable νqt follows a generalized inverse Gaussian (GIG) distribution: GIG(r, A, B)5 with p(νqt|•) ≈ GIG � � � � � 1 2, 2E � 1 σq � + θ2 q τ 2q E � 1 σq � � �� � Aq , E � 1 σq � τ 2q �� yt − f ′ tE( ˜βq) �2 + f ′ t ˆΣ ˜βqft � � �� � Bq � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The moments of νqt are given by E � νj qt � = �� Bq � Aq �j K1/2+j �� AqBq � K1/2 �� AqBq � , where Kx denotes the modified Bessel function of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Finally, we approximate p � 1 σq |• � ≈ G(cq1, dq1) 5We use the following parametrization of the GIG distribution: log (GIG(x)) ∝ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 − 1) log(x) − � Ax + 1 2 B x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 12 with cq1 = c0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5T, dq1 = d0 + T � t=1 E(νqt) + 1 2τ 2q T � t=1 (E � 1 νqt � � yt − f ′ tE( ˜βq) �2 + 2θq(f ′ tE( ˜βq) − yt) + E(νqt)θ2 q + E � 1 νqt � f ′ t ˆΣ ˜βqft), and E � 1 σq � = cq1 dq1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 Comparing computation times between VB and MCMC These steps, in combination with the updating steps for the priors detailed in the Online Ap- pendix, form the basis of our VB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As stated in the introduction, the key advantage of using VB instead of more precise MCMC-based techniques is computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Before we turn to our empirical work, we illustrate this point using synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 200 400 600 800 1000 1200 1400 1600 1800 2000 Number of Variables 0 5 10 15 20 25 30 35 40 45 Time in Minutes Runtime: VB vs MCMC VB MCMC Figure 1: Comparison of computation times against the number of covariates M + K To illustrate the computational merits of employing VB-based approximations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 1 shows the estimation times for different values of M + K using our VB-based QR (for a specific quantile) and the QR estimated through the Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The MCMC algorithm is repeated 10, 000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The figure shows that the computational burden increases lightly in the number of covariates for VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we focus on MCMC estimation, the computational requirements increase sharply in the number of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Especially in our empirical work, where K + M is often above 1, 000, VB proves to be a fast alternative to MCMC-based quantile regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' It is also worth stressing that if the number of quantiles to estimate is large (and no parallel computing facilities are available), MCMC-based estimation becomes excessively slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 13 3 Forecasting output growth using huge dimensional QRs In this section, we present our forecasting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The next sub-section provides information on the dataset and the forecasting setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We then proceed by discussing the results from QRs that exclude the nonlinear part in Sub-section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The question whether nonlinearities are important is investigated in Sub-section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3, and Sub-section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 deals with how forecast accuracy changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Sub-section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 discusses the determinants of the different tail forecasts and differences in the shrinkage properties across priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 Data overview and forecasting setup We use the quarterly version of the McCracken and Ng (2016) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The data set covers in- formation about the real economy (output, labor, consumption, orders and inventories), money, prices and financial markets (interest rates, exchange rates, stock market indexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' All series are seasonally adjusted and transformed to be approximately stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The set of variables included in xt and their transformation codes are described in Table 1 of the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' All models we consider also include the first lag of GDP growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 Forecasts are carried out using direct forecasting by appropriately lagging the elements in xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our sample runs from 1971Q1 to 2021Q3 and we use the period 1991Q2 to 2021Q3 as our hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The forecasting design is recursive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This implies that we estimate all our models on an initial training sample with data until 1991Q1 and produce one-quarter- and four-quarters-ahead predictive distributions for 1991Q2 and 1992Q1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' After obtaining these, we add the next observation (1991Q2) and recompute the models to obtain the corresponding predictive densities for 1991Q3 and 1992Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This procedure is repeated until we reach the end of the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As a measure of overall forecasting accuracy we focus on the continuous ranked probability score (CRPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The CRPS is a measure of density forecasting accuracy and generalizes the mean absolute error (MAE) to take into account how well a given model predicts higher order moments of a target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The CRPS measures overall density fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Considering overall CRPSs possibly masks rele- vant idiosyncrasies of model performance across quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' If a decision maker is interested in downside risks to GDP growth, she might value a model more that does well at the critical 5 or 10 percentiles as opposed to the remaining regions of the predictive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To shed light 6We find that including more lags of GDP growth only has small effects on the empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 14 on asymmetries across different predictive quantiles, we focus on the quantile score (QS): QSqt = (yt − Qqt)(q − 1{yt≤Qqt}), where Qqt is the forecast of the qth quantile of yt and 1{yt≤Qqt} denotes the indicator function that equals one if yt is below the forecast for the qth quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The QS can also be used to construct quantile-weighted (qw) CRPS scores (Gneiting and Ranjan, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These qw-CRPSs can be specified to put more weight on certain regions of the predictive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In general, the qw-CRPS is computed as: qw-CRPS = 2 J − 1 J−1 � j=1 ω(ζj)QSsjt, with ζj = j/J, J − 1 = 19 denoting the number of quantiles we use to set up the qw-CRPS and sj selects the jth element from the set of quantiles we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This set ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 and thus, s1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05, s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' , s19 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We use two weighting functions ω(ζj) that focus on different regions of the predictive density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These schemes are motivated in Gneiting and Ranjan (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first (CRPS-left) puts more weight on the left tail (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' downside risks) and is specified as ω(ζj) = (1 − ζj)2, while the second (CRPS-tails) puts more weight on both tails as opposed to the center of the distribution: ω(ζj) = (2ζj − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Notice that if we use equal weights we obtain a discrete approximation to the CRPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 Results based on linear QRs We start discussing the QRs that set g(xt) = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Here, our goal is to show that including more information pays off relative to the model proposed in Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, we benchmark the QR models to the model which only includes lagged GDP growth and the NFCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This model is henceforth called the ABG model and estimated in the same way as in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Table 1 shows average (over time) qw-CRPSs relative to the ABG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Numbers smaller than one suggest that a given model outperforms the ABG benchmark whereas numbers exceed- ing unity indicate that the model produces less precise density forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The table reveals a great deal of heterogeneity with respect to different priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Popular GL priors such as the HS, the NG or the DL lead to forecasts that are often slightly worse than the 15 Table 1: CRPS for linear models One-quarter-ahead Four-quarters-ahead Model CRPS CRPS-tails CRPS-left CRPS CRPS-tails CRPS-left HS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 RIDGE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='87 NG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='03 LASSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='88 DL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='14 Notes: We highlight in light gray (dark gray) rejection of equal forecasting accuracy against the benchmark model at significance level 10% (5%) using the test in Diebold and Mariano (1995) with adjustments proposed by Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Results are shown relative to the AGB model and are based on the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' ones obtained from the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, priors such as the Ridge or the LASSO (which is particular known for over-shrinking significant signals (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Griffin and Brown, 2010)) yield forecasts that are better than the benchmark forecasts for both forecast horizons and across the different variants of the CRPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our findings corroborate recent results in Carriero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2022) who show that large QRs with shrinkage improve upon the ABG benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is especially pronounced in the case of the Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, the accuracy gains vis-´a-vis the ABG benchmark reach 17 percent and, in most cases, accuracy differences are statistically significant according to the Diebold and Mariano (1995) test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Turning to the different forecast horizons reveals that specifications that do well in terms of short-term forecasting also produce precise longer-term predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For the LASSO-based model, four-quarters-ahead accuracy gains are slightly more pronounced whereas for the Ridge we do not find discernible differences across both forecast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Next, we drill deeper into the quantile-specific forecasting performance by considering QSs for q ranging from q ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These, for one-step-ahead forecasts, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3 provides the four-steps-ahead results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Before starting our discussion it is worth stressing that many of these differences are statistically significant with respect to the DM test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The corresponding results are provided in the Online Appendix (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 15 and 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Similar to the findings based on the CRPSs, there is a great deal of heterogeneity across priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Both the LASSO and the Ridge prior improve upon the ABG benchmark for all quantiles by relatively large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These gains appear to be more pronounced in the tails, reaching over 20 percent in terms of the QSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When focusing on the center of the distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', the median forecast), the gains are much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In general, the other priors perform considerably 16 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 HS RIDGE NG LASSO DL One-quarter-ahead quantile scores Figure 2: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 HS RIDGE NG LASSO DL One-year-ahead quantile scores Figure 3: Four-quarters-ahead quantile scores for different values of q, averaged over the hold- out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The only exception turns out to be the NG prior which, displays an excellent performance in the left tail, while being still outperformed by the LASSO and the Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Considering four-quarters-ahead tail forecasts yield a similar but less pronounced picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 17 For higher-order forecasts, priors that did well at the one-quarter-ahead horizon (LASSO and Ridge) also yield precise tail forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' One remarkable difference from short-term forecasts is that higher order median forecasts appear to be much more precise than the ones obtained from the ABG benchmark specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This brief discussion gives rise to a simple recommendation for practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' If interest is on producing precise tail forecasts (irrespective of the forecast horizon) it pays off to use large QRs coupled with either a LASSO or Ridge-type prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since the Ridge prior is much simpler (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', it only features a single hyperparameter) and the empirical performance is very similar to the LASSO, our focus from now on will be on comparing the Ridge-based QR with a range of non-linear specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 Allowing for nonlinearities in large scale QRs In the previous sub-section we have shown that using big QRs leads to tail forecasts that are superior to the ones of the benchmark ABG specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Conditional on the quantile, these models are linear in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, recent literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2022b) suggests that nonlinearities become more important in the tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, we now address this question within our approximate framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Table 2 shows relative CRPSs for the different nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As opposed to Table 1, all results are now benchmarked against the QR with the Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This allows us to directly measure the performance gains from introducing nonlinearities relative to setting gq(xt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Notice that the absence of gray shaded cells in the table indicates that the DM test does not point towards significant differences in forecast accuracy between the linear and the different nonlinear QRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Despite this, a few interesting insights emerge from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' First, many numbers in the table are close to unity and differences are not statistically significant from the best performing linear QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 This indicates that once we include many predictors, additionally controlling for nonlinearities of different forms only yields small positive (and sometimes negative) gains in terms of tail forecasting accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Second, the first finding strongly depends on the approxi- mation techniques chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Among all three specifications, using GPs is superior to using either polynomials or B-Splines to approximate the unknown function gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Second, and focusing on 7For the GP-QR specifications with ridge prior we obtain p-values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 using the test in Diebold and Mariano (1995) with adjustments proposed by Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 18 Table 2: CRPSs for nonlinear models One-quarter-ahead Four-quarters-ahead Model CRPS CRPS-tails CRPS-left CRPS CRPS-tails CRPS-left Polynomials HS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 RIDGE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 NG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 LASSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99 DL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='29 B-Splines HS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 RIDGE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 NG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='07 LASSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99 DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 Gaussian Processes HS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='10 RIDGE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 NG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='08 LASSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 DL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='29 Results are shown relative to the linear QR with a Ridge prior and are based on the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' GP-QR specifications, the specific prior chosen matters appreciably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Whereas the results for the conditionally linear models clearly suggest that the LASSO and Ridge priors are producing the most precise density forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The results for the nonlinear models tell a slightly different story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We observe that the Ridge does well again but, for one-quarter-ahead tail forecasts, is outper- formed by the HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The LASSO, by contrast, is the weakest specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since the LASSO is known to overshrink significant signals (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Griffin and Brown, 2010), it could be that it misses out important information arising from the GP-based basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Third, and finally, if we consider four-quarters-ahead predictions the QR coupled with a GP and a Ridge prior becomes the single best performing model again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To again gain a better understanding on which quantiles of the predictive distribution drive the CRPSs, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 4 and 5 are similar to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 2 and 3 and show the QSs for different quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These are normalized to the linear QR with ridge prior so that numbers smaller than one indicate that nonlinearities improve predictive accuracy for a given quantile and numbers exceeding one imply that nonlinearities decrease forecasting accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In general, both figures tell a consistent story: nonlinearities help in the right tail across 19 both forecast horizons, for all three nonlinear specifications, and for most priors considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The only exception to this pattern are four-quarters-ahead right tail forecasts of GDP growth when B-Splines are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When there are gains, they are often sizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For instance, in the case of the QR-GP model we observe accuracy improvements up to 25 percent relative to the linear QR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 Polynomials 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 B-Splines 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 Gaussian process HS RIDGE NG LASSO DL One-quarter-ahead quantile scores Figure 4: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out period and normalized to the QR with a Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we focus on the left tail, accuracy premia often turn negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In some cases (such as for GP models with Ridge, NG and HS priors) there are accuracy gains for predicting downside risks but these gains are only rather small (reaching five percent in the case of the QR-GP regression with a Ridge prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 Heterogeneity of forecast accuracy over time Up to this point, our analysis focused on averages over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In the next step we will focus on how forecasting performance changes over the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To shed light on the importance of nonlinearities over time, we again compare the different nonlinear specifications to the linear QR regression with a Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 5 and 6 show the cumulative CRPSs relative to the linear benchmark QR for one-quarter and four-quarters-ahead forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 20 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 Polynomials 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 B-Splines 5% 10% 25% 50% 75% 90% 95% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 Gaussian process HS RIDGE NG LASSO DL One-year-ahead quantile scores Figure 5: Four-quarters-ahead quantile scores for different values of q, averaged over the hold- out period and normalized to the QR with a Ridge prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' We start by focusing on the one-quarter-ahead forecasts first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For this specification, the density accuracy performance is heterogenous over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In the first part of the sample, models using either polynomials or Gaussian processes coupled with a DL prior yield CRPSs that are superior to the linear benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, these accuracy gains vanish during the GFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we put more weight on tail forecasting accuracy (and consider GP-QRs), the gains disappear as early as during the 2001 recession that followed the 9/11 terrorist attacks and the burst of the dot-com bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In the pandemic, we observe a sharp increase in predictive accuracy for several priors (most notably the Ridge and NG priors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This pattern is more pronounced for the weighted variants of the CRPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Considering the other nonlinear model specifications gives rise to similar insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Spline-based approximations to gq generally perform poorly up until the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' During the pandemic, even this specification improves sharply against the linear benchmark specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This pattern is particularly pronounced for the GP-QRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Considering the performance of the models and priors that did well on average (GP- QRs with Ridge and the HS) reveals that most of these gains are actually driven by superior performance during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 21 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 CRPS Polynomials 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 B-Splines 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 Gaussian process 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 CRPS-tails 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='15 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 CRPS-left 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 HS RIDGE NG LASSO DL CRPS over time One-Quarter-ahead Figure 6: Cumulative one-quarter-ahead CRPS relative to the linear QR with the Ridge prior over the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 CRPS Polynomials 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='9 2 B-Splines 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 3 Gaussian process 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 CRPS-tails 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 CRPS-left 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 HS RIDGE NG LASSO DL CRPS over time One-Year-ahead Figure 7: Cumulative four-quarter-ahead CRPS relative to linear QR with a Ridge prior over the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 22 Turning to four-quarters-ahead forecasts provides little new insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Models using the DL prior do not excel in the first part of the hold-out period and are generally outperformed by the linear QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, accuracy improvements during the GFC and the pandemic are quite pronounced for splines and the GP-QRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To sum up this discussion, our results indicate that forecast performance is heterogenous over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Different models such as the Polynomial-QR and the GP-QR with a DL prior out- perform in the early part of the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This performance premium vanishes during the first two recessions observed in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' By contrast, other models such as QR-GP with either the NG or the LASSO do not gain much in tranquil periods but excel during recessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 Properties and determinants of the quantile forecasts The previous sub-sections have outlined that QRs and QRs with nonlinear components perform well in terms of tail forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this sub-section, our goal is to investigate which variables determine the quantile forecasts and in what respect successful shrinkage priors differ from their less successful counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The presence of nonlinearities complicates our investigation since it is not clear on how to measure the effect of xt on a given quantile of yt in the presence of nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' As a simple solution, we follow Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2022a) and approximate the nonlinear, quantile-specific model using a linear posterior summary (see Woody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Specifically, we estimate the following regression model: Qq,t = x′ t ˆαq + ˆεt, ˆεt ∼ N(0, σ2 ˆεq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' On the linearized coefficients we use a Horseshoe prior and on the error variances an inverse Gamma prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To achieve interpretability and decouple shrinkage and selection (see Hahn and Carvalho, 2015), we then apply the SAVS estimator proposed in Ray and Bhattacharya (2018) to the posterior mean of ˆαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 This will yield a sparse variant of ˆαq, that is easy to interpret and can be understood as the best linear approximation to the corresponding quantile forecast arising from the nonlinear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For brevity, we focus on one-step-ahead forecasts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' xt includes a single lag of all variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Results for four-quarters-ahead are included in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 8Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2021) and Hauzenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2021) apply SAVS to multivariate time series models and show that it works well for forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 23 HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='1 0 Linear 10% M1REAL HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 50% PCESVx PAYEMS M1REAL PCESVx M1REAL PCESVx M1REAL HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 90% PCESVx M1REAL M1REAL PCESVx USSERV M1REAL USSERV M1REAL CLAIMS PCESVx USSERV M1REAL HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 Polynomials TCU CUMFNS PAYEMS CUMFNS M1REAL CUMFNS PAYEMS PRFIx HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 UNRATE M1REAL PCESVx PRFIx UEMP5T CLAIMS HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 PCESVx M1REAL TLBCBB TLBBBD PCESVx HWIx M1REAL TLBCBB TLBBBD UNRATE M1REAL TLBBBD PCECC9 UNRATE CLAIMS TNWMVB HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 B-Splines CUMFNS USPRIV CUMFNS USCONS M1REAL CUMFNS USPRIV SRVPRD FPIx M1REAL UMCSEN HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 UEMP15 UEMP15 UEMP15 M1REAL M1REAL HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 UEMP15 HWIx HWIURA TLBCBB TLBBBD UEMP15 HWIx HWIURA TLBCBB TLBBBD UEMP15 HWIx HWIURA TLBCBB TLBBBD USSERV M1REAL PCECC9 UEMP15 M1REAL TLBCBB TLBBBD HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 Gaussian process PRFIx PRFIx M1REAL SRVPRD PRFIx LNS120 HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 M1REAL M1REAL M1REAL M1REAL PCESVx M1REAL HS RIDGE NG LASSO DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1 PCESVx USSERV UEMP15 UEMP15 M1REAL PCESVx UEMP15 USSERV M1REAL CLAIMS PCESVx M1REAL Figure 8: One-quarter-ahead linearized posterior summaries across quantiles Figure 8 shows the results of this exercise across nonlinear specifications and priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Start- ing with the left tail forecasts and linear models suggests that most quantile forecasts are not related to elements in xt in a robust manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' There are only two exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first relates to the NG prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, real money growth (M1real) survives the sparsification step and the relationship indicates that declines in money growth imply an increase in tail risks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' a decline in GDP growth in the ten percent quantile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The other exception relates to the DL prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, employment growth in education and health services (USEHS) remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' If we focus on nonlinear models other variables appear to be correlated with forecasts of tail risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Among the different priors, we find some variables which show up repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Among these are all nonfarm employees (PAYEMS), money growth, capacity utilization in manufacturing (CUMFNS) and private fixed investment (both residential and non-residential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Most of these variables are forward looking in nature and thus consistent with our intuition that economic agents form expectations about the state of the economy in the future and thus change their investment decisions accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Notice that the relationship between private fixed investment is particularly pronounced for GP-QRs under the HS, NG and the DL prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Another pattern worth mentioning is that the LASSO-based forecasts are generated from sparse models across both linear and nonlinear specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Once we focus on the center of the distribution we find that forecasts from linear models 24 are driven by one or two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Most prominently, specifications that do well in terms of point forecasts (such as the Ridge and Lasso) yield point forecasts that display a strong relationship with (lagged) money growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In case we adopt a nonlinear specification, some differences arise across specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' For polynomials, median forecasts under all priors except the DL are related to very few predictors, with money growth and short-run unemployment showing up for the NG and LASSO models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The DL prior implies a more dense model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This could be a possible reason for the rather weak performance of this specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we turn to spline- based models we again find a similar pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Money growth shows up in the case of the NG and LASSO and short-run unemployment predicts median output growth if we adopt a HS, Ridge or NG prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Models that capture nonlinearities through GPs, our best performing nonlinear specifications, give rise to a very consistent pattern across priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In all cases, lagged money growth appears to be a robust predictor of GDP growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' And it impacts GDP growth forecasts negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Finally, when our focus is on right-tail forecasts, all models become much more dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Variables that have been showing up in the case of left-tail and point forecasts again show up (most notably money growth and short-term unemployment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Additional variables such as initial unemployment claims or prices remain in the sparse model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' But there is no clear pattern across models except for the fact that money growth also remains in the set of robust predictors even if much shrinkage is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The analysis based on linearized coefficients provides information on which variables are predictive for output growth forecasts across quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' However, the analysis in Sub-sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='3 suggests that differences in forecast performance are driven by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' To understand which properties of a given prior exert a positive effect on predictive accuracy, we now focus on the shrinkage hyperparameters of the different priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Comparing the amount of shrinkage introduced through the different priors is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Here, our measure of choice is based on using the re-scaled log determinant of the prior covariance matrices as a measure of overall shrinkage for each respective prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since all prior covariance matrices are diagonal this simply amounts to summing over the log of the diagonal elements of B0q and then normalizing through by the number of diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This constitutes a rough measure of overall shrinkage and we can compute it for each quarter in the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Again, we will focus on shrinkage introduced in one-quarter-ahead predictive regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The four-quarters-ahead results are qualitatively similar and included in the Online Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 25 Log-determinants of the prior covariance matrices over the hold-out period are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The figure includes (if applicable) solid lines which refer to the amount of shrinkage introduced on the linear coefficients and dashed lines which refer to the log-determinants of the prior covariances that relate to the shrinkage factors on the basis coefficients of the different nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 10% Linear 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 5 Polynomials 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 10 9 8 7 6 5 4 3 B-Splines 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 5 Gaussian process 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 50% 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9 8 7 6 5 4 3 2 1 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9 8 7 6 5 4 3 2 1 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 10 9 8 7 6 5 4 3 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='2 90% 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 10 9 8 7 6 5 4 3 1992 1995 1997 2000 2002 2005 2007 2010 2012 2015 2017 2020 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content='5 6 HS RIDGE NG LASSO DL HS-nonlinear RIDGE-nonlinear NG-nonlinear LASSO-nonlinear DL-nonlinear Figure 9: Overall shrinkage in one-step-ahead predictive QRs From this figure, a few interesting insights emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' First, the different priors introduce different degrees of shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Overall, two priors stand out in terms of the amount of shrinkage they introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The first one is the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is rather surprising given the fact that this prior performs worst in the forecasting horse race but also leads to posterior summaries which feature several non-zero coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our conjecture is that this prior forces the vast majority of coefficients to effectively zero but several coefficients remain sizable and the corresponding set of variables is still too large and overfitting issues arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The prior that introduces the largest amount of shrinkage is the LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, almost all coefficients are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These observations are corroborated by boxplots, included in the Online Appendix (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 3 to 6 in the Online Appendix), which show the scaling parameters over three sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Our results imply that models which feature a large number of shrunk coefficients provide better forecasts than models which feature many coefficients that are effectively zero and some coefficients that are non-zero and sizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is consistent with findings in Giannone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2021) who provide 26 empirical evidence that macroeconomic data is rather dense as opposed to sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Notice that the fact that dense models produce accurate tail forecasts is not inconsistent with our analysis based on linearized posterior summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This is because the linearized model under a shrinkage and sparsification approach strikes a balance between achieving a good model fit while keeping the model as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, if the covariates in the panel co-move, shrinkage and sparsification techniques will select one of these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Second, in almost all cases the amount of shrinkage introduced on the nonlinear part of the different models is much larger than the degree of shrinkage on linear coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' This holds for most priors, nonlinear methods and over all time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' One exception is the Spline-QR specification with a DL prior and when the right tail is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Interestingly, this specific combination of much stronger shrinkage on the linear part of the model and less shrinkage on the nonlinear part leads to good forecasts in the right tail (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Third, and finally, there is (with some notable exceptions) relatively little time-variation in the amount of shrinkage over the hold-out period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The only exception are the GP-QRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In this case, the amount of shrinkage decreases appreciably from 2013 onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 4 Concluding remarks In this paper, we have shown that combining QRs with nonlinear specifications and large datasets leads to precise quantile forecasts of GDP growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Since the resulting models are high di- mensional, we consider several popular shrinkage priors to regularize estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' MCMC-based estimation of these huge dimensional models is slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' Hence, we speed up computation by us- ing VB approximation methods that approximate the joint posterior distribution using simpler approximating densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' The empirical results indicate that our methods work remarkably well when the CRPS is taken under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' When we put more weight on the tail forecasting performance, we find that most of the overall gains are driven by a strong performance in both the left and right tail while the performance in the center of the distribution is close to the predictive accuracy of the simple quantile regression proposed in Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' These results, however, differ across priors and nonlinear specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' In principle, it can be said that models featuring simple shrinkage priors, such as the LASSO or Ridge, in combination with GPs to capture nonlinearities of arbitrary form yield the most precise forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 27 References Adams, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Adrian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', Boyarchenko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=', and Giannone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFRT4oBgHgl3EQfnDfl/content/2301.13604v1.pdf'} +page_content=' 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+2 Dept. of Mathematics +Simon Fraser University, Burnaby, B.C. Canada V5A 1S6 +3 Dept. of Computer Science and Engineering +University of Connecticut, Storrs, CT 06269, USA +Abstract +The reconstruction of phylogenetic networks is an important but challenging problem in phy- +logenetics and genome evolution, as the space of phylogenetic networks is vast and cannot be +sampled well. One approach to the problem is to solve the minimum phylogenetic network prob- +lem, in which phylogenetic trees are first inferred, then the smallest phylogenetic network that +displays all the trees is computed. The approach takes advantage of the fact that the theory of +phylogenetic trees is mature and there are excellent tools available for inferring phylogenetic trees +from a large number of biomolecular sequences. A tree-child network is a phylogenetic network +satisfying the condition that every non-leaf node has at least one child that is of indegree one. +Here, we develop a new method that infers the minimum tree-child network by aligning lineage +taxon strings in the phylogenetic trees. This algorithmic innovation enables us to get around the +limitations of the existing programs for phylogenetic network inference. Our new program, named +ALTS, is fast enough to infer a tree-child network with a large number of reticulations for a set of +up to 50 phylogenetic trees with 50 taxa that have only trivial common clusters in about a quarter +of an hour on average. +arXiv:2301.00992v1 [q-bio.PE] 3 Jan 2023 + +1 +Introduction +In this study, phylogenetic networks are rooted, +directed acyclic graphs in which the leaves are +labeled with taxa, the non-leaf indegree-1 nodes +represent speciation events and the nodes with +multiple incoming edges represent reticulation +events. The non-leaf indegree-1 nodes are called +tree nodes and the other non-leaf nodes are called +reticulate nodes. Phylogenetic trees are phyloge- +netic networks with no reticulate nodes. +Now that a variety of genomic projects have +been completed, reticulate evolutionary events +(e.g. horizontal gene transfer, introgression and +hybridization) have been demonstrated to play +important roles in genome evolution [9, 12, 19, +21, 26]. Although phylogenetic networks are ap- +pealing for modeling reticulate events [18], it is +extremely challenging to apply phylogenetic net- +works in the study of genome evolution. +One +reason for this is that a computer program has +yet to be made available for analyzing data as +large as what current research is interested in +[23, 31], although recently, Bayesian methods +have been used to reconstruct reassortment net- +works, which describe patterns of ancestry in +which lineages may have different parts of their +genomes inherited from distinct parents [24, 25]. +Here, we focus on reconstructing phylogenetic +networks from (phylogenetic) trees by comput- +ing the smallest phylogenetic network displaying +a given set of multiple trees [2, 8, 30, 28, 29]. +In this approach, trees are first inferred from +biomolecular sequences and then used to recon- +struct a phylogenetic network with the smallest +hybridization number (HN) that displays all the +trees (see [8]), where the HN is defined as the +sum over all the reticulate nodes of the differ- +ence between the indegree and outdegree of each +reticulate node. This approach takes advantage +of the fact that the theory of phylogenetic trees +is mature and there are excellent tools available +for inferring trees from a large number of se- +quences. Here, we focused on the parsimonious +inference of phylogenetic networks from multiple +trees, which computes a phylogenetic network +with the minimum HN that displays all the trees. +This problem is NP-hard even for the special case +when there are only two input trees [4]. +For +the two-tree case, the fastest programs include +MCTS-CHN [32] and HYBRIDIZATION NUM- +BER [29]. For the general case where there are +multiple input trees, HYBROSCALE [1] and its +predcessor [2], PRIN [30] and PRINs [22], have +been developed. +All these methods are based +on the process of searching through inserting +reticulate edges or other editing operations in +the space of phylogenetic networks, by reducing +the problem to the maximum acyclic agreement +forests of the input trees or both. Unfortunately, +none of them can be used for inferring a network +from a so-called irreducible set of 30 trees with +30 taxa in which the trees do not contain any +non-trivial common clusters. +Since the whole network space is vast and +cannot be fully sampled, attention has been +switched to the inference of the tree-child net- +works, in which every non-leaf node has at least +one child that is not reticulate [28], or, recently, +a member of a subclass of the tree-child network +[26]. Tree-child network [6] is a superclass of phy- +logenetic trees with a completeness property that +for any set of phylogenetic trees, there is always +a tree-child network (whose reticulate nodes can +be of indegree 2 or more) that displays all the +trees [20]. Other desired properties of tree-child +networks include the fact that all the tree-child +networks are efficiently enumerated [33]. Most +importantly, the validation results in [28] and our +results (reported in Section 4) suggest that the +HN of a tree-child network solution is close to +the optimal HN of a phylogenetic network that +displays the trees. +The program for inferring tree-child networks +that appears in [28] is based on a fixed-parameter +algorithm. The time-complexity of the algorithm +is O((8r)rpoly(k, n)), where k and n are, respec- +tively, the number of taxa and the input trees; r +is the HN of the network solution. +The new program we introduce here, ALTS, +1 + +takes a different approach that reduces the in- +ference problem to aligning the lineage taxon +strings of all the input trees. +Algorithmic in- +novations in ALTS enable us to get around some +of the limitations associated with parsimonious +inference by efficiently sampling the orderings of +the taxa and progressively computing the short- +est common supersequence (SCS) of the lineage +taxon strings derived for each taxon in all the +input trees. ALTS is fast enough to infer a par- +simonious tree-child network for a set of 50 trees +on 50 taxa in a quarter of an hour on average. +We also added a feature of inferring a weighted +tree-child network if the input trees are weighted. +2 +Concepts and notation +A directed graph G consists of a set V of nodes +and a set E of directed edges that are ordered +pairs of distinct nodes. Let e = (u, v) ∈ E. We +call e an outgoing edge of u and an incoming +edge of v. For a node v ∈ V , its outdegree and +indegree are defined as the number of outgoing +and incoming edges of v, respectively. +For a graph, subdividing an edge (u, v) involves +replacing it with a directed path from u to v +that passes one or more new nodes. Conversely, +an edge contraction at a node v of indegree one +and outdegree one is to remove v and replace +the path u → v → w with an edge (u, w), where +(u, v) and (v, w) are the unique incoming and +outgoing edge of v, respectively. +2.1 +Phylogenetic networks +A phylogenetic network on a set X of taxa is +a rooted, directed acyclic graph in which (i) all +the edges are oriented away from the root, which +is of indegree 0 and outdegree 1; (ii) the nodes +of indegree 1 and outdegree 0, called leaves, are +uniquely labeled with the taxa; and (iii) all the +non-root and non-leaf nodes are either tree nodes +that are of indegree 1 and outdegree 2 or reticu- +late nodes that are of indegree more than 1 and +outdegree 1. Reticulate nodes represent evolu- +tionary reticulation events. A phylogenetic net- +work is said to be binary if the indegree of every +reticulate node is exactly 2 (Figure 1). +Let N be a phylogenetic network. +We use +V(N) and E(N) to denote the node and edge set +of N, respectively. We also use R(N) to denote +the set of reticulate nodes, and use T (N) to de- +note the set of all non-reticulate nodes, including +the root, tree nodes and leaves. Let u, v ∈ V(N). +The node v is a child of u if (u, v) is an edge; v is +a descendant of u if there is a directed path from +u to v. If v is a descendant of u, v is said to be +below u. +A phylogenetic network N is a tree-child net- +work if every non-leaf node has a child that is +not reticulate. +Equivalently, N is a tree-child +network if and only if for every non-leaf node, +there is a path from that node to some leaf that +passes only tree nodes. Figure 1 presents a bi- +nary tree-child network (left) and two non-tree- +child networks. +Consider a tree-child network N with k retic- +ulate nodes. Let the root be r0 and let the retic- +ulate nodes be r1, r2, · · · , rk. After the removal +of the incoming edges of every ri, N becomes +the union of k + 1 subtrees, which are rooted +at r0, r1, · · · , rk, respectively, and have network +leaves as their leaves (see Figure 1). These sub- +trees are called the tree-node components of N. +Tree-node decomposition is a useful technique in +the study of phylogenetic networks [11, 13, 14]. +2.2 +Phylogenetic trees +A phylogenetic tree on X is a phylogenetic net- +work with no reticulate nodes. In fact, a tree is +a tree-child network. Let T be a phylogenetic +tree on X and u ∈ V (T). The node cluster of +u, denoted as C(u), is the subset of taxa that +are represented by the leaves below u. Clearly, +C(u) ∩ C(v) ∈ {C(u), C(v), ∅} for any two nodes +u and v. The node u and its descendants induce +a unique subtree on C(u). We use Tu or T(C(u)) +to denote the subtree. +Let S be a set of binary phylogenetic trees on +2 + +d +b +a +c +c +b +a +c +b +a +x +4 +2 +1 +3 +y +x +4 +2 +1 +3 +y +Edge insertion +Figure 1: A binary tree-child network (left) in which +there are four tree-node components (shaded grey) +and two non-tree-child networks (middle) and (right). +In the middle network, the child of the top reticulate +node is also reticulate. In the right network, the chil- +dren of a tree node in the middle are both reticulate. +X. A common cluster of S is a subset of X that is +a node cluster in every tree of S. Obviously, each +single taxon is common cluster of S, and so is X. +Any other common clusters of S are called non- +trivial common clusters. S is a reducible tree set +if there is a non-trivial common cluster for S, and +it is irreducible otherwise. A non-trivial common +cluster C of S is maximal if any subset C′ such +that C ⊂ C′ ⊂ X is not a common cluster of S. +Clearly, for any two maximal common cluster C1 +and C2 of S, C1 ∩ C2 = ∅; and any non-trivial +common cluster X′ of S must be contained in a +unique maximal cluster of S if X′ is not maximal. +2.3 +Tree display and network infer- +ence problems +Let T be a binary phylogenetic tree on X and let +N be a tree-child network with k reticulate nodes +on X. T is displayed by N if T can be obtained +from N by applying edge contraction from N +after the removal of all but one incoming edge +for each reticulation node (Figure 2). For any +set of binary phylogenetic trees over X, there is +always a tree-child network that displays all the +trees [20]. However, such a solution network may +not be binary. +Let P by a phylogenetic network. Its +c +d +a +b +c +d +a +b +c +d +b +a +A.) B.) C.) +Figure 2: (A.) A tree-child network with two retic- +ulate nodes on the taxa (a to d). (B.) A subtree of +the network in (A) that can be obtained by the re- +moval of the dashed incoming edges of the reticulate +nodes. (C.) A tree displayed in the network in (A), +which was obtained from the subtree in (B) by edge +contraction. +reticulate number is defined as the number of +reticulate nodes. Its HN, denoted as H(P), is +defined as the sum over all the reticulate nodes +of the difference between the indegree and the +outdegree of that reticulate node. If P is binary, +H(P) is equal to the reticulate number. Here, +we studied the following minimum tree-child +network inference problem: +Input: A set of phylogenetic trees on X. +Output: A parsimonious tree-child network P +on X (with the smallest H(P)) that displays +all input trees. +2.4 +The SCS problem +Let s and t be two sequences in an alphabet. The +sequence s is said to be a supersequence of t if t +can be obtained from s by the deletion of one or +more letters. The SCS problem is, given a set of +sequences, to find the shortest sequence that is +a supersequence of every given sequence. +The SCS problem can be solved in a quadratic +time for two sequences. However, it is NP-hard +in general. +3 + +3 +The methods +In this section, we assume that the input trees +are binary phylogenetic trees. +3.1 +The Inference Algorithm +Let X be a taxon set and let π = π1π2 · · · πn, +representing a (total) ordering of X by which +πi is ‘less than’ πi+1 for each i < n. +For any +non-empty subset X′ of X, we use minπ(X′) +and maxπ(X′) to denote the minimum and +maximum taxon of X′ with respect to (w.r.t.) +π, respectively. Consider a tree T on X. Since +the root of T is of outdegree 1, T has n non-leaf +nodes, called internal nodes. +We label the n +internal nodes one-to-one with X w.r.t. π using +the following algorithm: +Labeling +Input A tree T on X and an ordering π of X +1. Label the degree-1 root of T by minπ(X). +2. Label each internal node u with two +children v and w with +maxπ{minπ(C(v)), minπ(C(w))}, where +C(v) consists of all taxa below v in T. +For instance, let X = {a, b, c, d, e} and π be an +ordering of X such that b < c < a < d < e, +Figure 3B gives two trees in which their internal +nodes are labeled w.r.t. π by using Labeling. +For each taxon τ, there is a unique internal +node w that is labeled with τ, which is an an- +cestor of the leaf τ. The sequence of the taxon +labels appearing in the path from w to the leaf +τ exclusively is called the lineage taxon string +(LTS) of τ. +The LTSs computed in the trees +in Figure 3B are listed in Figure 3C. It is not +hard to see that a tree can be recovered by using +the LTSs derived from the given ordering of X +in the tree. In addition, we have the following +proposition, the proof of which appears in the +Supplementary Document. +Proposition 1 Let π be an ordering of X, +|X| = n. For a phylogenetic tree T on X, the +LTS sπ(i) of each taxon πi obtained by applying +the Labeling algorithm has the following prop- +erties: +(i) sπ(1) is always not empty; +(ii) sπ(n) is always empty; +(iii) every taxon πk (k > 1) appears in the LTS +2 +1 +4 +3 +5 +2 +5 +1 +3 +4 +1 +2 +3 +4 +5 +× +× +× +× +Ordered Taxa: b < c < a < d < e +b +a +d +c +e +b +c +a +e +d +b +e +a +c +d +b +c +e +d +a +The lineage taxon strings +b +c, a +c, e +Taxon Left tree Right tree +c +e, d +d, a +a +empty empty +d +empty empty +e +empty empty +A.) +B.) +D.) + E.) +b +c +a +d +e +c +e +a +d +e +a +b +c +a +d +e +C.) +Figure 3: The construction of a tree-child network +that displays two phylogenetic trees. (A) An order- +ing on {a, b, c, d, e}. (B) Two trees, where the inter- +nal nodes are labeled using the Labeling algorithm. +(C) The LTSs of the taxa obtained from the label- +ing in Panel B. (D) The rooted directed graph con- +structed from the shortest common supersequences +(SCS) of the LTSs of the taxa (in Panel C) using +the Tree-child Network Reconstruction algo- +rithm. Here, the SCS is [c, e, a] for [c, a] and [c, e], and +is [e, d, a] for [e, d] and [d, a]. (E) The tree-child net- +work obtained after contraction of the degree-2 nodes. +4 + +of πj for a unique j < k; +(iv) π1 does not appear in any LTS. +Let Ti (1 ≤ i ≤ k) be k trees on X and +let π = π1π2 · · · πn, an ordering of X, where +n = |X|. Let αij be the LTS of πj in Ti for each +j from 1 to n − 1. Assume that, for each j, βj +is a common supersequence of α1j, α2j, · · · , αkj +such that βj does not contain any symbol not +in X. +We can construct a tree-child network +Nπ(β1, β2, · · · , βn−1) on X using the following +algorithm. +Tree-Child Network Construction +1. (Vertical edges) For each βi, define a +path Pi with |βi| + 2 nodes: +hi, vi1, vi2, · · · , vi|βi|, ℓπi, +where βn is the empty sequence. +2. (Left–right edges) Arrange the n paths +from left to right as P1, P2, · · · , Pn. If +the m-th symbol of βi is πj, we add an +edge (vim, hj) for each i and each m. +3. Contract each hi if it is of indegree 1. +Tree-Child +Network +Construction +is +illustrated in Figure 3D, where the SCSs are +[c, e, a] and [e, d, a] for π1 = b and π2 = c, and +the empty sequence for π3 = a and π4 = d. +Clearly, the network output from the algorithm +is a tree-child network. +Proposition 2 Let Ti (1 ≤ i ≤ k) be k trees +on X such that |X| = n and let π be an order- +ing of X. Let αij be the LTS of πj in Ti with +respect to π, 1 ≤ j ≤ n − 1. If βj is a com- +mon supersequence of α1j, α2j, · · · , αkj on X for +each j from 1 to n − 1, the Tree-Child Net- +work Construction algorithm outputs a tree- +child network that displays all k trees. +Conversely, we assume that P is a tree-child +network with the smallest HN, H(P), compared +with those that displays all k trees Ti. The prop- +erty that P has the smallest HN implies that, for +each i, any display of Ti in P must use one in- +coming edge for each reticulate node of P. +A.) +B.) + C.) +D.) +b +a +d +c +b +c +a +d +b +d +a +c +d +a +b +d +a +a +b +c +d +a +b +c +a +b +b +c +The lineage taxon strings w.r.t. +the ordering: c < d < b < a +Taxon Left tree Middle tree Right tree +c +b, d +d, a +d, b +d +empty + b + a +b +a +empty +empty +a +empty empty +empty +Figure 4: Mapping a tree-child network on n taxa +that displays multiple trees to (n−1) common super- +sequences of the LTSs of the taxa in the trees with +respect to a selected ordering. +(A) Three trees on +taxa {a, b, c, d}. (B) A tree-child network with the +smallest HN (4) that displays all three trees. (C) De- +termine an ordering: c < d < b < a, label all internal +tree nodes, and derive the LTS for each taxon: [b, d, +a, b] (Taxon c), [a, b] (Taxon d), [a] (Taxon b), empty +(Taxon a). (D). The LTS of the taxa in the trees. For +each taxon, the LTS obtained in the network is the +SCS of the LTSs obtained in the trees. +Let P contain t reticulate nodes ri (1 ≤ +i ≤ t). +N has t + 1 tree-node components +C0, C1, C2, ..., Ct such that C0 is rooted at the +root r0 of P, and Ci is rooted at ri for i ≥ 1. +Since P is acyclic, its nodes can be topologically +5 + +sorted into a list such that u appears before v for +every edge (u, v) of P. By using such a topolog- +ical ordering of P, we can order all the taxa into +π1, π2, · · · , πn such that (i) all the taxa in each +tree-node component appear consecutively, and +(ii) if the reticulate node ri has a parent in Cj, +the taxa of Cj appear before the taxa of Ci in +the list. This is because there is a directed path +from rj to every node of Ci. For instance, for +the tree-child network in Figure 4B, C0 contains +Taxa c and d; the tree component rooted below +the left reticulate node contains Taxon b; the +tree component below the right reticulate node +contains Taxon a. Therefore, we can order the +taxa as either c < d < b < a or d < c < b < a, +where b must appear before a. +For an ordering π = π1π2 · · · πn satisfying the +property given in the last paragraph, we label the +tree nodes of P using the following algorithm. +• Label the network root with the smallest +taxon in C0 (i.e. π1). +Label each parent +of the reticulate node ri with the smallest +taxon in Ci for every i > 1. +• Let u be a tree node that is not a parent of +any reticulate node. In this case, u has two +children x and y in the same tree-node com- +ponent C. +We label u with maxπ(ax, ay), +where ax and ay are the smallest taxon be- +low x and y in C, respectively. (For exam- +ple, the tree-node component C0 contains +only one such tree node and this node is la- +beled with d in Figure 4C.) +As in the case of trees, we can obtain a LTS for +each taxon. +For the smallest taxon τ of each +tree-node component Ci, its LTS is composed of +the taxon labels of the tree nodes in the unique +path from ri to Leaf τ. For the other taxa τ of +Ci, there is a unique tree node w that is labeled +with τ. The LTS of τ is composed of the taxon +labels of the tree nodes (excluding w) in the path +from w to Leaf τ. (For example, in Figure 4C, +C0 contains Taxa c and d. The LTS for c and d +are [b, d, a, b] and [a, b], respectively. +Proposition 3 Let Ti (1 ≤ i ≤ k) be k trees on +X and let P be a tree-child network on X with +the smallest HN, compared with those that dis- +play all Ti. For any ordering π of X obtained +above and for each taxon τ, if we label the tree +nodes of P as described above, the LTS Sτ ob- +tained for τ is the SCS of the LTS obtained for +τ in the trees Ti. Moreover, applying the Tree- +child Construction algorithm to the obtained +supersequences Sτ gives the same network as P. +The proof of Proposition 3 appears in Sup- +plementary document. +By Propositions 2 and +3, we obtain the following exact algorithm for +inferring the minimum tree-child network that +displays the trees. +Algorithm A +Input: Trees T1, T2, · · · , Tk on X, |X| = n. +0. Define M = ∞ and n string variables +S1, S2, · · · , Sn−1; +1. For each ordering π1π2 · · · πn of X: +1.1. Call the Labeling algorithm to +label the internal nodes in each Ti; +1.2. For each taxon πj, compute its +LTS sij in each Ti; +1.3. Compute the SCS sj of +s1j, s2j, · · · , skj for each j < n; +1.4. If M > �n−1 +j=1 |sj|, update M to +the length sum; update Sj to sj +for each j; +2. Call the +Tree-Child Network Construction +algorithm to compute a tree-child network +P from the strings S1, S2, · · · , Sn−1. +Step 1.1 and Step 1.2 of Algorithm A take a +linear time of O(n). Note that the SCS problem +is a special case of the multiple sequence align- +ment problem. Since the the total length of the +(n − 1) LTSs computed in Step 1.2 is n − 1 for +each Ti, Step 1.3 takes a time of O((n − 1)k) at +most. Step 2 takes a quadratic time of O(n2). +Therefore, the worst-case time complexity of +6 + +Algorithm A is O +� +n!(n − 1)k� +. +3.2 +A Scalable Version +Since there are n! possible orderings of n taxa, +Algorithm A is not fast enough for a set of +multiple trees with 15 taxa if all the trees do not +have any common clusters other than the single- +ton cluster and the whole taxa. Another obstacle +to scalability is computing the SCS for the LTS +of each taxon. We achieved high scalability by +using an ordering sampling and a progressive ap- +proach for the SCS problem. +First, the ordering sampling starts with an ar- +bitrary ordering of the taxa and finishes in ⌊n/2⌋ +iterative steps. Assume that Πm is the set of or- +derings obtained in the m-th step (m ≥ 1), which +contains at most K orderings (K is a predefined +parameter to bound the running time). In the +(m+ 1) step, for each ordering π = π1π2 · · · πn ∈ +Πm, we generate (n−2m+1)(n−2m) orderings +by interchanging π2m−1 with πi and interchang- +ing π2m with πj for every possible i and j such +that i ̸= j, i > 2m and j > 2m. For each new or- +dering π′ = π′ +1π′ +2 · · · π′ +n, we compute a SCS si of +the LTSs of Taxon π′ +i in the input trees for each +i ≤ 2m. We compute Πm+1 by sampling at most +K orderings that have the smallest length sum +� +1≤i≤2m |si| from all the generated orderings of +the taxa. +Second, different progressive approaches can +be used to compute a short common superse- +quence for LTSs in each sampling step [10]. We +use the following approach, which had good per- +formance for our purposes according to our sim- +ulation test. +A common supersequence of n se- +quences is computed in n − 1 iterative +steps. In each step, a pair of sequences +si, sj for which the SCS of si and sj, +SCS(si, sj), has the minimum length, +over all possible pairs of sequences, is +selected and replaced with SCS(si, sj). +After the sampling process finishes, we obtain +a set Π⌊n/2⌋ of good ordering; for each ordering, +we obtain a short common supersequence of the +LTS of a taxon, which might not the shortest +one for each taxon. To further improve the tree- +child network solution, we also use the dynamic +programming algorithm to recalculate the SCS +for the LTS of each taxon w.r.t. each obtained +ordering, subject to the 1G memory usage limit. +We then use whichever is shorter to compute a +network. +3.3 +A program for network inference +Another technique for improving the scalability +is to decompose the input tree set into irreducible +sets of trees if the input trees are reducible [2, +30]. Let S be a reducible set of k trees on X, +which are ordered as: ⟨T1, T2, · · · Tk⟩. We assume +that C1, C2, · · · , Ct are all the maximal common +clusters of S. We introduce t new taxa yi and +let Y = {y1, y2, · · · , yt}. +By replacing Ti(Cj) +with yj in Ti for each i and j, we obtain a set +S′ of k trees T ′ +i on Y ∪ +� +X \ +� +∪t +i=1Ci +�� +. In this +way, we decompose S into an irreducible tree set +S′ = ⟨T ′ +1, T ′ +2, · · · , T ′ +k⟩ and t ordered sets of trees +S′ +i = ⟨T1(Ci), T2(Ci), · · · , Tk(Ci)⟩, 1 ≤ i ≤ t. +Combining the tree-child networks constructed +from S′ and all of S′ +i gives tree-child networks +that display all the trees of S. +Our program is named ALTS, an acronym +for “Aligning Lineage Taxon Strings”. +It +can +be +downloaded +from +the +Github +site +https://github.com/LX-Zhang/AAST. We also +developed a program that assigns a weight to +each edge of the obtained tree-child network if +the input trees are weighted. The least squares +method for estimating edge weights is presented +in Section B of the Supplementary Document. +In summary, the process of reconstructing a +parsimonious tree-child network involves the fol- +lowing steps. (i) Decompose the input tree set +S into irreducible tree sets, say S1, S2, · · · , St. +(ii) Infer a set Ni of tree-child networks for each +Si. +(iii) Assemble the tree-child networks in +N1, N2, · · · , Nt to obtain the networks that dis- +play all the trees in S. (iv) If the input trees are +7 + +weighted, the branch weights are estimated for +the output tree-child networks. +4 +Validation Experiments +We assessed the accuracy and scalability of +ALTS on a collection of simulated datasets that +were generated using an approach reported in +[30]. For each k ∈ {20, 30, 40, 50}, a phylogenetic +network on k taxa was first generated by simulat- +ing speciation and reticulation events backwards +in time with the weight ratio of reticulation to +speciation ratio being set to 3:1. Fifty trees dis- +played in the networks were then randomly sam- +pled. This process was repeated to generate 2500 +trees for each k. The test tree datasets are avail- +able together with the code for ALTS on Zhang’s +Github site mentioned in Section 3.3. +We compared ALTS with two heuristic net- +work inference programs: PRINs [22], which in- +fers an arbitrary phylogenetic network, and van +Iersel et al.’s method [28], which infers a tree- +child network. We tested the three methods on +50 sets of trees on 20 and 30 taxa, each con- +taining 10 trees. Van Iersel et al.’s program is a +parallel program. It could run successfully only +on 44 (of 50) tree sets in the 20-taxon case and +27 (of 50) tree sets in the 30-taxon case. It was +aborted for the remaining datasets after 24 hours +of clock time (or about 1000 CPU hours) had +elapsed. +To assess the scalability of ALTS, we further +ran it on 100 datasets, each containing 50 trees +on 40 or 50 taxa. PRINs finished on five 50-taxon +50-tree datasets. Van Iersel et al.’s method did +not run successfully on these datasets. +4.1 +The optimality evaluation +ALTS computed the same tree-child HN as van +Iersel et al.’s method on all but three datasets +where the latter ran successfully. +The HN of +the tree-child networks inferred with ALTS was +one more than that inferred with the latter on +two 20-taxon 10-tree datasets and three more +than that with the latter on one 30-taxon 10-tree +dataset. +Moreover, Van Iersel et al.’s method +only outputted a tree-child network, whereas +ALTS computed multiple tree-child networks +with the same HN. +PRINs ran successfully on all but one dataset +in the 20-taxon case. In theory, the HN is in- +herently equal to or less than the HN of the op- +timal tree-child networks for every tree set. In +the 20-taxon 10-tree case, the tree-child HN in- +ferred with ALTS was equal to that inferred with +PRINs on 20 datasets. The 29 discrepancy cases +are summarised in the row one of Table 1 +Table 1: Summary of the HN discrepancy be- +tween ALTS and PRINs in 20-taxon and 30- +taxon datasets each containing 10 trees. +HNALTS − HNPRINs +Date type +-1 +0 +1 +2 +3 +4 +20 taxa +20 +11 +9 +6 +3 +30 taxa +1 +5 +13 +14 +16 +1 +The HN discrepancies between the two pro- +grams in the 30-taxon case are summarised in +the row two of Table 1. Like the 20-taxon case, +the difference in HN was also at most four. The +tree-child HN inferred by ALTS was even one less +than the HN inferred by PRINs on one dataset. +We also noted that the difference in HN of- +ten occurred when the HNs inferred by the two +methods were greater than 15, when van Iersel’s +method could not run successfully. +In summary, ALTS is almost as accurate as +van Iersel et al.’s method in terms of minimizing +network HN. The comparison between ALTS and +PRINs indicated that the tree-child HN is rather +close to the HN for multiple trees. +4.2 +The scalability evaluation +The wall-clock time of the three methods on 100 +datasets, each having 10 trees on 20 or 30 taxa, +are summarized in Figure 5. In the 20-taxa 10- +tree case, the HN inferred by PRINs ranged from +8 + +1 +10 +100 +1000 +10000 +100000 +PRINs +van Iersel et al. +ALTS +0.01 +0.1 +1 +10 +100 +1000 +10000 +Prin +van Iersel et al. +ALTS +Run Time (log scale) +Fifty 20-taxon 10-tree Datasets +Run Time (log scale) +Fifty 30-taxon 10-tree Sets +1 +10 +100 +1000 +10000 +PRINs +AAST +Fifty 40-taxon 10-tree Sets +Run Time (log scale) +Figure 5: Run time (in seconds) of the three +methods on 100 datasets, each containing 10 +trees on 20 or 30 taxa. Here, the datasets are +sorted in the increasing order according to the +HN output from PRINs. Missing data points for +van Iersel et al.’s method are explained in the +main text. +5 to 17. The run time of ALTS ranged from 0.09 +s to 25 m 14 s (with the mean being 2 m 21 s). +On the 49 (out of 50) 20-taxa 10-tree datasets on +which PRINs finished, its run time ranges from +2.94 s to 17 m 19 s (with the mean being 2 m 58 +s). ALTS was faster than PRINs on 35 tree sets. +On average, PRINs and ALTS were comparable +in time. +On the 44 20-taxa 10-tree datasets on which +van Iersel et al.’s method finished, its run time +ranged from 0.07 s to 82 m 22 s (with the mean +being 13 m 3 s). Van Iersel et al.’s method ran +faster than ALTS on 26 datasets where the HN +inferred by PRINs was less than 11. One reason +for this is probably that the former is a parallel +program. However, ALTS was faster than van +Iersel et al.’s method on the remaining 18 tree +1 +10 +100 +1000 +10000 +100000 +PRINs +ALTS +1 +10 +100 +1000 +10000 +100000 +PRINs +ALTS +Fifty 50-taxon 50-Tree Sets +Run Time (log scale) +Fifty 40-taxon 50-Tree Sets +Run Time (log scale) +Figure 6: The run time (in seconds) of ALTS on +100 datasets, each containing 50 trees on 40 or +50 taxa. The datasets are sorted in the increas- +ing order according to the HN of the tree-child +networks inferred by ALTS. +sets where the HN inferred by PRINs was 12 or +more. +In the 30-taxon 10-tree case, the HN of the +solution from PRINs ranged from 8 to 21. As +shown in Figure 5, ALTS was faster than PRINs +on each of the 50 datasets. Van Iersel et al.’s +method finished on 31 (out of 50) datasets, for +which the HN of the solution obtained with +PRINs was 15 or more. +ALTS was faster on +23 datasets where the HN of the solution was +larger than 10. +Van Iersel et al.’s method +was faster than ALTS on the remaining eight +datasets where the HN ranged from 8 to 10. On +average, ALTS was 24 and 53 times faster than +PRINs and the van Iersel et al.’s method, respec- +tively, in the 30-taxon 10-tree case. +Lastly, ALTS was also able to infer tree-child +networks on 100 datasets, each containing 50 +trees with 40 or 50 taxa. In the 40-taxon 50-tree +case, the tree-child HN inferred by ALTS ranged +from 9 to 64. The run time of ALTS ranged from +9 + +Simplified network 1 +Network 1 +Network 2 +Simplified network 2 +20 trees +Figure 7: +The box and whisker plots for the +dissimilarity scores for the original network and +that inferred by ALTS in four cases. +In each +plot, the four bars from left to right summarize +the dissimilarity scores for the original network +and 10 networks inferred from 20-, 30-, 40-, and +50-tree sets, respectively. The four networks are +presented in Figure S3–S6. +3 s to 31 m 52 s (with the mean being 7 m 14 s). +On contrast, PRINs finished on 28 tree sets. Its +run time ranged from 3 m 19 s to 15 h 34 m 52 +s (with the mean being 3 h 49 m 46 s) (Fig. 6). +In the 50-taxon 50-tree case, the tree-child HN +inferred by ALTS ranged from 10 to 61. The run +time of ALTS ranged from 2 s to 45 m 12 s (with +the mean being 9 m 24 s) (Figure 6). In contrast, +van Lersel et al’s method could not finish on any +irreducible set of 50 trees on 50 taxa. +PRINs +finished on five tree sets in 2 h 25 m on average +(Fig. 6). +Taken together, these results suggest that +ALTS has high scalability and is fast enough to +infer a tree-child network on an irreducible tree +set with a size comparable with those of the cur- +rent focus of biological research. +4.3 +The accuracy evaluation +Evaluating the accuracy of ALTS (and the other +two methods) is not straightforward. The ran- +dom networks that were used to generate the tree +sets used in the last two subsections are not tree- +child networks and have frequently a large num- +ber of deep reticulation events. +On the other +hand, by the principle of parsimony, the net- +works inferred by the three programs contain far +fewer reticulation events. As such, we assessed +the accuracy of ALTS by considering the sym- +metric difference of the set of taxa clusters in +the original networks and the set of cluster in +the network inferred by ALTS [15]. Here, a clus- +ter in a network consists of all taxa below a tree +node in that network, as the cluster of a retic- +ulate node x is always equal to the cluster of +its child if x has only one child. Precisely, for +two phylogenetic networks N1 and N2 over X, +we use C(Ni) to denote the multiset of clusters +appearing in Ni for i = 1, 2, and define their +dissimilarity score s(N1, N2) as the Jaccard dis- +tance of C(N1) and C(N2), i.e. +s(N1, N2) = +1 − |C(N1) ∩ C(N2)|/|C(N1) ∪ C(N2)|. +We considered two simulated networks con- +taining 16 binary reticulations (network 1, Fig- +ure S3) and 19 binary reticulations (network +2, Figure S5) and their simplified version (Fig- +ure S4 and S6). The two networks were produced +using the same simulation method just with a +low rate of reticulation events; the two simpli- +fied networks were obtained by merging a retic- +ulate node and its child if the reticulate node +has a unique child and the child is also a reticu- +lation node, which have 9 and 10 multiple reticu- +lations, respectively. For each network and each +k = 20, 30, 40, 50, we generated 10 k-tree sets. +In total, we used 160 tree sets. +For each tree +set, we inferred a network using ALTS and com- +puted the dissimilarity score for it and the origi- +nal network. The dissimilarity score analyses are +summarised in Figure 7. +Network 1 (and its simplified version) contains +less reticulation events than Network 2. We had +slight better reconstruction accuracy for Net- +work 1 than Network 2 (mean dissimilarity score +range [0.3 to 0.45] vs. +[0.55, 0.65], Figure 7). +Also, the reconstruction from the trees sampled +from each network was not significantly better +than that from its simplified version. Given that +10 + +0.60 ++ +0.55 +0.50 +0.45 +0.40 +0.35 +0.300.70 +0.65 +0.60 +0.55 +0.50 +0.45 +0.400.70 +0.65 +X +0.60 +0.55 +0.50 +0.45 +0.400.55 +0.50 +0.45 +0.40 +X +X +0.35 +0.30all four networks can contain as many as 217 +trees, the results suggest that 50 trees are far +fewer than enough for accurate reconstruction of +both networks. +Since we could not run Iersel et al’s program +on the most of tree sets, we were unable to assess +its accuracy for comparison. +5 +A Phylogenetic Network for +Hominin Relationships +Hominins’ phylogenetic relationships are not +fully established. +As an application of ALTS, +we reconstructed a network model for hominin +species using 10 phylogenetic trees derived from +the Bayesian analysis of the morphological data +of hominin evolution presented in [7] (Fig. 8). +To choose 10 phylogenetic trees, we grouped the +posterior trees into five clusters using the dis- +tance metric and approach described in previ- +ous work [17, 16], using Ward clustering. +We +chose two trees from each of the five clusters. +Due to the nature of the morphological data, the +trees were discordant, and no single tree captures +a highly-supported pattern of ancestry among +the taxa. This motivates using a network to il- +lustrate the ancestral relationships among these +data. +The resulting network model contains 12 retic- +ulation events with the HN being 24. +The +top tree-node component contains the two out- +group species G. gorilla and P. troglodytes, as +well as the oldest hominin species, S. tchaden- +sis. +The three earliest members of the genus +Homo ( African H. erectus, H. rudolfensis and +H. habilis), together with Au. +africanus, ap- +pear in a tree-node component, whereas four re- +cent members of the genus Homo (H. heidelber- +gensis, H. neanderthalensis, H. sapiens and H. +naledi ) compose another tree-node component. +The three members of the genus Paranthropus, +together with Au. garhi, compose a tree-node +component. The model also reflects the high un- +certainty about the phylogenetic position of H. +floresiensis, who lived in the island of Flores, In- +donesia [3, 5, 27]. +This network provides an illustration of the +performance of ALTS on hominin morphological +data. We find that its HN is unexpectedly high. +Since the evolutionary time of hominin species +is relatively short, some discrepancies in the 10 +trees are perhaps a result of incomplete lineage +sorting (ILS) [31], (with impacts on morphology, +in order that they are implicitly detected in these +data), or of convergent evolution, ambiguity in +the morphological data, or other factors. With- +out genetic data, we cannot assess the extent to +which ILS or other factors affects the phyloge- +netic trees and consequently this network model. +6 +Conclusions +We have presented ALTS, a fast and scalable +method for inferring tree-child networks from +multiple trees. It is based on a novel algorith- +mic innovation that reduces the minimum tree- +child network problem to computing the SCS of +the LTSs obtained w.r.t. a predefined ordering +on the taxa in the input trees. Another contri- +bution is an algorithm for assigning weights to +the tree edges of the reconstructed tree-child net- +work if the input trees are weighted. Our work +makes network reconstruction more feasible in +the study of evolution. +The accuracy analyses in Section 4.3 suggest +that 50 trees are likely not enough for accurately +inferring a phylogenetic network model that has +10 or more reticulation events. +Therefore, a +program that can process over hundred trees is +definitely wanted. +We remark that ALTS can +be made even more scalable by distributing the +computing tasks for taxon orderings into a large +number of processors using the distributed com- +puting programming. This is because the com- +puting tasks for different orderings are indepen- +dent from each other. +We will further investigate how to improve the +accuracy of ALTS by incorporating the genomic +sequences of the taxa or/and ILS into network +11 + +4 +5 +7 +6 +20 +21 +22 +23 +3 +9 +17 +11 +16 +13 +15 +14 +12 +24 +1 +2 +18 +10 +19 +8 +Figure 8: A network model of hominin relationships. 1: G. gorilla; 2: P. troglodytes; 3: H. floresiensis; +4: Ar. ramidus; 5: Au. anamensis; 6: Au. afarensis; 7: K. platyops; 8: Au. africanus; 9: Au. sediba; 10: +African H. erectus; 11: Asian H. erectus; 12: H. heidelbergensis; 13: H. neanderthalensis; 14: H. sapiens; 15: +H. naledi; 16: H. antecessor; 17: Georgian H. erectus; 18: H. rudolfensis; 19: H. habilis; 20: Au. garhi; 21: +P. robustus; 22: P. boisei; 23: P. aethiopicus; 24: S. tchadensis. +inference. +Acknowledgements +We thank Cedric Chauve and Aniket Mane +for discussion in the beginning of this project. +We also thank anonymous reviewers for con- +structive comments on an earlier version of our +manuscript. +L. 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Total ordering, trees and child-tree networks +Let X be a set of taxa. A (total) ordering R on X is a binary relation on X such that (i) R is +anti-symmetric, i.e. if x1Rx2, then x2 ̸R x1. (ii) R is transitive, i.e., if x1Rx2 and x2Rx3, then +x1Rx3. (iii) For any x1, x2, x1Rx2 or x2Rx1. For convention, we write x 1. +For a phylogenetic tree T on X, +the ancestor sequence sπ(t) of each taxon t obtained by applying the Labeling algorithm to T +and π has the following properties: +(i) sπ(π1) is always not empty; +(ii) sπ(πn) is always empty; +(iii) for each 1 < i ≤ n, πi appears in the ancestor sequence of πj for a unique j such that j < i; +15 + +(iv) the smallest taxon π1 does not appear in any ancestor sequence. +Proof. Let the degree-1 root of T be ρ. Let the ancestors of Leaf π1 be: +ρ = u0, u1, u2, · · · , uk +and uk+1 = π1, where ui is the parent of ui+1 for 0 ≤ i ≤ k. Recall that each non-leaf, non-root +node has two children. We let u′ +i+1 be another child of ui for 0 ≤ i ≤ k. +(i) Since |X| > 1, k ≥ 1. Clearly, minπ C(ui) = π1 for each i ≤ k. Since π1 is the smallest taxon, in +Step 2 of the Labeling algorithm, ui is labeled with maxπ{minπ(ui+1), minπ(u′ +i+1)} = minπ(u′ +i+1) +for i = 1, 2, · · · , k. Therefore, that k ≥ 1 implies that sπ(π1) contains at least one taxon. +(ii) Let the parent and sibling of Leaf πn be v and v′. In Step 2 of the Labeling algorithm, v +is labeled with maxπ{minπ(v′), πn} = πn. Since there is no node between v and Leaf πn, sπ(πn) is +empty. +(iii) and (iv) We prove the statement by mathematical induction. If |X| = 2, clearly, the root ρT +is labeled with π1 and the other internal node is labeled with π2. In this case, sπ(1) contains only +π2 and sπ(2) is empty. Thus, the fact is true. +For |X| > 2, from the proof of Part (i), we have that ui is labeled with the minimum taxon +appearing in C(u′ +i+1) for i = 1, 2, · · · , k. Moreover, the internal nodes in each subtree T ′ +i rooted at +u′ +i are labeled with the taxa of C(u′ +i) \ { minπ C(u′ +i) } according to the algorithm. Since each T ′ +i is +a proper subtree of Ti, by induction, the fact holds. □ +Remark. +The ancestor sequences of the taxa obtained according to an ordering on X give +a unique phylogenetic tree T. This can be generalized to an algorithm to reconstruct a tree-child +network using ancestor sequences of taxa. +Tree-Child Network Construction +1. (Vertical edges) For each βi, define a path Pi with |βi| + 2 nodes: +hi, vi1, vi2, · · · , vi|βi|, ℓπi, where βn is the empty sequence. +2. (Left–right edges) Arrange the n paths from left to right as P1, P2, · · · , Pn. If the +m-th letter of βi is πj, we add an edge (vim, hj) for each m and each i. +3. Contract each hi (i > 1) if it is of indegree 1 and outdegree 1. +Proposition 2. +Let Ti (1 ≤ i ≤ k) be k trees on X such that |X| = n and π be an +ordering on X. +Let αij += βTi,π(πj), the ancestor sequences of πj in Ti with respect to +π, 1 ≤ j ≤ n − 1. If βj is a common supersequence of α1j, α2j, · · · , αkj for each j, the Tree- +Child Network Construction algorithm outputs a tree-child network that displays the k trees. +Proof. Let N be the directed network constructed by applying the algorithm to β1, β2, · · · , βk. +First, N is acyclic due to the two facts: (i) the edges of each path Pi are oriented downwards, and +(ii) the so-called left–right edges (u, v) are oriented from a node u in a path defined for πi to a +node v in a path defined for πj such that i < j. +Second, N is tree-child. This is because all the nodes of each Pi are tree nodes except hi for each +i > 1 (see Figure 3 in main text). The node h1 is the network root. For i > 1, hi may or may not +be a reticulation node. Therefore, every non-leaf node has a child that is not reticulate. +16 + +Lastly, we prove that Ti is displayed by N as follows. By assumption, βj is a supersequence of +{αij | i = 1, 2, · · · , k} for each j = 1, 2, · · · , n − 1. Following the notation used in the Tree-Child +Network Construction algorithm, we let: +βj = βj1βj2 · · · βjtj, tj ≥ 1, +where tj is the length of βj. Since αij is a subsequence of βj, there is an increasing subsequence +1 ≤ m1 < m2 < · · · < mℓj ≤ tj such that +αij = βim1βim2 · · · βimℓj +and ℓj = |αij| ≥ 1. +According to Step 1 of the algorithm, in N, each taxon βjx of βj corresponds one-to-one a node +vjx in the path Pj; and there is a (left-right) edge from vjx to the first node hy(x) of the path Py(x) +that ends with the taxon πy(x) = βjx, where y(x) ≥ j. +Conversely, after removing the edge (vjx, hy(x)) for each x ̸= m1, m2, · · · , mℓj, we obtain a +subtree T ′ +i of N. This is because each taxon πt appears exactly once in αi1, αi2, · · · , αi(n−1) and +thus the node ht is of indegree 1 in the resulting subgraph, where t = 2, 3, · · · , n. It is not hard +to see that after contracting degree-2 nodes of T ′ +i, the resulting subtree T ′′ +i has the same ancestor +sequence as Ti for each πj. Thus T ′′ +i is equal to Ti. □ +The proof of Proposition 3 is divided into several lemmas. +Lemma 1. +Let π be an ordering on X and let T1, T2, · · · , Tk be k phylogenetic trees on +X. +For each x ∈ X and each Ti, we use βx(Ti, π) to denote the ancestor sequence of x ob- +tained from π using the Labeling algorithm on Ti. Assume βx is a common supersequence of +{βx(T1, π), βx(T2, π), · · · , βx(Tk, π)} for each x ∈ X. +For the tree-child network P constructed +from {βx | x ∈ X} and π using the Tree-Child Network Construction algorithm, +H(P) = � +x∈X |βx| − |X| + 1. +Proof. Since only the first node hi of each path can be a reticulate node and that each node in +the middle of each path is a parent of some hi, H(P) = �|X| +i=2(din(hi) − 1) = � +x∈X |βx| − |X| + 1, +where din(hi) is the indegree of hi. □ +Definition 1. +Let P be a phylogenetic network on X, where |X| > 1 and π be an order- +ing on X. P is said to be compatible with π if for each reticulate edge (s, r) of P, the minimum +taxon below s in the tree-node component Cs is less than the minimum taxon in the tree-node +component Cr. +Remark. +For a tree-child network P, we can construct a compatible ordering π as follows. +We first compute a topological sorting on the vertices of P. Assume the reticulate nodes and the +network root ρ appear in the sorted list as: r0 = ρ, r1, r2, · · · , rk. We construct a desired ordering +by listing the taxa in the tree-node component Cri before the taxa in the tree-node component +Cri+1 for every i ≤ k − 1. +Let π be an ordering on X and P be a tree-child network on X that is compatible with π. The +compatibility property implies that the smallest taxon is in the tree-node component Cρ that is +17 + +rooted at the network root ρ. We use the following generalized Labelling algorithm to label all +the tree nodes of P, which is identical to Labelling when P is a phylogenetic tree. +Generalized Labelling +S1: For every reticulate node r, label all parents of r with the smallest taxon in +the tree-node component Cr. Similarly, the network root ρ is labeled with +the smallest taxon in Cρ. +S2: For each tree node z that is not a parent of any reticulate node, label x with +maxπ(minπ(C(x)), minπ(C(y)), where x and y are the two children of z, and +C(x) and C(y) are the set of taxa below x and y in the tree-node component +where they belong to. +Lemma 2. Let C be a tree-node component of P and let it contain t taxa x1, x2, · · · , xt in P. +All t − 1 tree nodes that are not a parent of any reticulate node are uniquely labeled with some +xj ̸= minπ{xi | 1 ≤ i ≤ t} (blue labels in Figure S1B). +Proof. This can be proved using the same mathematical induction as in Prop. 1.iii. □ +Definition 2. +Let π be an ordering on X and N be a tree-child network on X that is +compatible with π. +Assume the tree nodes of N are labeled by using the Generalized La- +belling algorithm. The ancestor sequence of a taxon x obtained according to π is defined to +be the sequence of the labels of the x’s ancestors that are in Cx, if x is the smallest taxon in +C; it is the sequence of the labels of the x’s ancestors that are below the unique tree node la- +beled with x in Cx otherwise. The ancestor sequence of x obtained in this way is denoted by βN,π(x). +Definition 3. +Let P be a tree-child network on X and let (s, r) be a reticulate edge. +P − (r, s) is defined to be the tree-child network obtained through the removal of (s, r) and +contraction of s (and also r if r is of indegree 2 in N). +Lemma 3. +Let π be an ordering on X and P be a tree-child network on X such that +H(P) ≥ 1 and P is compatible with π. For any reticulate node r and each parent s of r, the +tree-child network P − (s, r) has the following properties: +1. P − (s, r) is also compatible with π; +2. For each taxon x, βP,π(x) is a supersequence of βP−(s,r),π(x). +Proof. These properties are illustrated in Figure S1. Let (s, r) be a reticulate edge. We have that +s is a tree node, and r is a reticulate node. Recall that CN(z) denotes the tree-node component +containing z for each node z and for N = P, or P − (s, r). We consider the two cases. +Case 1. The r is of indegree 3 or more. +In this case, after (s, r) is removed, s will be contracted and all the other nodes remains the same +in P − (s, r). Moreover, P − (s, r) has the same tree-nodes components as P and also has the same +18 + +labelling as P. For any reticulate edge (s′, r′), CP−(s,r)(s′) = CP (s′) and CP−(s,r)(r′) = CP (r′). As +such, the constraint is also satisfied for (s′, r′) in P − (s, r). Therefore, the first fact holds. +Let x be a taxon. +If βP,π(x) contains the label y of s, say βP,π(x) += +β1yβ2, then, +βP−(s,r),π(x) = β1β2. +If βP,π(x) does not contain the label of s, βP−(s,r),π(x) = βP,π(x). +This concludes that βP,π(x) is a supersequence of βP−(s,r),π(x). Therefore the second fact is true. +Case 2. The r is of indegree 2. +This case is illustrated in Figure S1b. Let s′ be another parent of r. After (s, r) is removed, the +r becomes a node of indegree 1 and outdegree 1 and thus is contracted, together with s. All the +other nodes remains in P − (s, r). Therefore, s′ becomes a tree node in P − (s, r). The tree-node +component CP−(s,r)(s′) is the merge of CP (s′) and CP (r). Assume (s′′, r′) be a reticulate edge of +P − (s, r). +If CP−(s,r)(s′′) ̸= CP−(s,r)(s′) and CP−(s,r)(r′) ̸= CP−(s,r)(s′), then, CP−(s,r)(s′′) = CP (s′′) and +1 +3 +2 +4 +5 +6 +7 +8 +9 +r +1 +3 +2 +4 +5 +6 +7 +8 +9 +9 +8 +8 +7 +7 +6 +6 +5 +5 +4 +3 +2 +1 +1 +3 +2 +4 +5 +6 +7 +8 +9 +9 +8 +8 +7 +7 +6 +6 +5 +5 +4 +3 +2 +1 +(a) + (b) +(c) +Figure S1: Illustration of the Generalized Labelling algorithm and the proof of Lemma 3. (a) +A tree-child network on the taxa from 1 to 9, which has two tree-node components each containing +at least two taxa. (b) Labelling all the tree nodes in a tree-child network using the increasing order +of taxa: i < i + 1, i = 1, 2, ..., 8, which is compatible. The labels of the parents of a reticulation +node are in blue; while the labels of other tree-nodes are in red. (c) the resulting network after the +removal of the left incoming edge of the reticulation node r, in which the tree-nodes are labeled +identically if the same ordering is used. +19 + +CP−(s,r)(r′) = CP (r′). The constraint is satisfied for (s′′, r′). +If CP−(s,r)(s′′) ̸= CP−(s,r)(s′) and CP−(s,r)(r′) = CP−(s,r)(s′), the constraint is satisfied for s′′, r′ +because of the fact that minπ CP−(s,r)(r′) = minπ CP (r′). +If CP−(s,r)(s′′) = CP−(s,r)(s′) and CP−(s,r)(r′) ̸= CP−(s,r)(s′), then the minimum taxon below s′′ +in CP−(s,r)(s′′) is equal to that in CP (s′′), the constraint is satisfied for (s′′, r′). +We have proved the first statement. We prove the second statement as follows. To this end, we +use cP (r) to denote the unique child of r in P. +Recall that after (s, r) was removed, s and r were contracted to obtain P − (r, s). Note that in +P − (r, s), s′ becomes the parent of cP (r). Since P is compatible with π, the minimum taxon y +below cP (r) is larger than the minimum taxon below s′ in π. This implies that s′ is labeled with +y, as s′ is not a parent of any reticulate node in P − (s, r). Therefore, for any taxon x ∈ X, if +βP,π(x) contains the label y of s, say βP,π(x) = β1yβ2, then, βP−(s,r),π(x) = β1β2. If βP,π(x) does +not contain the label of s, βP−(s,r),π(x) = βP,π(x). This concludes that βP,π(x) is a supersequence +of βP−(s,r),π(x) for each x ∈ X. □ +Proposition 3. +Let T1, T2, · · · , Tk be k trees on X and P be a tree-child network on X +with the smallest H(P), compared with those displaying all Ti. For any ordering Π of X such +that P is compatible with it, if we label the tree nodes of P using the Generalized Labelling +algorithm, the ancestor sequence βP,Π(x) of each taxon x is a shortest common supersequence of +{βTi,Π(x) | i = 1, 2, · · · , k}. Moreover, applying the Tree-child Construction algorithm to the +obtained supersequences βP,Π(x) produces the same network as P. +Proof. +Let P be a tree-child network on X with the smallest H(P), compared with those +displaying all Ti. For each i, Ti can be obtained from P by deleting all but one incoming edge +for each reticulate node. For convention, we assume that all removed reticulate edges are (sj, rj), +1 ≤ j ≤ H(P). Let x be a taxon. By Lemma 3, βP,Π(x) is a supersequence of βP−(s1,r1),Π(x) and +βP−�j +t=1(st,rt),Π(x) is a supersequence of βP−�j+1 +t=1(st,rt),Π(x) for each j = 1, .., H(P) − 1. Therefore, +for any x, βP,Π(x) is a supersequence of βTi,π(x) for each Ti, as Ti = P − �H(P) +j=1 (sj, rj). +Let P contain m reticulate nodes. P has m+1 tree-node components. In a tree-node component +C, there are |X(C)| − 1 tree nodes that are not the parents of any reticulation nodes, where X(C) +is the set of taxa in C. Hence +� +x∈X +|βP,Π(x)| += +� +C +(|X(C)| − 1) + +� +r∈R(P) +din(r) += +|X| − (m + 1) + H(P) + m += +|X| − 1 + H(P). +This implies that H(P) = � +x∈X |βP,Π(x)| − |X| + 1. +Assume βP,Π(x) is not a shortest supersequence of βTi,Π(x) (i = 1, 2, · · · , k) for some x. Let βx +be a shortest supersequence of βTi,Π(x) (i = 1, 2, · · · , k). Then, |βx| < |βP,Π(x)|. By Lemma 1, +we can use the Tree-Child Network Construction algorithm to obtain a tree-child network +with the HN smaller than H(P), a contradiction. +It is obvious that the Tree-Child Network Construction algorithm to obtain P. □ +20 + +B. Computing the branch weights of inferred tree-child network +A phylogenetic network is weighted if every branch has a non-negative value, which represents +time or other evolutionary measures. A weighted phylogenetic tree T is said to be displayed in a +weighted network N if the tree is displayed in the network when the branch weights are ignored. For +a display T ′ of T in N, its fitness score ||T −T ′||2 is defined as +�� +e∈E(T) |wT (e) − wT ′(P(u′, v′))|2, +where wT (e) is the weight of e = (u, v) in T and wT ′(P(u′, v′)) is the weight of the unique path +between the images u′ and v′ of u and v under the display mapping, respectively. +Recall that a tree can be displayed multiple times in a network. The score of the display of T in +N is the smallest fitness score which a display of T in N can have, denoted d(T, N). If d(T, N) = 0, +we say that N perfectly displays T. +If the input trees are weighted, we will first compute tree-child networks that each display all the +trees. We then use branch weights of trees and the information on how the trees are displayed in +a tree-child network to compute the weights of the network branches. +We model the branch weight assignment problem as an optimization problem with the following +assumption on the inferred tree-child network N that displays all the trees: +For any reticulate edge e, the tree-child network P − e obtained after removal of e fails to +display one input tree at least. +By ordering the edges of N on X, we may assume +E(N) = {e1, e2, · · · , em}. +Let S = {T1, T2, · · · , Ts}, where |S| = s. We further assume that T ′ +k is a display of Tk in N. Then, +each edge e′ +i of Tk is mapped to a path P ′ +i of T ′ +k, where 1 ≤ i ≤ 2|X| − 2. Since N displays Ti, we +derive the following linear equation system from the display of Tk: +� +1≤j≤m +aijw(ej) = w(e′ +i), i = 1, 2, · · · , 2|X| − 2, +(1) +where +aij = +� 1 +ej ∈ E(P ′ +i); +0 +ej ̸∈ E(P ′ +i). +Let the coefficient matrix of Eqn. (1) be Ak = (aij), which is a (2|X| − 2) × m matrix, and let: +Wk = +� +� +� +� +� +� +w(e′ +1) +w(e′ +2) +... +w +� +e′ +2|X|−2 +� +� +� +� +� +� +� +. +Since N displays every tree of S, we then determine the edge weights of N by solving the following +linear equation system: +� +� +� +� +� +A1 +A2 +... +As +� +� +� +� +� × +� +� +� +� +� +x1 +x2 +... +xm +� +� +� +� +� = +� +� +� +� +� +W1 +W2 +... +Ws +� +� +� +� +� +(2) +21 + +e1: ( 0, 13) +e2: (10, 12) +e3: (10, 16) +e4: (11, 12) +e5: (11, 1) +e6: (12, 18) +e7: (13, 10) +e8: (14, 3) +e9: (13, 15) +e10: (14, 15) +e11: (15, 17) +e12: (16, 11) +e13: (16, 4) +e14: (22, 2) +e15: (17, 19) +e16: (18, 14) +e17: (19, 6) +e18: (18, 20) +e19: (19, 20) +e20: (20, 21) +e21: (21, 5) +e22: (21, 22) +e23: (17, 22) +e2 +e6 +e16 +e10 +e11 +e15 +e19 +e20 +e22 +e14 +e21 +e17 +e8 +e3 +e12 +e13 +e5 +e’1: (11, 3) +e’2: (10, 11) +e’3: (10, 12) +e’4: (11, 13) +e’5: (13, 6) +e’6: (12, 1) +e’7: (13, 14) +e’8: (14, 2) +e’9: (12, 4) +e’10: (14, 5) +A B C +D +Figure S2: An illustration of how to derive linear equations from a tree display. (A) The list of the +edges of a tree-child network. (B) A display of the tree in C. (C) a phylogenetic tree on six taxa (1 +to 6). (D) the list of the edges of the tree in C. +Note that Eqn. (2) is a linear equation system that contains 2s(|X| − 1) equations and at most +5|X| − 4 variable, as each Ti contains 2|X| − 2 edges and N contains 3r + 2|X| − 1, where r is the +number of reticulations, which is at most |X| − 1. +Example 1. The edge list of a tree-child network is given in Figure S2A, where the full network is +not given here. Figure S2B presents a particular display of the tree in Figure S2C, whose edges are +listed in Figure S2D. In the display of the tree, the edge e′ +2 is mapped to the path from the node +10 to the node 14, which consists of three edges e2, e6, e16 (Figure S2B). From e′ +2 and its image, +we obtain the following equation in the linear equation system Eqn. (2): +x2 + x4 + x16 = w(e′ +2). +In general, N may not perfectly display every T when branch weights are considered. Therefore, +let us set: +A = +� +� +� +� +� +A1 +A2 +... +As +� +� +� +� +� +(3) +W = +� +� +� +� +� +W1 +W2 +... +Ws +� +� +� +� +� . +(4) +22 + +10 +1416 +18 +1 +3 +15 +20Noticing that +s +� +i=1 +||T ′ +i − Ti||2 +2 = ||AX − W||2 +2, +we determine the branch weights of N by solving the following quadratic optimization problem: +min ||AX − W||2 +2 +(5) +subject to: +xj ≥ 0, 1 ≤ j ≤ m. +(6) +Remark. Let r be a reticulation node that has incoming e1, e2, · · · , ed and the outgoing ed+1. +For each input tree Ti, one of edge pairs (e1, ed+1), (e2, ed+1), ..., (ed, ed+1) appears in the display +of Ti exclusively. Thus, solving the above optimization problem can only determine the value of +w(ei) + w(ed+1) for i ≤ d. +C. Tree distance and clustering in the hominin analysis +We analysed the morphological data in [7] by sampling 500 phylogenetic trees from a posterior +collection of trees estimated from the morphological data. We computed the distance between each +pair of trees using the rooted tree metric described in [17]. Briefly, this metric is the Euclidean +distance between two vectors (one for each tree). The vector captures the amount of shared ancestry +between each pair of tips, as well as each tip’s distance from its parent. We used the tree topology +only (λ = 0 in the tree metric in the ‘treespace’ function in the ‘treespace‘ package in R [16]). The +amount of shared ancestry is the length of the path (in a phylogeny) between the root and the most +recent common ancestor of a pair of tips. Having found pairwise distances between all pairs of trees +in our sample of 500, we clustered the trees into five clusters using Ward clustering. We chose two +trees uniformly at random from each of the five clusters, as input for the analysis presented here. +23 + +Figure S1. Network 1 used in the accuracy assessment in Section 4.3. +It has 16 binary reticulation events. +Figure S3: Network 1 used in the accuracy assessment in Section 4.3. It has 16 binary reticulation +events. +Figure S2. Simplified network 1 used in the accuracy assessment in +Section 4.3. It has 9 reticulation events. +Figure S4: Simplified network 1 used in the accuracy assessment in Section 4.3. It has 9 reticulation +events. +24 + +Figure S3. Network 2 used in the accuracy assessment in Section 4.3. +It has 19 binary reticulation events. +Figure S5: Network 2 used in the accuracy assessment in Section 4.3. It has 19 binary reticulation +events. +Figure S4. Simplified network 2 used in the accuracy assessment in +Section 4.3. It has 10 reticulation events. +Figure S6: Simplified network 2 used in the accuracy assessment in Section 4.3. It has 10 reticulation +events. +25 + diff --git a/5dAzT4oBgHgl3EQfEfoE/content/tmp_files/load_file.txt b/5dAzT4oBgHgl3EQfEfoE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6a4456f8beacd4e9a249aef9d85f3194d2f0c57 --- /dev/null +++ b/5dAzT4oBgHgl3EQfEfoE/content/tmp_files/load_file.txt @@ -0,0 +1,999 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf,len=998 +page_content='A Fast and Scalable Method for Inferring Phylogenetic Networks from Trees by Aligning Lineage Taxon Strings Louxin Zhang1 ∗, Niloufar Abhari2, Caroline Colijn2, Yufeng Wu3 1 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' of Mathematics and Centre for Data Science and Machine Learning National University of Singapore, Singapore 119076 Corresponding author: matzlx@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' +65-65166579 2 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' of Mathematics Simon Fraser University, Burnaby, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Canada V5A 1S6 3 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' of Computer Science and Engineering University of Connecticut, Storrs, CT 06269, USA Abstract The reconstruction of phylogenetic networks is an important but challenging problem in phy- logenetics and genome evolution, as the space of phylogenetic networks is vast and cannot be sampled well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' One approach to the problem is to solve the minimum phylogenetic network prob- lem, in which phylogenetic trees are first inferred, then the smallest phylogenetic network that displays all the trees is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The approach takes advantage of the fact that the theory of phylogenetic trees is mature and there are excellent tools available for inferring phylogenetic trees from a large number of biomolecular sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A tree-child network is a phylogenetic network satisfying the condition that every non-leaf node has at least one child that is of indegree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, we develop a new method that infers the minimum tree-child network by aligning lineage taxon strings in the phylogenetic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This algorithmic innovation enables us to get around the limitations of the existing programs for phylogenetic network inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Our new program, named ALTS, is fast enough to infer a tree-child network with a large number of reticulations for a set of up to 50 phylogenetic trees with 50 taxa that have only trivial common clusters in about a quarter of an hour on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='00992v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='PE] 3 Jan 2023 1 Introduction In this study, phylogenetic networks are rooted, directed acyclic graphs in which the leaves are labeled with taxa, the non-leaf indegree-1 nodes represent speciation events and the nodes with multiple incoming edges represent reticulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The non-leaf indegree-1 nodes are called tree nodes and the other non-leaf nodes are called reticulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Phylogenetic trees are phyloge- netic networks with no reticulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Now that a variety of genomic projects have been completed, reticulate evolutionary events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' horizontal gene transfer, introgression and hybridization) have been demonstrated to play important roles in genome evolution [9, 12, 19, 21, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Although phylogenetic networks are ap- pealing for modeling reticulate events [18], it is extremely challenging to apply phylogenetic net- works in the study of genome evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' One reason for this is that a computer program has yet to be made available for analyzing data as large as what current research is interested in [23, 31], although recently, Bayesian methods have been used to reconstruct reassortment net- works, which describe patterns of ancestry in which lineages may have different parts of their genomes inherited from distinct parents [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, we focus on reconstructing phylogenetic networks from (phylogenetic) trees by comput- ing the smallest phylogenetic network displaying a given set of multiple trees [2, 8, 30, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In this approach, trees are first inferred from biomolecular sequences and then used to recon- struct a phylogenetic network with the smallest hybridization number (HN) that displays all the trees (see [8]), where the HN is defined as the sum over all the reticulate nodes of the differ- ence between the indegree and outdegree of each reticulate node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This approach takes advantage of the fact that the theory of phylogenetic trees is mature and there are excellent tools available for inferring trees from a large number of se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, we focused on the parsimonious inference of phylogenetic networks from multiple trees, which computes a phylogenetic network with the minimum HN that displays all the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This problem is NP-hard even for the special case when there are only two input trees [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For the two-tree case, the fastest programs include MCTS-CHN [32] and HYBRIDIZATION NUM- BER [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For the general case where there are multiple input trees, HYBROSCALE [1] and its predcessor [2], PRIN [30] and PRINs [22], have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' All these methods are based on the process of searching through inserting reticulate edges or other editing operations in the space of phylogenetic networks, by reducing the problem to the maximum acyclic agreement forests of the input trees or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Unfortunately, none of them can be used for inferring a network from a so-called irreducible set of 30 trees with 30 taxa in which the trees do not contain any non-trivial common clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since the whole network space is vast and cannot be fully sampled, attention has been switched to the inference of the tree-child net- works, in which every non-leaf node has at least one child that is not reticulate [28], or, recently, a member of a subclass of the tree-child network [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Tree-child network [6] is a superclass of phy- logenetic trees with a completeness property that for any set of phylogenetic trees, there is always a tree-child network (whose reticulate nodes can be of indegree 2 or more) that displays all the trees [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Other desired properties of tree-child networks include the fact that all the tree-child networks are efficiently enumerated [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Most importantly, the validation results in [28] and our results (reported in Section 4) suggest that the HN of a tree-child network solution is close to the optimal HN of a phylogenetic network that displays the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The program for inferring tree-child networks that appears in [28] is based on a fixed-parameter algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The time-complexity of the algorithm is O((8r)rpoly(k, n)), where k and n are, respec- tively, the number of taxa and the input trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' r is the HN of the network solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The new program we introduce here, ALTS, 1 takes a different approach that reduces the in- ference problem to aligning the lineage taxon strings of all the input trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Algorithmic in- novations in ALTS enable us to get around some of the limitations associated with parsimonious inference by efficiently sampling the orderings of the taxa and progressively computing the short- est common supersequence (SCS) of the lineage taxon strings derived for each taxon in all the input trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ALTS is fast enough to infer a par- simonious tree-child network for a set of 50 trees on 50 taxa in a quarter of an hour on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We also added a feature of inferring a weighted tree-child network if the input trees are weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2 Concepts and notation A directed graph G consists of a set V of nodes and a set E of directed edges that are ordered pairs of distinct nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let e = (u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We call e an outgoing edge of u and an incoming edge of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For a node v ∈ V , its outdegree and indegree are defined as the number of outgoing and incoming edges of v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For a graph, subdividing an edge (u, v) involves replacing it with a directed path from u to v that passes one or more new nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Conversely, an edge contraction at a node v of indegree one and outdegree one is to remove v and replace the path u → v → w with an edge (u, w), where (u, v) and (v, w) are the unique incoming and outgoing edge of v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1 Phylogenetic networks A phylogenetic network on a set X of taxa is a rooted, directed acyclic graph in which (i) all the edges are oriented away from the root, which is of indegree 0 and outdegree 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (ii) the nodes of indegree 1 and outdegree 0, called leaves, are uniquely labeled with the taxa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' and (iii) all the non-root and non-leaf nodes are either tree nodes that are of indegree 1 and outdegree 2 or reticu- late nodes that are of indegree more than 1 and outdegree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Reticulate nodes represent evolu- tionary reticulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A phylogenetic net- work is said to be binary if the indegree of every reticulate node is exactly 2 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let N be a phylogenetic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We use V(N) and E(N) to denote the node and edge set of N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We also use R(N) to denote the set of reticulate nodes, and use T (N) to de- note the set of all non-reticulate nodes, including the root, tree nodes and leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let u, v ∈ V(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The node v is a child of u if (u, v) is an edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' v is a descendant of u if there is a directed path from u to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' If v is a descendant of u, v is said to be below u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A phylogenetic network N is a tree-child net- work if every non-leaf node has a child that is not reticulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Equivalently, N is a tree-child network if and only if for every non-leaf node, there is a path from that node to some leaf that passes only tree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Figure 1 presents a bi- nary tree-child network (left) and two non-tree- child networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Consider a tree-child network N with k retic- ulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let the root be r0 and let the retic- ulate nodes be r1, r2, · · · , rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' After the removal of the incoming edges of every ri, N becomes the union of k + 1 subtrees, which are rooted at r0, r1, · · · , rk, respectively, and have network leaves as their leaves (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' These sub- trees are called the tree-node components of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Tree-node decomposition is a useful technique in the study of phylogenetic networks [11, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2 Phylogenetic trees A phylogenetic tree on X is a phylogenetic net- work with no reticulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In fact, a tree is a tree-child network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let T be a phylogenetic tree on X and u ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The node cluster of u, denoted as C(u), is the subset of taxa that are represented by the leaves below u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Clearly, C(u) ∩ C(v) ∈ {C(u), C(v), ∅} for any two nodes u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The node u and its descendants induce a unique subtree on C(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We use Tu or T(C(u)) to denote the subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let S be a set of binary phylogenetic trees on 2 d b a c c b a c b a x 4 2 1 3 y x 4 2 1 3 y Edge insertion Figure 1: A binary tree-child network (left) in which there are four tree-node components (shaded grey) and two non-tree-child networks (middle) and (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the middle network, the child of the top reticulate node is also reticulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the right network, the chil- dren of a tree node in the middle are both reticulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A common cluster of S is a subset of X that is a node cluster in every tree of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Obviously, each single taxon is common cluster of S, and so is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Any other common clusters of S are called non- trivial common clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' S is a reducible tree set if there is a non-trivial common cluster for S, and it is irreducible otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A non-trivial common cluster C of S is maximal if any subset C′ such that C ⊂ C′ ⊂ X is not a common cluster of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Clearly, for any two maximal common cluster C1 and C2 of S, C1 ∩ C2 = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' and any non-trivial common cluster X′ of S must be contained in a unique maximal cluster of S if X′ is not maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 Tree display and network infer- ence problems Let T be a binary phylogenetic tree on X and let N be a tree-child network with k reticulate nodes on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' T is displayed by N if T can be obtained from N by applying edge contraction from N after the removal of all but one incoming edge for each reticulation node (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For any set of binary phylogenetic trees over X, there is always a tree-child network that displays all the trees [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' However, such a solution network may not be binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let P by a phylogenetic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Its c d a b c d a b c d b a A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') Figure 2: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') A tree-child network with two retic- ulate nodes on the taxa (a to d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') A subtree of the network in (A) that can be obtained by the re- moval of the dashed incoming edges of the reticulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') A tree displayed in the network in (A), which was obtained from the subtree in (B) by edge contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' reticulate number is defined as the number of reticulate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Its HN, denoted as H(P), is defined as the sum over all the reticulate nodes of the difference between the indegree and the outdegree of that reticulate node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' If P is binary, H(P) is equal to the reticulate number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, we studied the following minimum tree-child network inference problem: Input: A set of phylogenetic trees on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Output: A parsimonious tree-child network P on X (with the smallest H(P)) that displays all input trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='4 The SCS problem Let s and t be two sequences in an alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The sequence s is said to be a supersequence of t if t can be obtained from s by the deletion of one or more letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The SCS problem is, given a set of sequences, to find the shortest sequence that is a supersequence of every given sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The SCS problem can be solved in a quadratic time for two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' However, it is NP-hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3 3 The methods In this section, we assume that the input trees are binary phylogenetic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1 The Inference Algorithm Let X be a taxon set and let π = π1π2 · · · πn, representing a (total) ordering of X by which πi is ‘less than’ πi+1 for each i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For any non-empty subset X′ of X, we use minπ(X′) and maxπ(X′) to denote the minimum and maximum taxon of X′ with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Consider a tree T on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since the root of T is of outdegree 1, T has n non-leaf nodes, called internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We label the n internal nodes one-to-one with X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' π using the following algorithm: Labeling Input A tree T on X and an ordering π of X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Label the degree-1 root of T by minπ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Label each internal node u with two children v and w with maxπ{minπ(C(v)), minπ(C(w))}, where C(v) consists of all taxa below v in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For instance, let X = {a, b, c, d, e} and π be an ordering of X such that b < c < a < d < e, Figure 3B gives two trees in which their internal nodes are labeled w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' π by using Labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each taxon τ, there is a unique internal node w that is labeled with τ, which is an an- cestor of the leaf τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The sequence of the taxon labels appearing in the path from w to the leaf τ exclusively is called the lineage taxon string (LTS) of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The LTSs computed in the trees in Figure 3B are listed in Figure 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' It is not hard to see that a tree can be recovered by using the LTSs derived from the given ordering of X in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In addition, we have the following proposition, the proof of which appears in the Supplementary Document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Proposition 1 Let π be an ordering of X, |X| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For a phylogenetic tree T on X, the LTS sπ(i) of each taxon πi obtained by applying the Labeling algorithm has the following prop- erties: (i) sπ(1) is always not empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (ii) sπ(n) is always empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (iii) every taxon πk (k > 1) appears in the LTS 2 1 4 3 5 2 5 1 3 4 1 2 3 4 5 × × × × Ordered Taxa: b < c < a < d < e b a d c e b c a e d b e a c d b c e d a The lineage taxon strings b c, a c, e Taxon Left tree Right tree c e, d d, a a empty empty d empty empty e empty empty A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') b c a d e c e a d e a b c a d e C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') Figure 3: The construction of a tree-child network that displays two phylogenetic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (A) An order- ing on {a, b, c, d, e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (B) Two trees, where the inter- nal nodes are labeled using the Labeling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (C) The LTSs of the taxa obtained from the label- ing in Panel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (D) The rooted directed graph con- structed from the shortest common supersequences (SCS) of the LTSs of the taxa (in Panel C) using the Tree-child Network Reconstruction algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, the SCS is [c, e, a] for [c, a] and [c, e], and is [e, d, a] for [e, d] and [d, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (E) The tree-child net- work obtained after contraction of the degree-2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4 of πj for a unique j < k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (iv) π1 does not appear in any LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let Ti (1 ≤ i ≤ k) be k trees on X and let π = π1π2 · · · πn, an ordering of X, where n = |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let αij be the LTS of πj in Ti for each j from 1 to n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Assume that, for each j, βj is a common supersequence of α1j, α2j, · · · , αkj such that βj does not contain any symbol not in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We can construct a tree-child network Nπ(β1, β2, · · · , βn−1) on X using the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Tree-Child Network Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (Vertical edges) For each βi, define a path Pi with |βi| + 2 nodes: hi, vi1, vi2, · · · , vi|βi|, ℓπi, where βn is the empty sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (Left–right edges) Arrange the n paths from left to right as P1, P2, · · · , Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' If the m-th symbol of βi is πj, we add an edge (vim, hj) for each i and each m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Contract each hi if it is of indegree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Tree-Child Network Construction is illustrated in Figure 3D, where the SCSs are [c, e, a] and [e, d, a] for π1 = b and π2 = c, and the empty sequence for π3 = a and π4 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Clearly, the network output from the algorithm is a tree-child network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Proposition 2 Let Ti (1 ≤ i ≤ k) be k trees on X such that |X| = n and let π be an order- ing of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let αij be the LTS of πj in Ti with respect to π, 1 ≤ j ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' If βj is a com- mon supersequence of α1j, α2j, · · · , αkj on X for each j from 1 to n − 1, the Tree-Child Net- work Construction algorithm outputs a tree- child network that displays all k trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Conversely, we assume that P is a tree-child network with the smallest HN, H(P), compared with those that displays all k trees Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The prop- erty that P has the smallest HN implies that, for each i, any display of Ti in P must use one in- coming edge for each reticulate node of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') b a d c b c a d b d a c d a b d a a b c d a b c a b b c The lineage taxon strings w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' the ordering: c < d < b < a Taxon Left tree Middle tree Right tree c b, d d, a d, b d empty b a b a empty empty a empty empty empty Figure 4: Mapping a tree-child network on n taxa that displays multiple trees to (n−1) common super- sequences of the LTSs of the taxa in the trees with respect to a selected ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (A) Three trees on taxa {a, b, c, d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (B) A tree-child network with the smallest HN (4) that displays all three trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (C) De- termine an ordering: c < d < b < a, label all internal tree nodes, and derive the LTS for each taxon: [b, d, a, b] (Taxon c), [a, b] (Taxon d), [a] (Taxon b), empty (Taxon a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The LTS of the taxa in the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each taxon, the LTS obtained in the network is the SCS of the LTSs obtained in the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let P contain t reticulate nodes ri (1 ≤ i ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' N has t + 1 tree-node components C0, C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=', Ct such that C0 is rooted at the root r0 of P, and Ci is rooted at ri for i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since P is acyclic, its nodes can be topologically 5 sorted into a list such that u appears before v for every edge (u, v) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' By using such a topolog- ical ordering of P, we can order all the taxa into π1, π2, · · · , πn such that (i) all the taxa in each tree-node component appear consecutively, and (ii) if the reticulate node ri has a parent in Cj, the taxa of Cj appear before the taxa of Ci in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This is because there is a directed path from rj to every node of Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For instance, for the tree-child network in Figure 4B, C0 contains Taxa c and d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' the tree component rooted below the left reticulate node contains Taxon b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' the tree component below the right reticulate node contains Taxon a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Therefore, we can order the taxa as either c < d < b < a or d < c < b < a, where b must appear before a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For an ordering π = π1π2 · · · πn satisfying the property given in the last paragraph, we label the tree nodes of P using the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Label the network root with the smallest taxon in C0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' π1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Label each parent of the reticulate node ri with the smallest taxon in Ci for every i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let u be a tree node that is not a parent of any reticulate node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In this case, u has two children x and y in the same tree-node com- ponent C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We label u with maxπ(ax, ay), where ax and ay are the smallest taxon be- low x and y in C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (For exam- ple, the tree-node component C0 contains only one such tree node and this node is la- beled with d in Figure 4C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=') As in the case of trees, we can obtain a LTS for each taxon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For the smallest taxon τ of each tree-node component Ci, its LTS is composed of the taxon labels of the tree nodes in the unique path from ri to Leaf τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For the other taxa τ of Ci, there is a unique tree node w that is labeled with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The LTS of τ is composed of the taxon labels of the tree nodes (excluding w) in the path from w to Leaf τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (For example, in Figure 4C, C0 contains Taxa c and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The LTS for c and d are [b, d, a, b] and [a, b], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Proposition 3 Let Ti (1 ≤ i ≤ k) be k trees on X and let P be a tree-child network on X with the smallest HN, compared with those that dis- play all Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For any ordering π of X obtained above and for each taxon τ, if we label the tree nodes of P as described above, the LTS Sτ ob- tained for τ is the SCS of the LTS obtained for τ in the trees Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Moreover, applying the Tree- child Construction algorithm to the obtained supersequences Sτ gives the same network as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The proof of Proposition 3 appears in Sup- plementary document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' By Propositions 2 and 3, we obtain the following exact algorithm for inferring the minimum tree-child network that displays the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Algorithm A Input: Trees T1, T2, · · · , Tk on X, |X| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Define M = ∞ and n string variables S1, S2, · · · , Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each ordering π1π2 · · · πn of X: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Call the Labeling algorithm to label the internal nodes in each Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each taxon πj, compute its LTS sij in each Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Compute the SCS sj of s1j, s2j, · · · , skj for each j < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' If M > �n−1 j=1 |sj|, update M to the length sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' update Sj to sj for each j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Call the Tree-Child Network Construction algorithm to compute a tree-child network P from the strings S1, S2, · · · , Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1 and Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2 of Algorithm A take a linear time of O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Note that the SCS problem is a special case of the multiple sequence align- ment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since the the total length of the (n − 1) LTSs computed in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2 is n − 1 for each Ti, Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 takes a time of O((n − 1)k) at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Step 2 takes a quadratic time of O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Therefore, the worst-case time complexity of 6 Algorithm A is O � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (n − 1)k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2 A Scalable Version Since there are n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' possible orderings of n taxa, Algorithm A is not fast enough for a set of multiple trees with 15 taxa if all the trees do not have any common clusters other than the single- ton cluster and the whole taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Another obstacle to scalability is computing the SCS for the LTS of each taxon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We achieved high scalability by using an ordering sampling and a progressive ap- proach for the SCS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' First, the ordering sampling starts with an ar- bitrary ordering of the taxa and finishes in ⌊n/2⌋ iterative steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Assume that Πm is the set of or- derings obtained in the m-th step (m ≥ 1), which contains at most K orderings (K is a predefined parameter to bound the running time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the (m+ 1) step, for each ordering π = π1π2 · · · πn ∈ Πm, we generate (n−2m+1)(n−2m) orderings by interchanging π2m−1 with πi and interchang- ing π2m with πj for every possible i and j such that i ̸= j, i > 2m and j > 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each new or- dering π′ = π′ 1π′ 2 · · · π′ n, we compute a SCS si of the LTSs of Taxon π′ i in the input trees for each i ≤ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We compute Πm+1 by sampling at most K orderings that have the smallest length sum � 1≤i≤2m |si| from all the generated orderings of the taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Second, different progressive approaches can be used to compute a short common superse- quence for LTSs in each sampling step [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We use the following approach, which had good per- formance for our purposes according to our sim- ulation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A common supersequence of n se- quences is computed in n − 1 iterative steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In each step, a pair of sequences si, sj for which the SCS of si and sj, SCS(si, sj), has the minimum length, over all possible pairs of sequences, is selected and replaced with SCS(si, sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' After the sampling process finishes, we obtain a set Π⌊n/2⌋ of good ordering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' for each ordering, we obtain a short common supersequence of the LTS of a taxon, which might not the shortest one for each taxon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' To further improve the tree- child network solution, we also use the dynamic programming algorithm to recalculate the SCS for the LTS of each taxon w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' each obtained ordering, subject to the 1G memory usage limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We then use whichever is shorter to compute a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 A program for network inference Another technique for improving the scalability is to decompose the input tree set into irreducible sets of trees if the input trees are reducible [2, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Let S be a reducible set of k trees on X, which are ordered as: ⟨T1, T2, · · · Tk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We assume that C1, C2, · · · , Ct are all the maximal common clusters of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We introduce t new taxa yi and let Y = {y1, y2, · · · , yt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' By replacing Ti(Cj) with yj in Ti for each i and j, we obtain a set S′ of k trees T ′ i on Y ∪ � X \\ � ∪t i=1Ci �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In this way, we decompose S into an irreducible tree set S′ = ⟨T ′ 1, T ′ 2, · · · , T ′ k⟩ and t ordered sets of trees S′ i = ⟨T1(Ci), T2(Ci), · · · , Tk(Ci)⟩, 1 ≤ i ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Combining the tree-child networks constructed from S′ and all of S′ i gives tree-child networks that display all the trees of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Our program is named ALTS, an acronym for “Aligning Lineage Taxon Strings”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' It can be downloaded from the Github site https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='com/LX-Zhang/AAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We also developed a program that assigns a weight to each edge of the obtained tree-child network if the input trees are weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The least squares method for estimating edge weights is presented in Section B of the Supplementary Document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In summary, the process of reconstructing a parsimonious tree-child network involves the fol- lowing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (i) Decompose the input tree set S into irreducible tree sets, say S1, S2, · · · , St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (ii) Infer a set Ni of tree-child networks for each Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (iii) Assemble the tree-child networks in N1, N2, · · · , Nt to obtain the networks that dis- play all the trees in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (iv) If the input trees are 7 weighted, the branch weights are estimated for the output tree-child networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4 Validation Experiments We assessed the accuracy and scalability of ALTS on a collection of simulated datasets that were generated using an approach reported in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each k ∈ {20, 30, 40, 50}, a phylogenetic network on k taxa was first generated by simulat- ing speciation and reticulation events backwards in time with the weight ratio of reticulation to speciation ratio being set to 3:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Fifty trees dis- played in the networks were then randomly sam- pled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This process was repeated to generate 2500 trees for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The test tree datasets are avail- able together with the code for ALTS on Zhang’s Github site mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We compared ALTS with two heuristic net- work inference programs: PRINs [22], which in- fers an arbitrary phylogenetic network, and van Iersel et al.’s method [28], which infers a tree- child network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We tested the three methods on 50 sets of trees on 20 and 30 taxa, each con- taining 10 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Van Iersel et al.’s program is a parallel program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' It could run successfully only on 44 (of 50) tree sets in the 20-taxon case and 27 (of 50) tree sets in the 30-taxon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' It was aborted for the remaining datasets after 24 hours of clock time (or about 1000 CPU hours) had elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' To assess the scalability of ALTS, we further ran it on 100 datasets, each containing 50 trees on 40 or 50 taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' PRINs finished on five 50-taxon 50-tree datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Van Iersel et al.’s method did not run successfully on these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1 The optimality evaluation ALTS computed the same tree-child HN as van Iersel et al.’s method on all but three datasets where the latter ran successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The HN of the tree-child networks inferred with ALTS was one more than that inferred with the latter on two 20-taxon 10-tree datasets and three more than that with the latter on one 30-taxon 10-tree dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Moreover, Van Iersel et al.’s method only outputted a tree-child network, whereas ALTS computed multiple tree-child networks with the same HN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' PRINs ran successfully on all but one dataset in the 20-taxon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In theory, the HN is in- herently equal to or less than the HN of the op- timal tree-child networks for every tree set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the 20-taxon 10-tree case, the tree-child HN in- ferred with ALTS was equal to that inferred with PRINs on 20 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The 29 discrepancy cases are summarised in the row one of Table 1 Table 1: Summary of the HN discrepancy be- tween ALTS and PRINs in 20-taxon and 30- taxon datasets each containing 10 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' HNALTS − HNPRINs Date type 1 0 1 2 3 4 20 taxa 20 11 9 6 3 30 taxa 1 5 13 14 16 1 The HN discrepancies between the two pro- grams in the 30-taxon case are summarised in the row two of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Like the 20-taxon case, the difference in HN was also at most four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The tree-child HN inferred by ALTS was even one less than the HN inferred by PRINs on one dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We also noted that the difference in HN of- ten occurred when the HNs inferred by the two methods were greater than 15, when van Iersel’s method could not run successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In summary, ALTS is almost as accurate as van Iersel et al.’s method in terms of minimizing network HN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The comparison between ALTS and PRINs indicated that the tree-child HN is rather close to the HN for multiple trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='2 The scalability evaluation The wall-clock time of the three methods on 100 datasets, each having 10 trees on 20 or 30 taxa, are summarized in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the 20-taxa 10- tree case, the HN inferred by PRINs ranged from 8 1 10 100 1000 10000 100000 PRINs van Iersel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ALTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='1 1 10 100 1000 10000 Prin van Iersel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ALTS Run Time (log scale) Fifty 20-taxon 10-tree Datasets Run Time (log scale) Fifty 30-taxon 10-tree Sets 1 10 100 1000 10000 PRINs AAST Fifty 40-taxon 10-tree Sets Run Time (log scale) Figure 5: Run time (in seconds) of the three methods on 100 datasets, each containing 10 trees on 20 or 30 taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, the datasets are sorted in the increasing order according to the HN output from PRINs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Missing data points for van Iersel et al.’s method are explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 5 to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The run time of ALTS ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='09 s to 25 m 14 s (with the mean being 2 m 21 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On the 49 (out of 50) 20-taxa 10-tree datasets on which PRINs finished, its run time ranges from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='94 s to 17 m 19 s (with the mean being 2 m 58 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ALTS was faster than PRINs on 35 tree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On average, PRINs and ALTS were comparable in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On the 44 20-taxa 10-tree datasets on which van Iersel et al.’s method finished, its run time ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='07 s to 82 m 22 s (with the mean being 13 m 3 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Van Iersel et al.’s method ran faster than ALTS on 26 datasets where the HN inferred by PRINs was less than 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' One reason for this is probably that the former is a parallel program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' However, ALTS was faster than van Iersel et al.’s method on the remaining 18 tree 1 10 100 1000 10000 100000 PRINs ALTS 1 10 100 1000 10000 100000 PRINs ALTS Fifty 50-taxon 50-Tree Sets Run Time (log scale) Fifty 40-taxon 50-Tree Sets Run Time (log scale) Figure 6: The run time (in seconds) of ALTS on 100 datasets, each containing 50 trees on 40 or 50 taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The datasets are sorted in the increas- ing order according to the HN of the tree-child networks inferred by ALTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' sets where the HN inferred by PRINs was 12 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the 30-taxon 10-tree case, the HN of the solution from PRINs ranged from 8 to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' As shown in Figure 5, ALTS was faster than PRINs on each of the 50 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Van Iersel et al.’s method finished on 31 (out of 50) datasets, for which the HN of the solution obtained with PRINs was 15 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ALTS was faster on 23 datasets where the HN of the solution was larger than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Van Iersel et al.’s method was faster than ALTS on the remaining eight datasets where the HN ranged from 8 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On average, ALTS was 24 and 53 times faster than PRINs and the van Iersel et al.’s method, respec- tively, in the 30-taxon 10-tree case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Lastly, ALTS was also able to infer tree-child networks on 100 datasets, each containing 50 trees with 40 or 50 taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the 40-taxon 50-tree case, the tree-child HN inferred by ALTS ranged from 9 to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The run time of ALTS ranged from 9 Simplified network 1 Network 1 Network 2 Simplified network 2 20 trees Figure 7: The box and whisker plots for the dissimilarity scores for the original network and that inferred by ALTS in four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In each plot, the four bars from left to right summarize the dissimilarity scores for the original network and 10 networks inferred from 20-, 30-, 40-, and 50-tree sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The four networks are presented in Figure S3–S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3 s to 31 m 52 s (with the mean being 7 m 14 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On contrast, PRINs finished on 28 tree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Its run time ranged from 3 m 19 s to 15 h 34 m 52 s (with the mean being 3 h 49 m 46 s) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In the 50-taxon 50-tree case, the tree-child HN inferred by ALTS ranged from 10 to 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The run time of ALTS ranged from 2 s to 45 m 12 s (with the mean being 9 m 24 s) (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In contrast, van Lersel et al’s method could not finish on any irreducible set of 50 trees on 50 taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' PRINs finished on five tree sets in 2 h 25 m on average (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Taken together, these results suggest that ALTS has high scalability and is fast enough to infer a tree-child network on an irreducible tree set with a size comparable with those of the cur- rent focus of biological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 The accuracy evaluation Evaluating the accuracy of ALTS (and the other two methods) is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The ran- dom networks that were used to generate the tree sets used in the last two subsections are not tree- child networks and have frequently a large num- ber of deep reticulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' On the other hand, by the principle of parsimony, the net- works inferred by the three programs contain far fewer reticulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' As such, we assessed the accuracy of ALTS by considering the sym- metric difference of the set of taxa clusters in the original networks and the set of cluster in the network inferred by ALTS [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Here, a clus- ter in a network consists of all taxa below a tree node in that network, as the cluster of a retic- ulate node x is always equal to the cluster of its child if x has only one child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Precisely, for two phylogenetic networks N1 and N2 over X, we use C(Ni) to denote the multiset of clusters appearing in Ni for i = 1, 2, and define their dissimilarity score s(N1, N2) as the Jaccard dis- tance of C(N1) and C(N2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' s(N1, N2) = 1 − |C(N1) ∩ C(N2)|/|C(N1) ∪ C(N2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We considered two simulated networks con- taining 16 binary reticulations (network 1, Fig- ure S3) and 19 binary reticulations (network 2, Figure S5) and their simplified version (Fig- ure S4 and S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The two networks were produced using the same simulation method just with a low rate of reticulation events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' the two simpli- fied networks were obtained by merging a retic- ulate node and its child if the reticulate node has a unique child and the child is also a reticu- lation node, which have 9 and 10 multiple reticu- lations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each network and each k = 20, 30, 40, 50, we generated 10 k-tree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' In total, we used 160 tree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For each tree set, we inferred a network using ALTS and com- puted the dissimilarity score for it and the origi- nal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The dissimilarity score analyses are summarised in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Network 1 (and its simplified version) contains less reticulation events than Network 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We had slight better reconstruction accuracy for Net- work 1 than Network 2 (mean dissimilarity score range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='45] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='65], Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Also, the reconstruction from the trees sampled from each network was not significantly better than that from its simplified version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Given that 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='60 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='65 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='40 X X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='30all four networks can contain as many as 217 trees, the results suggest that 50 trees are far fewer than enough for accurate reconstruction of both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since we could not run Iersel et al’s program on the most of tree sets, we were unable to assess its accuracy for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 5 A Phylogenetic Network for Hominin Relationships Hominins’ phylogenetic relationships are not fully established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' As an application of ALTS, we reconstructed a network model for hominin species using 10 phylogenetic trees derived from the Bayesian analysis of the morphological data of hominin evolution presented in [7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' To choose 10 phylogenetic trees, we grouped the posterior trees into five clusters using the dis- tance metric and approach described in previ- ous work [17, 16], using Ward clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We chose two trees from each of the five clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Due to the nature of the morphological data, the trees were discordant, and no single tree captures a highly-supported pattern of ancestry among the taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This motivates using a network to il- lustrate the ancestral relationships among these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The resulting network model contains 12 retic- ulation events with the HN being 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The top tree-node component contains the two out- group species G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' gorilla and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' troglodytes, as well as the oldest hominin species, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' tchaden- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The three earliest members of the genus Homo ( African H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' erectus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' rudolfensis and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' habilis), together with Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' africanus, ap- pear in a tree-node component, whereas four re- cent members of the genus Homo (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' heidelber- gensis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' neanderthalensis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' sapiens and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' naledi ) compose another tree-node component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The three members of the genus Paranthropus, together with Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' garhi, compose a tree-node component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The model also reflects the high un- certainty about the phylogenetic position of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' floresiensis, who lived in the island of Flores, In- donesia [3, 5, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This network provides an illustration of the performance of ALTS on hominin morphological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We find that its HN is unexpectedly high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Since the evolutionary time of hominin species is relatively short, some discrepancies in the 10 trees are perhaps a result of incomplete lineage sorting (ILS) [31], (with impacts on morphology, in order that they are implicitly detected in these data), or of convergent evolution, ambiguity in the morphological data, or other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' With- out genetic data, we cannot assess the extent to which ILS or other factors affects the phyloge- netic trees and consequently this network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 6 Conclusions We have presented ALTS, a fast and scalable method for inferring tree-child networks from multiple trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' It is based on a novel algorith- mic innovation that reduces the minimum tree- child network problem to computing the SCS of the LTSs obtained w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' a predefined ordering on the taxa in the input trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Another contri- bution is an algorithm for assigning weights to the tree edges of the reconstructed tree-child net- work if the input trees are weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Our work makes network reconstruction more feasible in the study of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' The accuracy analyses in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='3 suggest that 50 trees are likely not enough for accurately inferring a phylogenetic network model that has 10 or more reticulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Therefore, a program that can process over hundred trees is definitely wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We remark that ALTS can be made even more scalable by distributing the computing tasks for taxon orderings into a large number of processors using the distributed com- puting programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' This is because the com- puting tasks for different orderings are indepen- dent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We will further investigate how to improve the accuracy of ALTS by incorporating the genomic sequences of the taxa or/and ILS into network 11 4 5 7 6 20 21 22 23 3 9 17 11 16 13 15 14 12 24 1 2 18 10 19 8 Figure 8: A network model of hominin relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 1: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' gorilla;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 2: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' troglodytes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 3: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' floresiensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 4: Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' ramidus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 5: Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' anamensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 6: Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' afarensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 7: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' platyops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 8: Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' africanus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 9: Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' sediba;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 10: African H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' erectus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 11: Asian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' erectus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 12: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' heidelbergensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 13: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' neanderthalensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 14: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' sapiens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 15: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' naledi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 16: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' antecessor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 17: Georgian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' erectus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 18: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' rudolfensis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 19: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' habilis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 20: Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' garhi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 21: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' robustus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 22: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' boisei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 23: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' aethiopicus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 24: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' tchadensis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Acknowledgements We thank Cedric Chauve and Aniket Mane for discussion in the beginning of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' We also thank anonymous reviewers for con- structive comments on an earlier version of our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Zhang was partly supported by Singapore MOE Tier 1 grant R-146-000-318- 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Wu was partly supported by U.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' 14 Supplementary Document A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Propositions and their proof A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' Total ordering, trees and child-tree networks Let X be a set of taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' A (total) ordering R on X is a binary relation on X such that (i) R is anti-symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' if x1Rx2, then x2 ̸R x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (ii) R is transitive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=', if x1Rx2 and x2Rx3, then x1Rx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' (iii) For any x1, x2, x1Rx2 or x2Rx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfEfoE/content/2301.00992v1.pdf'} +page_content=' For convention, we write x 𝜅) and are upper bounded by their +respective variance curves at the EP. Moreover, +log4⟨∆𝑝𝑠,𝐿 +2 ⟩ always grows monotonically above the +vacuum-noise level; while +log4⟨∆𝑞𝑖,0 +2 ⟩ invariably +exhibits quantum squeezing, and the larger 𝛾/𝜅 the +larger the squeezing and 𝐿. What’s more, the completely +incompatible nature, quantum versus classical, of the + +same physical observable 𝑞𝑖(0) before and after the PT +phase transition renders our system a unique candidate to +study the transition between these two different worlds, +whose boundary is physically defined by the EP curve. +Contrarily, since {𝑝𝑖(0), 𝑞𝑠(𝐿)} are decoupled from +{𝑞𝑖(0), 𝑝𝑠(𝐿)}, their variances fluctuate periodically, akin +to the TMSV case. As shown in Figs. 2(c) and (d), though +notably affected by 𝑄𝑠 and 𝛾 , log4⟨∆𝑞𝑠,𝐿 +2 ⟩ resembles +the regular quadrature squeezing, but not log4⟨∆𝑝𝑖,0 +2 ⟩. +From the above analysis, we learned that for the same +single-mode quadrature, PT results in a nontrivial +fundamental transition from quantum to classical when +the non-Hermitian parameter 𝛾 oversteps a threshold. +One may wonder whether this exotic phenomenon can +also take place in a two-mode quadrature measurement. +The answer is affirmative. To see how this works, we pay +attention +to +𝑑1 = [𝑞𝑖(0) + 𝑞𝑠(𝐿)]/√2 and +𝑑2 = +[𝑝𝑖(0) + 𝑝𝑠(𝐿)]/√2, which satisfy [𝑑1, 𝑑2] = 𝑖/2. For +the vacuum input, it is easy to check that their variances +are simply the sum of the single-mode ones (5a)–(5d), +〈∆𝑑1 +2〉 = +⟨∆𝑞𝑖,0 +2 ⟩+⟨∆𝑞𝑠,𝐿 +2 ⟩ +2 +, 〈∆𝑑2 +2〉 = +⟨∆𝑝𝑖,0 +2 ⟩+⟨∆𝑝𝑠,𝐿 +2 ⟩ +2 + . (6) +Based on Figs. 2(b) and (d), 〈∆𝑑2 +2〉 is expected to be +distributed above the vacuum noise all the time. +Moreover, because ⟨∆𝑝𝑠,𝐿 +2 ⟩ and ⟨∆𝑝𝑖,0 +2 ⟩ have different +fluctuation periods before the phase transition, we +envision that 〈∆𝑑2 +2〉 will exhibit interleaved dual periodic +oscillations but reduce to a single period after the phase +breaking. Though the situation becomes somewhat subtle +for 〈∆𝑑1 +2〉 , its layout can be deduced similarly by +compromising Figs. 2(a) and (c). To be specific, in the +PT-phase unbroken region, it is a double-cycle growth +fluctuation staggered on top of the vacuum noise (except +the very short distance case). When PT symmetry +spontaneously breaks down, counterintuitively, the +single-period oscillating 〈∆𝑑1 +2〉 will always return certain +squeezing at some effective distances, and these distances +will be extended for a bigger 𝛾/𝜅. Same as 𝑞𝑖(0), 𝑑1 +can serve as another physical probe to visualize the +quantum-to-classical transition induced by quadrature PT +symmetry, too, with the boundary defined by the EP +curve. All these statements excellently agree with our +numerical simulations given in Figs. 3(a) and (b). +RISM.—Other than homodyne detection, there is one +additional means to explore quadrature PT, the so-called +relative intensity squeezing measurement (RISM). +Traditionally, this method enables the shot-noise of one +beam to be measured and subtracted from the other so as +to attain lower-noise differential measurement of a signal +of interest. To this end, we begin with our own relative- +intensity +operator, +𝑁𝑖,0 − 𝑁𝑠,𝐿 = 𝑎𝑖 +†(0)𝑎𝑖(0) − +𝑎𝑠 +†(𝐿)𝑎𝑠(𝐿) . The degree of squeezing is then +characterized by the noise figure (NF), which is +determined +by +the +relative-intensity +variance. +Mathematically, it takes the form [38] +Fig. 2. PT-manifested log4⟨∆𝑞𝑖,0 +2 ⟩ (a) and log4⟨∆𝑝𝑠,𝐿 +2 ⟩ +(b) in the presence of quantum Langevin noise. Non-PT- +symmetric but loss-noise-mediated log4⟨∆𝑞𝑠,𝐿 +2 ⟩ (c) and +log4⟨∆𝑝𝑖,0 +2 ⟩ (d). As the references, the black solid and +dashed lines represent the regular TMSV (𝛾/𝜅 = 0) and +vacuum noise, respectively. + +Fig. 3. PT-symmetric log4⟨∆𝑑1 +2⟩ (a) and log4⟨∆𝑑2 +2⟩ (b) +with account of quantum noise. Again, as the references, +the black solid and dashed curves are respectively the +ideal TMSV (𝛾/𝜅 = 0) and vacuum noise. + + +10 +10 +(a) +(b) +8 +8 +人人人 +人人人人人人 +log4(Aqi,o) +4 +2 +2 +0 +0 +2 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +2KL +2KL +8 +25 +(c) +(d) +6 +20 +vacuumnoise +Y/x=O +-y/x=1 +0.5 +y/x=0.6.y/x=1.4 +2 +1.5 +log4( +10 +5 +0 +2 +5 +10 +15 +20 +0 +5 +10 +15 +20 +2KL +2kL10 +25 +(a) 0.8 +(b) +8 +从人人 +20 +vacuum noise +6 +-y/x=0y/K=1 +《pv) +15 +人人人 +Y/x = 0.6—/x = 1.4 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +2K +2KLNF = +Var[𝑁𝑖,0−𝑁𝑠,𝐿] +〈𝑁𝑖(0)〉+〈𝑁𝑠(𝐿)〉 . (7) +Here, the average photon numbers are computed by +plugging Eqs. (4a) and (4b) to 〈𝑁𝑖,0〉 = ⟨𝑞𝑖 +2(0)⟩ + +⟨𝑝𝑖 +2(0)⟩ − 1/2 and 〈𝑁𝑠,𝐿〉 = ⟨𝑞𝑠2(𝐿)⟩ + ⟨𝑝𝑠2(𝐿)⟩ − 1/2 . +In stark contrast to the quadrature variances discussed +earlier, the NF, while bringing about some alike +characteristics, clearly reveals some quite opposite +peculiarities. For 𝛾/𝜅 ≥ 1, as demonstrated in Fig. 4(a), +in addition to the incremental single-period fluctuation +log10(NF≥0 + 1) grows along with the increment of +2𝜅𝐿, and the larger γ/κ is, the noisier it is. From the plot, +it is not difficult to conclude that in the PT-phase broken +region, NF is essentially occupied by the noise anti- +squeezing. However, NF behaves highly complex as +𝛾/𝜅 < 1. Although it is still an interleaved double-period +oscillation within this range, the EP curve is no longer the +partition to separate the classical and quantum +fluctuations. In line with the numerical simulations, we +find that quantum squeezing materializes when 𝛾/𝜅 < +0.52. Some representative examples are depicted in Fig. +4(b) by plotting −log10(NF<0 + 1) for different γ/κ. +Their comparison suggests that the smaller the value of +γ/κ, the more pronounced the achievable squeezing over +a longer distance 2𝜅𝐿. As a matter of fact, the RISM +obviously supplies certain sharp signatures unreachable to +the homodyne detection, regardless of the two highly +unbalanced channels. +Before proceeding, a few remarks are ready here. First, +even in the presence of Langevin noise, utilizing PSA +instead of PIA is practicable to accomplish quantum +optical PT under fair sampling measurement. Second, +contrary to PIA, PSA arouses the unusual quadrature PT +and licenses the singular quantum-to-classical transition +accompanied by the PT phase transition. Last but not +least, quadrature PT sheds new light on protecting +continuous-variable (CV) qubits from decoherence in +inevitable lossy transmission, a long-standing conundrum +for various CV-based quantum technologies [39]. +Quantum sensing.—Being a discipline of practical +application, quantum sensing [40-42] exploits quantum +properties, effects, or systems to fulfill high-resolution +and super-sensitive measurements of physical parameters +over the similar measurements performed within a +classical framework. For this, quantum squeezing has +long been recognized as one of the indispensable +nonclassical resources for ultra-precision estimations. +Among them, one far-reaching example is its recent +adoption by the Laser Interferometer Gravitational-Wave +Observatory (LIGO) for gravitational wave detection. +Nevertheless, the inevitable propagation loss often +degrades the available squeezing and compromises the +promised sensitivity. We note that in recent non- +Hermitian studies, the abrupt change near EP has been +capitalized for enhanced sensing in classical settings [43- +47]. Yet, its extension to the quantum level turns out to be +problematic because of quantum noise [48]. To avoid +such noise, one usually resorts to either ideal anti-PT +systems or post-selection measurement [25,34,35]. +Unlike these studies, here we directly confront Langevin +noise and explore the opportunity of quadrature PT in +quantum sensing under fair sampling measurement. We +are particularly interested to know whether the system +could have any advantage in improving sensitivity. As +shown below, the PT-quadrature observables can yield +the best performance of classical sensing before the phase +transition but departing far from the EP; while the non- +PT-quadrature observables are capable of optimal +quantum sensing by noise-mediated squeezing for 𝛾/𝜅 +less than 1. This distinguishes our work from the previous +anti-PT-, squeezing-, or EP-based proposals. Our analysis +is carried out by estimating 𝜅 (or 𝛾) and comparing the +achievable precision with the quantum Cramér-Rao +bound (set by the quantum Fisher information of the +Fig. 4. (a) PT-regulated noise figure (NF) via relative +intensity squeezing measurement. (b) Representative +examples of quantum PT-symmetric NF. + + +40 ++ +(a) +log10(NFzo+ +10 +(b) +30 +-Y/x=0.8 +-/x=1 +20 +-Y/x=0.2 +-Y/x=2 +10 ++1) +0 +Y/x=0.2 ++ +07 +107 +Y/x=0.1 +AN +INF. +/x=0.05 +10 +log10l +5 +-l0g10() +0 +10 +15 +20 +25 +30 +35 +0 +5 +10 +15 +20 +25 +30 +35 +2kL +2kLquantum state). +Suppose that the two bosonic modes are initially +prepared in a coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩. An optimal +homodyne measurement is then implemented on, say, +𝑞𝑖(0)after an evolution distance 𝐿. Using Eq. (4a), one +can readily have its mean value and variance, +⟨𝑞𝑖(0)⟩ = +𝑐𝑜𝑠 𝜖⟨𝑞𝑖(𝐿)⟩−𝑠𝑖𝑛(𝛽𝐿)⟨𝑝𝑠(0)⟩ +𝑐𝑜𝑠(𝛽𝐿−𝜖) + , (8) +⟨∆𝑞𝑖,0 +2 ⟩ = +3+𝑤[2𝛾𝐿−tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖 +8cos2(𝛽𝐿−𝜖) + , (9) +where 𝑤 = 2𝑛𝑡ℎ + 1 and 𝑛𝑡ℎ is the average thermal +boson number. From Eqs. (8) and (9), it is clear that the +Langevin noise shifts the peaks of ⟨∆𝑞𝑖,0 +2 ⟩ away from the +troughs of ⟨𝑞𝑖(0)⟩, so that they do not coincide at all. To +ease the subsequent derivations, hereafter we assume +𝛼𝑖 = 𝑖𝛼𝑠∗ = √2𝛼𝑒𝑖𝜋/4. The estimation precision relies on +measuring the change of ⟨𝑞𝑖(0)⟩ due to a tiny +perturbation 𝛿𝜅 on a preset 𝜅 . This alternatively +suggests to examine the system response to ⟨𝑞𝑖(0)⟩ for a +small variation 𝛿𝜅 around 𝜅 . We thus define the +susceptibility to capture such a response, +𝜒𝜅 +𝑞𝑖(0) ≡ 𝜕⟨𝑞𝑖(0)⟩ +𝜕𝜅 += +𝛼{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]} +2𝛽cos2(𝛽𝐿−𝜖) + . (10) +When 𝜅 → 𝛾 or 𝛽 → 0 , 𝜒𝜅 +𝑞𝑖(0) → 𝛼𝐿(3 + 𝜅2𝐿2)/ +[3(1 + 𝜅𝐿)2] is a curvetureless constant, implying the +loss of the sensing ability in the EP vicinity for the chosen +observable. On the other hand, the κ-estimation is jointly +determined by the variance and susceptibility, ∆𝜅,𝑞𝑖,0 +2 += +⟨∆𝑞𝑖,0 +2 ⟩/ [𝜒𝜅 +𝑞𝑖(0)] +2 +, whose inverse dictates the accuracy, +∆𝜅,𝑞𝑖,0 +−2 += +2𝛼2{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]}2 +𝛽2cos2(𝛽𝐿−𝜖){3+𝑤[2𝛾𝐿− tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖} .(11) +The sensing fulfillment is to compare ∆𝜅,𝑞𝑖,0 +−2 +,0with the +quantum Fisher information, 𝐹𝜅, which sets the ultimate +precision, i.e., the lower quantum Cram´ er-Rao bound for +any optimal measurement, 𝐹𝜅 ≥ ∆𝜅,𝑞𝑖,0 +−2 +. Fκcan be +accordingly derived from +𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − + ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩) . (12) +for the final state |𝜓𝑓⟩ of the system in the Schrödinger +representation. +As +detailed +in +Supplementary +Information, one can perform similar sensing evaluations +to the rest three quadratures. Basing on the calculations +shown in Figs. 5(a) and (b), we note that in the current +arrangement, 𝑞𝑖(0) and 𝑝𝑠(𝐿) can only permit optimal +classical sensing for a moderate medium length in the +phase-unbroken regime but away from the EP, as revealed +by the ratios of ∆𝜅,𝑞𝑖,0 +−2 +/𝐹𝜅 and ∆𝜅,𝑝𝑠,𝐿 +−2 +/𝐹𝜅 . On the +contrary, 𝑞𝑠(𝐿) and 𝑝𝑖(0) behave distinctively in the +sense that, despite the impacts of the photon loss and +Langevin noise, they are still able to offer optimal +quantum sensing over a relatively longer distance for +smaller 𝛾/𝜅, as sketched in Figs. 5(c) and (d). +In short, fundamentally different from all previous +research, the PSA-induced quadrature PT enables a +unique way to observe a quantum-to-classical transition +for a physical observable at the breakdown of symmetry. +Such quadrature PT radically reshapes the dynamics of +two-mode squeezing with a striking phase transition +never seen before. Unfortunately, optical loss and +Langevin noise hinder the PT quadrature pair to confer a +quantum advantage in improving sensitivity, although the +non-PT pair can still offer optimal quantum sensing. +Besides of nonlinear wave mixing, our model can be also +Fig. 5. Quantum sensing in quadrature PT symmetry +characterized by the ratios of ∆𝜅,𝑞𝑖,0 +−2 + (a), ∆𝜅,𝑝𝑠,𝐿 +−2 + (b), +∆𝜅,𝑞𝑠,𝐿 +−2 + (c), and ∆𝜅,𝑝𝑖,0 +−2 + (d) to the quantum Fisher +information 𝐹𝜅 for the parameters {𝛼 = 2, 𝛾 = 0.2, 𝜅 = +1 } (blue), {2,1,1} (red), and {2,2,1} (orange), +respectively. + + +0.25 +0.25 +(a) +(b) +0.2 +X10-3 +0.2 +×10-3 +10 +10 +0.15 +-Y/x=0.2 +6 +Y/x=1 +6 +0.1 +4 +0.1 +Y/x=2 +4 +2 +0.05 +0.05 +0 +5 +1015 +202530 +0 +5 +1015202530 +0 +0 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +2KL +2KL +0.25 +0.25 +(c) +(d) +X10-3 +X10-3 +0.2 +10 +0.2 +10 +8 +0.15 +6 +6 +2 +4 +AK +0.1 +N +0.05 +0 +0 +51015202530 +0.05 +0 +5 +1015202530 +0 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +2KL +2KLachieved in other platforms such as superconducting +circuits. Most importantly, our work forges a new avenue +to explore the long-sought, nontrivial quantum-to- +classical transition utilizing non-Hermitian physics. + +Acknowledgements.—This work was supported by NSF +1806519 and NSF EFMA-1741693. X.J. acknowledges +the support by the National Key R&D Program of China +(2021YF A1400803). D.L. was supported by the Nature +Science +Foundation +of +Guangdong +Province +(2019A1515011401). + +AUTHOR CONTRIBUTIONS.—J.W. conceived the +theoretical scheme and supervised the whole project with +the help of D.L. and X.J. W.W., supervised by J.W., +carried out the whole calculations with the assistance of +Y.Z. and S.V.G. All authors contributed to the discussions +and writing of the manuscript. + +* dmliu@scnu.edu.cn +† jxs@nju.edu.cn +‡ jianming.wen@kennesaw.edu +[1] Bender, C. M. & Boettcher, S. Real spectra in non- +Hermitian Hamiltonians having PT symmetry. Phys. +Rev. Lett. 80, 5243-5246 (1998). +[2] El-Ganainy, R., Makris, K. 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Derivation of the Heisenberg-Langevin equations +In our previous work [1], we have theoretically proved that a forward parametric optical process +may lead to anti-PT symmetry while a backward parametric optical process can result in PT symmetry. +For this reason, as schematic in Fig. 1 in the main text [2], we are interested in a backward nonlinear +parametric optical process such as backward four-wave mixing, where the two counter-propagating +parametric modes, idler and signal, respectively experience balanced phase-sensitive linear quantum +amplification (PSA) and attenuation in their own channels within the medium of length 𝐿. For such +an open system, the evolution of the paired idler and signal field operators is effectively determined +by a non-Hermitian Hamiltonian, + +𝐻 = 𝑖 +ℏ𝑔 +2 (𝑎𝑖 +†2 − 𝑎𝑖 +2) − 𝑖ℏ𝛾𝑎𝑠 +†𝑎𝑠 + ℏ𝜅(𝑎𝑖 +†𝑎𝑠 +† + 𝑎𝑖𝑎𝑠), +(S1.1) +where 𝑔 and 𝛾 respectively denote the PSA rate and loss rate. From Eq. (S1.1), one can readily +obtain the Heisenberg equations of the idler-signal field operators, + +𝑖ℏ +𝜕𝑎𝑖 +𝜕(−𝑧) = [𝑎𝑖, 𝐻], +(S1.2a) + +𝑖ℏ +𝜕𝑎𝑠 +𝜕𝑧 = [𝑎𝑠, 𝐻]. +(S1.2b) +Thanks to the noiseless amplification empowered by the PSA, the idler dynamics is not subject to the +additive noise and the commutation relation can be always satisfied throughout the whole process. +However, this is not true for the lossy signal propagation. To restore the commutation relation, one has +to introduce the quantum Langevin noise in the Heisenberg equation of the signal field operator. In +this way, we arrive at the following coupled Heisenberg-Langevin equations for the system of interest, + +𝜕𝑎𝑖 +𝜕𝑧 = 𝑔𝑎𝑖 +† + 𝑖𝜅𝑎𝑠 +†, +(S1.3a) + +𝜕𝑎𝑠 +𝜕𝑧 = −𝛾𝑎𝑠 − 𝑖𝜅𝑎𝑖 +† + 𝑓𝑠. +(S1.3b) +Though Eqs. (S1.3a) and (S1.3b) seem to have nothing to do with PT symmetry at first glance, as +pointed out in the main text, the hidden PT symmetry arises if transforming both equations into the +dynamics of the corresponding quadrature operators, 𝑞𝑗 = (𝑎𝑗 +† + 𝑎𝑗)/2 and 𝑝𝑗 = 𝑖(𝑎𝑗 +† − 𝑎𝑗)/2 +(𝑗 = 𝑖, 𝑠) with [𝑞𝑗, 𝑝𝑗] = 𝑖/2. For simplicity, we concentrate on the case of the balanced PSA and loss, +𝑔 = 𝛾. With these preparations, one can easily attain the following sets of the coupled-quadrature +equations + +𝑑 +𝑑𝑧 [𝑞𝑖 +𝑝𝑠] = [ 𝛾 +𝜅 +−𝜅 +−𝛾] [𝑞𝑖 +𝑝𝑠] + [0 +𝑃𝑠], +(S1.4a) + +𝑑 +𝑑𝑧 [𝑝𝑖 +𝑞𝑠] = [−𝛾 +𝜅 +−𝜅 +−𝛾] [𝑝𝑖 +𝑞𝑠] + [ 0 +𝑄𝑠], +(S1.4b) +with 𝑃𝑠 = 𝑖(𝑓𝑠 +† − 𝑓𝑠)/2 and 𝑄𝑠 = (𝑓𝑠 +† + 𝑓𝑠)/2 being the Langevin-noise quadrature operators. +From Eq. (S1.4a), one can derive the effective Hamiltonian matrix for the quadrature pair (𝑞𝑖, 𝑝𝑠), +which reads + +𝐻(𝑞𝑖,𝑝𝑠) = [ 𝑖𝛾 +𝑖𝜅 +−𝑖𝜅 +−𝑖𝛾] . +(S1.5) +It is straightforward to show that 𝐻(𝑞𝑖,𝑝𝑠) is indeed PT-symmetric, because it satisfies 𝑃𝑇𝐻(𝑞𝑖,𝑝𝑠) = +𝐻(𝑞𝑖,𝑝𝑠)𝑃𝑇 for the combined PT operation with the parity operator being 𝑃 = [0 +1 +1 +0] and the time- +reversal operator assuming the complex conjugation. For this reason, we call (𝑞𝑖, 𝑝𝑠) the PT- + +2 + +quadrature pair. This is in a sharp contrast to the effective Hamiltonian matrix 𝐻(𝑝𝑖,𝑞𝑠) = [−𝑖𝛾 +𝑖𝜅 +−𝑖𝜅 +−𝑖𝛾] +in Eq. (S1.4b) for the other conjugate quadrature pair (𝑝𝑖, 𝑞𝑠), which is apparently irrelevant to PT +symmetry. The two eigenvalues of 𝐻(𝑞𝑖,𝑝𝑠) are 𝛽± = ±√𝜅2 − 𝛾2. Akin to the classical PT symmetry, +𝛾 +𝜅 < 1 corresponds to the quadrature PT-phase unbroken regime while for +𝛾 +𝜅 > 1 , quadrature PT +symmetry spontaneously breaks down. The quadrature PT phase transition occurs at the singular or +exceptional point (EP), +𝛾 +𝜅 = 1. In terms of the initial boundary conditions, the general solutions of Eqs. +(S1.4a) and (S1.4b) are readily found to be + +[𝑞𝑖(0) +𝑝𝑠(𝐿)] = sec(𝛽𝐿 − 𝜖) [ +cos 𝜖 +−sin(𝛽𝐿) +−sin(𝛽𝐿) +cos 𝜖 +] [𝑞𝑖(𝐿) +𝑝𝑠(0)] ++ sec(𝛽𝐿 − 𝜖) ∫ 𝑑𝑧𝑃𝑠(𝑧) +𝐿 +0 +[−sin(𝛽(𝐿 − 𝑧)) +cos(𝛽𝑧 − 𝜖) ], +(S1.6a) + +[𝑝𝑖(0) +𝑞𝑠(𝐿)] = sec(𝜅𝐿) [ +𝑒𝛾𝐿 +−sin(𝜅𝐿) +−sin(𝜅𝐿) +𝑒−𝛾𝐿 +] [𝑝𝑖(𝐿) +𝑞𝑠(0)] ++ sec(𝜅𝐿) ∫ 𝑑𝑧𝑄𝑠(𝑧) [−𝑒𝛾𝑧sin(𝜅(𝐿 − 𝑧)) +𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧) +] +𝐿 +0 +, +(S1.6b) +with 𝜖 = arctan ( +𝛾 +𝛽). It is not difficult to prove that the dynamical solutions (S1.6a) and (S1.6b) well +maintain the commutation relations at all times, + +[𝑞𝑖(0), 𝑝𝑖(0)] = +𝑒𝛾𝐿cos 𝜖 +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑖(𝐿), 𝑝𝑖(𝐿)] ++ +sin(𝛽𝐿)sin(𝜅𝐿) +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑠(0), 𝑞𝑠(0)] ++ ∫ 𝑑𝑧 +𝐿 +0 +𝑒𝛾𝑧sin(𝛽(𝐿 − 𝑧))sin(𝜅(𝐿 − 𝑧)) +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) +[𝑃𝑠, 𝑄𝑠] = 𝑖 +2, +(S1.7a) + +[𝑞𝑠(𝐿), 𝑝𝑠(𝐿)] = +sin(𝛽𝐿)sin(𝜅𝐿) +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑖(𝐿), 𝑞𝑖(𝐿)] ++ +𝑒−𝛾𝐿 cos 𝜖 +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑠(0), 𝑝𝑠(0)] ++ ∫ 𝑑𝑧 +𝐿 +0 +𝑒𝛾(𝑧−𝐿)cos(𝛽𝑧 − 𝜖)cos(𝜅𝑧) +cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) +[𝑄𝑠, 𝑃𝑠] = 𝑖 +2. +(S1.7b) +Note that the quantum Langevin noise of zero mean satisfies 〈𝑓𝑠(𝑧)𝑓𝑠 +†(𝑧′)〉 = 2𝛾𝛿(𝑧 − 𝑧′) and +〈𝑓𝑠 +†(𝑧)𝑓𝑠(𝑧′)〉 = 0 . The quantumness of quadrature PT symmetry can be further approached by +analyzing the variances (or noise fluctuations) of 𝑞𝑗 and 𝑝𝑗 for the vacuum input state. After some +algebra, we have reached the following important results: + +⟨∆𝑞𝑖,0 +2 ⟩ = ⟨𝑞𝑖 +2(0)⟩ − ⟨𝑞𝑖(0)⟩2 = ℎ(𝐿) − 2sin2𝜖 − sec 𝜖 cos(2𝛽𝐿 − 𝜖) +8cos2(𝛽𝐿 − 𝜖) +, +(S1.8a) + +⟨∆𝑝𝑠,𝐿 +2 ⟩ = ⟨𝑝𝑠 +2(𝐿)⟩ − ⟨𝑝𝑠(𝐿)⟩2 += ℎ(𝐿) cos 𝜖 − cos(2𝛽𝐿 + 𝜖) − 2sin2𝜖 cos(2𝛽𝐿 − 𝜖) +8 cos 𝜖 cos2(𝛽𝐿 − 𝜖) + , +(S1.8b) + +⟨∆𝑞𝑠,𝐿 +2 ⟩ = ⟨𝑞𝑠 +2(𝐿)⟩ − ⟨𝑞𝑠(𝐿)⟩2 = 2 + cos2𝜑𝑒−2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 + 𝜑) +8 cos2(𝜅𝐿) + , +(S1.8c) + +⟨∆𝑝𝑖,0 +2 ⟩ = ⟨𝑝𝑖 +2(0)⟩ − ⟨𝑝𝑖(0)⟩2 = +(2 + cos2𝜑)𝑒2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 − 𝜑) +8 cos2(𝜅𝐿) + . +(S1.8d) +where ℎ(𝐿) = 3 + 2𝛾𝐿 and 𝜑 = arctan ( +𝛾 +𝜅) . We notice from Eqs. (S1.8a)—(S1.8d) that although + +3 + +these variances contain linear terms, they do not affect the periodic characteristics of the noise +fluctuations. As revealed in Fig. 2 in the main text, when taking 2𝜅𝐿 as the dimensionless variable, +we find that the oscillation period of the variances of the PT-quadrature pair (𝑞𝑖, 𝑝𝑠) is approximately +to be +2𝜅𝜋 +𝛽 while the fluctuation period of the variances of the non-PT quadrature pair (𝑝𝑖, 𝑞𝑠) simply +assumes 2𝜋 for the parameter space in the PT-phase intact region. +As emphasized in the main text, the ultimate novelty of our work is not just to find a system +capable of the observation of genuine quantum optical PT symmetry under fair sampling measurement, +but to unearth an extraordinary phenomenon that has never been discovered. That is, the PT-quadrature +observable enables one to witness a compelling quantum-to-classical transition perfectly coinciding +with the PT phase transition by varying the non-Hermitian parameter 𝛾, and the transition boundary +is physically defined by the EP curve. To the best of our knowledge, this is the first proposal on +exploring the untrivial transition between two incompatible worlds, classical and quantum, with a well- +defined physical boundary by measuring the same quantum observable. We are aware that there exists +a parallel way in the literature that exploits some massive quantum systems such as cooled cavity +optomechanical structures to probe the quantum-to-classical transition by constantly checking the +decoherence of a quantum state when manipulating some system parameters. However, even if these +proposals are viable in the lab, they have to face an inescapable conundrum, that is, in these systems +it becomes extremely challenging to determine the exact transition boundary. In other words, observing +the sharp transition would be exceedingly difficult and even impossible for these protocols. In contrast, +these difficulties do not appear in our system. Moreover, our method aims to measure the expectation +of a quantum observable while the existing protocols concentrate on studying the state of the system. +This difference fundamentally distinguishes our work from all others. All in all, our work for the first +time presents a new way to explore the quantum-to-classical transition by taking advantage of non- +Hermiticity and symmetry. +Before ending this part of discussion, here we would like to add a couple of additional comments +on the following important issues on quantum optical PT symmetry raised in the literature. One is in +response to the quantum noncloning theorem. In fact, the amplification won’t violate the quantum +noncloning theorem at all in our proposal, because the PSA here only acts on the single-mode idler +field. This also concurs with the well-established knowledge in the field of quantum optics, especially +in quantum squeezing. The second question concerns the law of causality. Although the gain may lead +to the fast-light or superluminal effect, since the quantum noise introduced by the transmission loss is +inseparable from the actual signal of interest, the causality proves not to be a problem when considering +PT symmetry at the quantum level (as we did here). +II. Derivation of quantum sensing +For the quantum sensor application, we consider the generation of the idler-signal bosonic modes +from a seeding two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩. If taking into account the thermal reservoir +with an average thermal bosonic number 𝑛th, the quantum Langevin noise in Eq. (S1.3b) obeys the +following properties, 〈𝑓𝑠(𝑧)𝑓𝑠 +†(𝑧′)〉 = 2𝛾(𝑛th + 1)𝛿(𝑧 − 𝑧′) and 〈𝑓𝑠 +†(𝑧)𝑓𝑠(𝑧′)〉 = 2𝛾𝑛th𝛿(𝑧 − 𝑧′). +After an interaction length 𝐿, the mean values and variances of the quadrature measurements on 𝑞𝑗 +and 𝑝𝑗 with respect to the final state |𝜓𝑓⟩ of the system can be obtained by using Eqs. (S1.6a) and +(S1.6b), + +⟨𝑞𝑖(0)⟩ = +cos 𝜖 +cos(𝛽𝐿 − 𝜖) ⟨𝑞𝑖(𝐿)⟩ − +sin(𝛽𝐿) +cos(𝛽𝐿 − 𝜖) ⟨𝑝𝑠(0)⟩, +(S2.1a) + +⟨𝑝𝑠(𝐿)⟩ = +− sin(𝛽𝐿) +cos(𝛽𝐿−𝜖) ⟨𝑞𝑖(𝐿)⟩ + +cos 𝜖 +cos(𝛽𝐿−𝜖) ⟨𝑝𝑠(0)⟩, +(S2.1b) + +⟨𝑞𝑠(𝐿)⟩ = − +sin(𝜅𝐿) +cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ + +𝑒−𝛾𝐿 +cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, +(S2.1c) + +4 + + +⟨𝑝𝑖(0)⟩ = +𝑒𝛾𝐿 +cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ − +sin(𝜅𝐿) +cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, +(S2.1d) +and + ⟨∆𝑞𝑖,0 +2 ⟩ = 3 + 𝑤[2𝛾𝐿 − tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖 +8cos2(𝛽𝐿 − 𝜖) +, +(S2.2a) + ⟨∆𝑝𝑠,𝐿 +2 ⟩ = 3 + 𝑤[2𝛾𝐿 + tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖 +8cos2(𝛽𝐿 − 𝜖) + , +(S2.2b) + ⟨∆𝑞𝑠,𝐿 +2 ⟩ += 𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿) +8cos2(𝜅𝐿) + , +(S2.2c) + ⟨∆𝑝𝑖,0 +2 ⟩ += 𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)] +8cos2(𝜅𝐿) + . +(S2.2d) +Here, 𝑤 = 2𝑛th + 1 with the thermal average boson number 𝑛th = [Exp ( +ℎ𝑣𝜆 +𝑘𝐵𝑇) − 1] +−1 + . One can +easily check that the thermal photon number becomes infinitesimal at the room temperature (~300 K) +in the visible spectral range. Alternatively, the above Langevin noise properties reduce to the simpler +formats mentioned in Section I. +The ultimate precision of the parameter estimation of 𝜅 is essentially determined by the variance +∆𝜅 +2 in terms of a targeted physical observable. The performance of the proposed quadrature-PT sensing +scheme can be however evaluated by comparing the inverse variances ∆𝜅,𝑞𝑗 +−2 = +(𝜒𝜅 +𝑞𝑗) +2 +〈∆𝑞𝑗 +2〉 and ∆𝜅,𝑝𝑗 +−2 = +(𝜒𝜅 +𝑝𝑗) +2 +〈∆𝑝𝑗 +2〉 with the quantum Fisher information 𝐹𝜅 at the system’s final state |𝜓𝑓⟩ . Here, we have +introduced the susceptibilities 𝜒𝜅 +𝑞𝑗 = 𝜕𝜅⟨𝑞𝑗⟩ and 𝜒𝜅 +𝑝𝑗 = 𝜕𝜅⟨𝑝𝑗⟩ to capture the system response to +⟨𝑞𝑗⟩ and ⟨𝑝𝑗⟩ for a small perturbation 𝛿𝜅 about the preset 𝜅. With the help of Eqs. (S2.1a)—(S2.1d), +after some labor one can show that 𝜒𝜅 +𝑞𝑗 and 𝜒𝜅 +𝑝𝑗 (𝑗 = 𝑖, 𝑠) take the form of, + +𝜒𝜅 +𝑞𝑖(0) = 𝛼{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]} +2𝛽cos2(𝛽𝐿 − 𝜖) +, +(S2.3a) + +𝜒𝜅 +𝑝𝑠(𝐿) = 𝜒𝜅 +𝑞𝑖(0), +(S2.3b) + +𝜒𝜅 +𝑞𝑠(𝐿) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1], +(S2.3c) + +𝜒𝜅 +𝑝𝑖(0) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1]. +(S2.3d) +By plugging Eqs. (S2.2a)—(S2.2d) and Eqs. (S2.3a)—(S2.3d) into the inverse variances ∆𝜅,𝑞𝑗 +−2 = +(𝜒𝜅 +𝑞𝑗) +2 +〈∆𝑞𝑗 +2〉 +and ∆𝜅,𝑝𝑗 +−2 = +(𝜒𝜅 +𝑝𝑗) +2 +〈∆𝑝𝑗 +2〉 , we arrive at the following key results: + ⟨∆𝜅,𝑞𝑖,0 +−2 +⟩ = 2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2 +𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 − tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖} , +(S2.4a) + ⟨∆𝜅,𝑝𝑠,𝐿 +−2 +⟩ = +2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2 +𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 + tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖}, +(S2.4b) + ⟨∆𝜅,𝑞𝑠,𝐿 +−2 +⟩ += +8𝛼2𝐿2sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1]2 +𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿) , +(S2.4c) + +5 + + +⟨∆𝜅,𝑝𝑖,0 +−2 +⟩ += +8𝛼2𝐿2sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1]2 +𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)] . +(S2.4d) +After having the inverse variances (S2.4a)—(S2.4d), now let us turn our attention to the quantum +Fisher information 𝐹𝜅, i.e., the quantum Cramér-Rao bound, which demands the optimal measurement +to satisfy the inequality ∆𝜅 +−2≤ 𝐹𝜅. In this sensing protocol, we start with the initial system state to be +in a two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩ for the sake of simplicity. Then, the final state of the +system evolves as |𝜓𝑓⟩ = +𝑈 +√𝜇 |𝜓𝑖⟩, where 𝑈 = 𝑒−𝑖𝐻𝐿 is the evolution operator and 𝜇 = ⟨𝜓𝑓|𝜓𝑓⟩ is +the normalization coefficient. By working in the Schrödinger picture and treating the idler-signal field +operators 𝑎𝑖 = 𝑎𝑖(𝐿) and 𝑎𝑠 = 𝑎𝑠(0) as constant operators, the quantum Fisher information can be +calculated by the definition of 𝐹𝜅 = 4 (⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ − |⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩| +2) for a parameter 𝜅 that +controls the strength of the system’s Hamiltonian 𝐻 (S1.1) with respect to a known physical +observable (in our case, it can be any of the four quadratures). To this end, let us give a detailed +examination on the first term in 𝐹𝜅: + +⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ += ⟨𝜓𝑖| +√𝜇(𝜕𝜅𝑈†) − 𝜕𝜅𝜇 +2√𝜇 𝑈† +𝜇 +√𝜇(𝜕𝜅𝑈) − 𝜕𝜅𝜇 +2√𝜇 𝑈 +𝜇 +|𝜓𝑖⟩ += 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿 +𝜕𝜅𝜇 +2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿 +𝜕𝜅𝜇 +2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ + +(𝜕𝜅𝜇)2 +4𝜇2 , +(S2.5) +where 𝜕𝜅𝑈 = −𝑖𝐿(𝜕𝜅𝐻)𝑈. Note that 𝜕𝜅𝐻 commutes with 𝑈. In the same way, we can also obtain +the exact expression for the second term as follows: + +|⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩| +2 += 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿 +𝜕𝜅𝜇 +2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿 +𝜕𝜅𝜇 +2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ + +(𝜕𝜅𝜇)2 +4𝜇2 . +(S2.6) +Substituting these two results into 𝐹𝜅 yields the concise and intuitive expression of the quantum +Fisher information, which is + +𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩). +(S2.7) +Since 𝜕𝜅𝐻 = 𝜕𝜅𝐻† = ℏ(𝑎𝑖 +†𝑎𝑠 +† + 𝑎𝑖𝑎𝑠) = 2ℏ(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖) in the Schrödinger picture and the +expectation value of an operator does not change along with the picture transformation, we can +transform the above formulae of the quantum Fisher information into the Heisenberg representation to +ease the calculations. That is, + +𝐹𝜅 = 16𝐿2 (⟨𝜓𝑓|(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖)2|𝜓𝑓⟩ − (⟨𝜓𝑓|𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖|𝜓𝑓⟩) +2) + = 16𝐿2{⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2|𝜓𝑖⟩ +− [⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]|𝜓𝑖⟩]2} + = 16𝐿2{〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2}. +(S2.8) +From Eq. (S2.8), one can easily evaluate the term 𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) using Eqs. (S1.6a) and +(S1.6b). After some lengthy derivations, we eventually get + +𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) += 𝐴𝑞𝑖(𝐿)𝑝𝑖(𝐿) + 𝐵𝑝𝑠(0)𝑝𝑖(𝐿) + ∫ 𝑑𝑧𝐶𝑃𝑠𝑝𝑖(𝐿) +𝐿 +0 ++ 𝐷𝑞𝑖(𝐿)𝑞𝑠(0) ++ 𝐸𝑝𝑠(0)𝑞𝑠(0) + ∫ 𝑑𝑧𝐹𝑃𝑠𝑞𝑠(0) +𝐿 +0 ++ ∫ 𝑑𝑧𝐺𝑄𝑠𝑞𝑖(𝐿) +𝐿 +0 ++ ∫ 𝑑𝑧𝐽𝑄𝑠𝑝𝑠(0) +𝐿 +0 ++ ∫ 𝑑𝑧𝑅𝑃𝑠𝑄𝑠 +𝐿 +0 +, +(S2.9) +where all the involved coefficients are + +6 + + +𝐴 = +𝑒𝛾𝐿sin(𝛽𝐿)−cos𝜖sin(𝜅𝐿) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10a) + +𝐵 = +sin(𝛽𝐿)sin(𝜅𝐿)−𝑒𝛾𝐿 cos𝜖 +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10b) + +𝐶 = +sin[𝛽(𝐿−𝑧)]sin(𝜅𝐿)−𝑒𝛾𝐿cos(𝛽𝑧−𝜖) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10c) + +𝐷 = +𝑒−𝛾𝐿 cos 𝜖−sin(𝛽𝐿)sin(𝜅𝐿) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10d) + +𝐸 = +cos𝜖sin(𝜅𝐿)−𝑒−𝛾𝐿sin(𝛽𝐿) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10e) + +𝐹 = +sin (𝜅𝐿)cos(𝛽𝑧−𝜖)−𝑒−𝛾𝐿sin(𝛽(𝐿−𝑧)) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10f) + +𝐺 = +𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧) cos𝜖−𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]sin(𝛽𝐿) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10g) + +𝐽 = +𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos 𝜖−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin(𝛽𝐿) +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +, +(S2.10h) + +𝑅 = +𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos(𝛽𝑧−𝜖)−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin[𝛽(𝐿−𝑧)] +cos(𝛽𝐿−𝜖)cos(𝜅𝐿) +. +(S2.10i) +With these results, we are ready to work out the following step, + +〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2 += 𝐴2(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ − ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩2) ++ 𝐵2(⟨𝑝𝑖 +2(𝐿)𝑝𝑠2(0)⟩ − ⟨𝑝𝑠(0)𝑝𝑖(𝐿)⟩2) + 𝐷2(⟨𝑞𝑖 +2(𝐿)𝑞𝑠2(0)⟩ − ⟨𝑞𝑠(0)𝑞𝑖(𝐿)⟩2) ++ 𝐸2(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ − ⟨𝑝𝑠(0)𝑞𝑠(0)⟩2) ++ 𝐴𝐵(⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑝𝑖(𝐿)⟩ +− 2⟨𝑝𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑝𝑠(0)⟩ ++ 𝐴𝐷(⟨𝑞𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩ +− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑞𝑠(0)⟩ ++ 𝐵𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ +− 2⟨𝑝𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑝𝑖(𝐿)⟩ ++ 𝐷𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑞𝑠(0)⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ +− 2⟨𝑞𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑞𝑖(𝐿)⟩ ++ 𝐵𝐷(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑠(0)𝑞𝑠(0)⟩ +− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑠(0)⟩⟨𝑝𝑖(𝐿)⟩⟨𝑝𝑠(0)⟩) ++ ∫ 𝑑𝑧[𝐶2⟨𝑝𝑖 +2(𝐿)⟩ + 𝐹2⟨𝑞𝑠2(0)⟩ + 2𝐶𝐹⟨𝑞𝑠(0)𝑝𝑖(𝐿)⟩]⟨𝑃𝑠 +2⟩ +𝐿 +0 ++ ∫ 𝑑𝑧[𝐺2⟨𝑞𝑖 +2(𝐿)⟩ + 𝐽2⟨𝑝𝑠2(0)⟩ + 2𝐺𝐽⟨𝑝𝑠(0)𝑞𝑖(𝐿)⟩]⟨𝑄𝑠 +2⟩ +𝐿 +0 ++ ∫ 𝑑𝑧𝐶𝐺[⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩⟨𝑃𝑠𝑄𝑠⟩] +𝐿 +0 ++ ∫ 𝑑𝑧𝐹𝐽[⟨𝑝𝑠(0)𝑞𝑠(0)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)⟩⟨𝑃𝑠𝑄𝑠⟩] +𝐿 +0 ++ ∫ 𝑑𝑧𝑅2⟨𝑃𝑠𝑄𝑠𝑃𝑠𝑄𝑠⟩ +𝐿 +0 +− (∫ 𝑑𝑧𝑅⟨𝑃𝑠𝑄𝑠⟩ +𝐿 +0 +) +2 +, + +(S2.11) + +7 + +in terms of the quadrature operators at the initial boundary conditions. By further simplification, we +finally approach the following resultant function for the quantum Fisher information, + +𝐹𝜅 = 16𝐿2 [(𝐴2 + 𝐸2) ( +𝛼2 +2 + +1 +8) + (𝐴𝐵 + 𝐴𝐷 + 𝐵𝐸 + 𝐷𝐸) ( +𝛼2 +2 ) + ( +1 +16 + +𝛼2 +2 ) (𝐵2 + 𝐷2) − +1 +8 𝐵𝐷 + ( +1 +4 + 𝛼2) +𝛾 +2 (2𝑛th + 1) ∫ 𝑑𝑧(𝐶2 + 𝐹2 + 𝐺2 + 𝐽2) +𝐿 +0 ++ 𝛼2𝛾(2𝑛th + 1) ∫ 𝑑𝑧(𝐶𝐹 + +𝐿 +0 +𝐽𝐺) + +𝛾 +4 ∫ 𝑑𝑧(𝐽𝐹 − 𝐶𝐺) +𝐿 +0 ++ +𝛾 +2 (𝑛th + +𝛾 +2) ∫ 𝑑𝑧𝑅2 +𝐿 +0 ++ +𝛾2 +4 (∫ 𝑑𝑧𝑅 +𝐿 +0 +) +2 +] . +(S2.12) +Obviously, the quantum Fisher information 𝐹𝜅 (S2.12) will feature different manifestations in +response to the contrasting PT domains of the system. As a representative example, in Fig. S1 we +accordingly present the quantum Fisher information log4𝐹𝜅 for three distinct scenarios: +𝛾 +𝜅 = 0.8 +unbroken quadrature-PT phase), +𝛾 +𝜅 = 1 (EP point), and +𝛾 +𝜅 = 1.2 (breaking quadrature-PT phase). +FIG. S1. The quantum Fisher information at different quadrature PT states. Blue, red, and +orange lines are, respectively, corresponding to 𝛾 𝜅 +Τ += 0.8 (unbroken quadrature-PT phase), +𝛾 𝜅 +Τ += 1 (EP point), and 𝛾 𝜅 +Τ += 1.2 (broken quadrature-PT phase). +FIG. S2. Quantum sensing performance by comparing log4(Δ𝜅𝑞𝑖,0 +−2 ) (a), log4(Δ𝜅𝑝𝑠,𝐿 +−2 ) +(b), log4(Δ𝜅𝑞𝑠,𝐿 +−2 ) (c), and log4(Δ𝜅𝑝𝑖,0 +−2 ) (d) with log4 𝐹𝑘 in the quadrature-PT phase +unbroken region for the parameters (𝛼 = 2, 𝛾 = 0.2, 𝜅 = 1). + +60 +50 +y +y +- +0.8 +1 +1.2 +K +K +t +40 +30 +20 +10 +0 +-10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +2KL30 +30 +(a) +(b) +—log4((Fk) +25 +25 +log4((Ago) +log4((Akpz,)) +20 +20 +15 +15 +人儿从 +10 +10 +in +0 +0 +-5 +-5 +10 +10 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +30 +30 +(c) +(d) +25 +log4(Fk) +25 +log4(Fk)) +log4((Akgz)) +20 +log4((Akpz0)) +20 +15 +15 +10 +10 +5 +5 +0 +0 +-5 +-5 +-10 +10 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +308 + + + + + + +Different from other existing (quantum) sensing protocols based on PT or EP enhancement, we find +that the PT-quadrature variables permit optimal classical sensor performance in the PT phase unbroken +regime but far away from the EP. This observation is strongly supported by analyzing the quantum +Fisher information with respect to the inverse variances across the parameter space. In the main text, +FIG. S4. Evaluating quantum sensing performance (FIGs. S3(a)—(d)) near the exceptional +point by examining the ratios of ∆𝜅𝑞𝑖,0 +−2 (a), ∆𝜅𝑝𝑠,𝐿 +−2 (b), ∆𝜅𝑞𝑖,𝐿 +−2 (c), and ∆𝜅𝑝𝑖,0 +−2 (d) to 𝐹𝜅 for +𝛾 +𝜅 = 0.94 (red curve) and +𝛾 +𝜅 = 0.95 (black curve), respectively. The other parameters are 𝛼 = +2 and 𝜅 = 1. +FIG. S3. Quantum sensing performed near the exceptional point by comparing log4(Δ𝜅𝑞𝑖,0 +−2 ) +(a), log4(Δ𝜅𝑝𝑠,𝐿 +−2 ) (b), log4(Δ𝜅𝑞𝑠,𝐿 +−2 ) (c), and log4(Δ𝜅𝑝𝑖,0 +−2 ) (d) with log4 𝐹𝑘 for +𝛾 +𝜅 = 0.94 +and +𝛾 +𝜅 = 0.95 for the parameters 𝛼 = 2 and 𝜅 = 1. + +0.25 +0.25 +(a) +(b) + 10-4 +×10-4 +0.2 +10 +0.2 +10 +8 +Y/K= 0.94 +8 +三 0.15 +0.15 +6 +—y/k=0.95 +6 +25 +4 +4 +0.1 +2 +0.1 +2 +0 +0.05 +2 +4 +6 +8 +0.05 +2 +4 +9 +8 +0 +0 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +2KL +2KL +0.25 +0.25 +(c) +×10-4 +(d) +×10-4 +10 +10 +0.2 +0.2 +8 +8 +6 +K +6 +0.15 +0.15 +4 +4 +/T'S +2 +2 +2 +20 +0.1 +0.1 +0 +6 +7 +8 +5 +6 +7 +8 +0.05 +0.05 +0 +0 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +2kL +2KL60 +60 +(a) +(b) +50 +0.94 +0.95log4Fx +50 +0.94 +0.95log4F +K +40 +Y +=0.94 +Y +40 +=0.94 +30 +30 +20 +20 +10 +10 +0 +0 +10 +10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +2KL +2KL +60 +60 +(c) +(d) +50 +0.951og4Fx +50 +0.951og4Fx +=0.94 +K +0.94 +K +40 +Y +=0.94 +Y +=0.95 log4Axqz +40 +Y +=0.94 +Y +=0.95 +30 +30 +20 +20 +10 +10 +0 +0 +-10 +10 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +2KL +2KL9 + +we have shown the precision of the 𝜅 -parameter estimation by looking at the ratio of the inverse +variance ∆𝜅−2 to the quantum Fisher information 𝐹𝜅 in Figs. 5(a)—(d), which should be bounded in +the range of [0, 1]. In Fig. S2(a)—(d), we have further given these inverse variances in comparison +with 𝐹𝜅 . To make the point more straightforward and convincing, in Figs. S3(a)—(d) we have +particularly examined the measurement schemes implemented very close to the EP for +𝛾 +𝜅 = 0.94 and +FIG. S5. Quantum sensing implemented exactly at the exceptional point by comparing +log4(Δ𝜅𝑞𝑖,0 +−2 ) (a), log4(Δ𝜅𝑝𝑠,𝐿 +−2 ) (b), log4(Δ𝜅𝑞𝑠,𝐿 +−2 ) (c), and log4(Δ𝜅𝑝𝑖,0 +−2 ) (d) with log4 𝐹𝑘 for +𝛼 = 2 and 𝜅 = 1. +FIG. S6. Evaluating quantum sensing performance (FIG. S5(a)—(d)) at the exceptional point +by examining the ratios of ∆𝜅𝑞𝑖,0 +−2 (a), ∆𝜅𝑝𝑠,𝐿 +−2 (b), ∆𝜅𝑞𝑖,𝐿 +−2 (c), and ∆𝜅𝑝𝑖,0 +−2 (d) to 𝐹𝜅 for the +parameters 𝛼 = 2 and 𝜅 = 1. + +40 +40 +(a) +log4Fx +(b) +log4Fx +30 +30 +20 +20 +10 +10 +0 +0 +-10 +10 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +2KL +2kL +40 +40 +(c) +log4Fx +(d) +log4Fk +30 +30 +20 +20 +10 +10 +0 +0 +-10 +10 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +2KL +2KL0.25 +0.25 +(a) +(q) +×10-4 +× 10-4 +0.2 +10 +0.2 +10 +8 +K +Y/K= 1 +8 +三 0.15 +0.15 +6 +6 +4 +2弘 +4 +AK +0.1 +2 +0.1 +2 +0. +0.05 +2 +4 +6 +8 +0.05 +2 +4 +6 +8 +0 +0 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +2KL +2kL +0.25 +0.25 +(c) +× 10-4 +(d) +× 10-4 +10 +0.2 +10 +0.2 +8 +8 +K +6 +E +K +0.15 +6 +0.15 +4 +4 +2 +2 +/0 +2 +0.1 +1 0.1 +2 +0 +AK +0 +6 +7 +8 +5 +6 +7 +8 +0.05 +0.05 +0 +0 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +2kL +2KL10 + +𝛾 +𝜅 = 0.95 by plotting log4 Δ𝜅𝑞𝑖,0 +−2 , log4 Δ𝜅𝑝𝑠,𝐿 +−2 , log4 Δ𝜅𝑞𝑠,𝐿 +−2 , and log4 Δ𝜅𝑝𝑖,0 +−2 . Similarly, the +quantitative sensing performance offered by each quadrature can be well assessed by evaluating the +corresponding ratio of ∆𝜅𝑞𝑖,0 +−2 (Fig. S4(a)), ∆𝜅𝑝𝑠,𝐿 +−2 (Fig. S4(b)), ∆𝜅𝑞𝑠,𝐿 +−2 (Fig. S4(c)), and ∆𝜅𝑝𝑖,0 +−2 (Fig. +S4(d)) to 𝐹𝜅 for the same parameters used in Figs. S3(a)—(d). By comparing these figures with Figs. +5(a)—(d) in the main text, it is not difficult to conclude that indeed, the presence of gain and loss in +gain-loss-coupled PT symmetry can substantially diminish the EP-based super-sensitivity promised in +the classical settings and make it unavailable in the quantum level. Moreover, even if one still insists +on performing any quantum sensing measurement in the vicinity of the EP (e.g., +𝛾 +𝜅 = 0.94 and +𝛾 +𝜅 = +0.95), it would become highly challenging due to the vast difference between the peak values of ∆𝜅 +−2 +and 𝐹𝜅 spanning over many orders of magnitude, regardless of whether the quadrature observables +are associated with the characteristics of PT symmetry. This is especially true if comparing with the +measurement carried out at +𝛾 +𝜅 = 0.2. +What happens if one attempts to fulfill the quantum sensing at the phase transition point? In such +a case, unfortunately, the paired PT quadratures will cease to showcase any response to the parameter +precision estimation, thereby making them fully unsuitable for quantum sensor applications when the +symmetry spontaneously breaks down. As demonstrated in Figs. S5(a) and (b), one can clearly see that +log4 Δ𝜅𝑞𝑖,0 +−2 and log4 Δ𝜅𝑝𝑠,𝐿 +−2 for the PT-symmetric quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿)) become smooth +and curvatureless, indicating that they are completely insensitive to any perturbation on an unknown +parameter yet to be estimated. Alternatively, no gain on parameter estimation will be accessed at the +EP. On the other hand, we notice from Figs. S5(c) and (d) that the non-PT-symmetric quadrature pair +(𝑝𝑖(0), 𝑞𝑠(𝐿)) enables best sensing measurement only near the first peaks of the inverse variances +log4 Δ𝜅𝑞𝑠,𝐿 +−2 , and log4 Δ𝜅𝑝𝑖,0 +−2 , in accordance with the quantum Fisher information log4 𝐹𝑘. Obviously, +this behaves differently from the cases of +𝛾 +𝜅 = 0.2 , where the supersensitive measurements are +available near the first two peaks of ∆𝜅 +−2 and even more peaks (Figs. S2(c) and (d)). In fact, when PT +symmetry disappears, the quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿)) lose to offer any sensing capabilities, despite +(sub)optimal sensing may be accessible to the other non-PT-symmetric conjugate pair (𝑝𝑖(0), 𝑞𝑠(𝐿)), +according to our numerical simulations. To have a more intuitive evaluation on the quantum sensing +performance exactly at the EP, it is better to look at the ratios of ∆𝜅𝑞𝑖,0 +−2 (Fig. S6(a)), ∆𝜅𝑝𝑠,𝐿 +−2 (Fig. +S6(b)), ∆𝜅𝑞𝑖,𝐿 +−2 (Fig. S6(c)), and ∆𝜅𝑝𝑖,0 +−2 (Fig. S6(d)) to 𝐹𝜅 in the same way as we did above. From +Figs. S6(a)—(d), we can easily find that these ratios quickly approach zero for the longer medium +length 𝐿, implying that the system loses its sensing ability at the EP. + +References: +[1] Jiang, Y., Mei, Y., Zuo, Y., Zhai, Y., Li, J., Wen, J. & Du, S. Anti-parity-time symmetry optical +four-wave mixing in cold atoms. Phys. Rev. Lett. 123, 193604 (2019). +[2] Wang, W., Zhai, Y., Liu, D., Jiang, X., Ghamsari, S. V. & Wen, J. Quadrature parity-time +symmetry. (2022) + + diff --git a/7tE5T4oBgHgl3EQfQQ5g/content/tmp_files/load_file.txt b/7tE5T4oBgHgl3EQfQQ5g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..adf9f1a2f8e67ff47e117e651ba7f7f503adb24f --- /dev/null +++ b/7tE5T4oBgHgl3EQfQQ5g/content/tmp_files/load_file.txt @@ -0,0 +1,1118 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf,len=1117 +page_content='Quantum-to-classical transition enabled by quadrature-PT symmetry Wencong Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 Yanhua Zhai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 Dongmei Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1* Xiaoshun Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 3† Saeid Vashahri Ghamsari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4 and Jianming Wen4‡ 1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' School of Physics and Telecommunication Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' China 2 Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Spelman College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Atlanta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Georgia 30314,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' USA 3National Laboratory of Solid State Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' College of Engineering and Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' China 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Kennesaw State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Marietta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Georgia 30060,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' USA Quantum Langevin noise makes experimental realization of genuine quantum-optical parity-time (PT) symmetry in a gain-loss-coupled open system elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Here, we challenge this puzzle by exploiting twin beams produced from a nonlinear parametric process, one undergoing phase-sensitive linear quantum amplification (PSA) and the other engaging balanced loss merely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Unlike all previous studies involving phase-insensitive amplification (PIA), our PSA-loss scheme allows one quadrature pair to experience PT symmetry, a unique quantum effect without any classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Such symmetry showcases many radical noise behaviors beyond conventional quantum squeezing and inaccessible to any PIA-based platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Importantly, it is the only non-Hermitian system hitherto that enables the emergence of non-Hermiticity-induced quantum-to-classical transition for the same quantum observable when crossing exceptional point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Utilizing this quadrature-PT structure, we have further studied its potential in quantum sensing by exploring the quantum Cramér-Rao bound or Fisher information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Besides, the proposed quadrature PT symmetry also sheds new light on protecting continuous-variable (CV) qubits from decoherence in lossy transmission, a long-standing conundrum for various CV-based quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—In canonical quantum mechanics, the system Hamiltonian as a physical observable is required to be Hermitian to ensure the realness of associated eigenspectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Yet, it has long been known that the Hermiticity is just a sufficient but not necessary condition for a Hamiltonian to have real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This radical change of view stems from the seminal work by Bender and Boettcher in 1998, where a large class of non- Hermitian quantum Hamiltonians enjoying the joint parity-time (PT) symmetry was discovered to possess entirely real eigenvalues below a phase-transition point or exceptional point (EP) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' However, it remains elusive to probe such a non-Hermitian but PT-symmetric quantum Hamiltonian experimentally due to the lack of complex quantum potential in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Nevertheless, the notion of PT symmetry [2-9] has successfully survived in many other physical branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Thanks to the mathematical equivalence between the quantum Schrödinger and paraxial light propagation equation, classical optics was first suggested to simulate the wave properties of PT- symmetric quantum mechanics in synthesized settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' By incorporating linear gain and loss, optics has become a fertile ground for exploring PT symmetry [2-17] with an iconic feature of pair of eigenvalues phase transitioning from purely real to complex conjugate when a parameter crosses the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In this regard, a plethora of intriguing phenomena have been uncovered by utilizing various linear and nonlinear optical materials to control and engineer light for practical applications [2-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Despite the impressive progress, to date the PT studies have been mostly limited to a mean-field approach that encapsulates all quantum dissipation in an ‘effective Hamiltonian’ [2-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This method treats light essentially as a (semi) classical electromagnetic (EM) field and only retains the minimum number of degrees of freedom to describe an open system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As a result, it fails to yield valid results when the nonclassicality of light are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Notably, if the EM field is quantized, one must introduce the Langevin noise operator to preserve its corresponding commutation relation [18], though the introduction of quantum Langevin noise is generally thought to prevent the system from approaching quantum optical PT symmetry [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This is especially true when a phase- insensitive linear quantum amplifier (PIA) serves as an optical gain resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Unfortunately, so far the research on PT optics has all been confined to PIA-based systems, thereby rendering the observation of quantum signatures highly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The gain involvement also encounters another fundamental issue from the famous quantum noncloning theorem [20], namely how to maintain the integrity of the signal state, let alone the limitation further dictated by the Kramers-Kronig relation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Therefore, it becomes a legitimate question whether gain-loss- coupled PT symmetry is viable quantum optically [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We notice that two alternative means have recently been implemented by using either a passive scheme or a non- Hermitian subset Hamiltonian in a large Hermitian system [22-27] to bypass the noise issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' These efforts have unearthed some quantum features, but they are incapable of providing a conclusive picture to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We are also aware that a distinct trajectory is devoted to exploiting anti-PT symmetry [28-34], a counterpart of PT, to avoid the adverse effect of Langevin noise [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' By overcoming the aforementioned obstacles, here we propose a novel and experimentally feasible platform utilizing twin beams generated from a nonlinear optical parametric process such as parametric down conversion (PDC) and four-wave mixing (FWM) [36], with the signal arm experiencing pure loss while the idler channel undergoing phase-sensitive linear quantum amplification (PSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Thanks to PSA-empowered noiseless amplification [37], our architecture not only makes the observation of true quantum optical PT a reality, but also displays distinctive features unapproachable to any previous scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Opposite to PIA equally amplifying paired quadratures with additive noise, PSA maximally amplifies some of them but inversely attenuates the rest without adding extra noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This asymmetric amplification naturally gives rise to the so-called quadrature PT symmetry, a unique quantum effect without any classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Given various never- before-seen attributes, our design further opens a door to uncovering the stunning classical-to-quantum transition for the same physical observable when a non-Hermitian parameter passes through the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—For simplicity, let us focus on the quadrature-PT setup schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 1, where under perfect phase matching, nondegenerate paired signal-idler waves are parametrically created from vacuum in a counterpropagating geometry by driving an FWM medium of length 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' During their propagation, the idler and signal channels are respectively subject to PSA and loss with the rates of 𝑔 and 𝛾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For nondepleted and classical input pump lasers, the system evolves along the ∓𝑧 direction under the influence of the non-Hermitian Hamiltonian 𝐻 = 𝑖ℏ𝑔(𝑎𝑖 2−𝑎𝑖 †2) 2 − 𝑖ℏ𝛾𝑎𝑠 †𝑎𝑠 + ℏ𝜅(𝑎𝑖 †𝑎𝑠 † + 𝑎𝑖𝑎𝑠) , (1) and the signal-idler field operators (as, ai) obey the Heisenberg-Langevin equations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' PT symmetry undergone by twin beams from a backward-FWM process, where the −𝑧 idler mode experiences PSA and the +𝑧 signal faces equal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' PSAdler Four-wave mixing Signal𝑑𝑎𝑖 𝑑𝑧 = 𝑔𝑎𝑖 † + 𝑖𝜅𝑎𝑠 †, 𝑑𝑎𝑠 𝑑𝑧 = −𝛾𝑎𝑠 − 𝑖𝜅𝑎𝑖 † + 𝑓𝑠 , (2) with † denoting Hermitian conjugate, 𝜅 the parametric conversion strength, and 𝑓𝑠 the quantum Langevin noise of zero mean satisfying 〈𝑓𝑠(𝑧)𝑓𝑠 †(𝑧′)〉 = 2𝛾𝛿(𝑧 − 𝑧′) and 〈𝑓𝑠 †(𝑧)𝑓𝑠(𝑧′)〉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' At first glance, the dynamics (2) seems PT-irrelevant even if 𝑔 = 𝛾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Surprisingly, the hidden PT arises if one transforms Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (2) into the corresponding quadrature-operator evolution by defining 𝑞𝑗 = (𝑎𝑗 † + 𝑎𝑗)/2 and 𝑝𝑗 = 𝑖(𝑎𝑗 † − 𝑎𝑗)/2 (𝑗 = 𝑖, 𝑠) with [𝑞𝑗, 𝑝𝑗] = 𝑖/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' That is 𝑑 𝑑𝑧 [𝑞𝑖 𝑝𝑠] = [ 𝑔 𝜅 −𝜅 −𝛾] [𝑞𝑖 𝑝𝑠] + [0 𝑃𝑠] , (3a) 𝑑 𝑑𝑧 [𝑝𝑖 𝑞𝑠] = [−𝑔 𝜅 −𝜅 −𝛾] [𝑝𝑖 𝑞𝑠] + [ 0 𝑄𝑠] , (3b) where 𝑃𝑠 = 𝑖(𝑓𝑠 † − 𝑓𝑠)/2 and 𝑄𝑠 = (𝑓𝑠 † + 𝑓𝑠)/2 are the Langevin-noise quadrature operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The underlying physics now becomes apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The 𝑔-𝛾 introduction fundamentally intervenes the evolution of the usual two- mode squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Specifically, for 𝑔 = 𝛾 , albeit the impact of 𝑃𝑠 , {𝑞𝑖, 𝑝𝑠} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (3a) become a PT- quadrature pair and adhere to PT-manifested noise reduction with the advent of nontrivial phase transition at the EP (𝛾 = 𝜅) for the pair of eigen-propagation constants ( 𝛽 = √𝜅2 − 𝛾2, −𝛽 ) transiting from purely real to conjugate imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In contrast, the conjugate pair {𝑝𝑖, 𝑞𝑠} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (3b) simply follow 𝑄𝑠-mediated two-mode quadrature squeezing, with their propagation decoupled from {𝑞𝑖, 𝑝𝑠}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Such asymmetric and contrasting dynamics are unavailable to any existing setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' More strikingly, the system will facilitate dual opposing quadrature PT symmetry–{𝑞𝑖, 𝑝𝑠} for active while {𝑝𝑖, 𝑞𝑠} for passive, if without 𝑓𝑠 and 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The general solutions to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (3a) and (3b) are [𝑞𝑖(0) 𝑝𝑠(𝐿)] = 𝑠𝑒𝑐(𝛽𝐿 − 𝜖) [ 𝑐𝑜𝑠 𝜖 − 𝑠𝑖𝑛(𝛽𝐿) − 𝑠𝑖𝑛(𝛽𝐿) 𝑐𝑜𝑠 𝜖 ] [𝑞𝑖(𝐿) 𝑝𝑠(0)] +𝑠𝑒𝑐(𝛽𝐿 − 𝜖) ∫ 𝑑𝑧𝑃𝑠(𝑧) 𝐿 0 [−𝑠𝑖𝑛(𝛽(𝐿 − 𝑧)) 𝑐𝑜𝑠(𝛽𝑧 − 𝜖) ] , (4a) [𝑝𝑖(0) 𝑞𝑠(𝐿)] = 𝑠𝑒𝑐(𝜅𝐿) [ 𝑒𝛾𝐿 − 𝑠𝑖𝑛(𝜅𝐿) − 𝑠𝑖𝑛(𝜅𝐿) 𝑒−𝛾𝐿 ] [𝑝𝑖(𝐿) 𝑞𝑠(0)] +𝑠𝑒𝑐(𝜅𝐿) ∫ 𝑑𝑧𝑄𝑠(𝑧) [−𝑒𝛾𝑧𝑠𝑖𝑛(𝜅(𝐿 − 𝑧)) 𝑒𝛾(𝑧−𝐿)𝑐𝑜𝑠(𝜅𝑧) ] 𝐿 0 , (4b) with 𝜖 = 𝑎𝑟𝑐𝑡𝑎𝑛(𝛾/𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From these solutions, indeed, {𝑞𝑖(0), 𝑝𝑠(𝐿)} but not {𝑝𝑖(0), 𝑞𝑠(𝐿)} carry on the PT- adjusted squeezing and anti-squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—To disclose quadrature PT symmetry, one straightforward way is to analyze the noise behaviors across the phase transition by homodyne detecting quadrature variances in comparison with the ideal squeezed vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This in turn encourages us to look at the following four variances: ⟨∆𝑞𝑗 2⟩ = ⟨𝑞𝑗 2⟩ − ⟨𝑞𝑗⟩ 2and ⟨∆𝑝𝑗 2⟩ = ⟨𝑝𝑗 2⟩ − ⟨𝑝𝑗⟩ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (4a) and (4b), after some lengthy algebra, one reaches ⟨∆𝑞𝑖,0 2 ⟩ = ℎ(𝐿)−2𝑠𝑖𝑛2𝜖−𝑠𝑒𝑐 𝜖𝑐𝑜𝑠(2𝛽𝐿−𝜖) 8𝑐𝑜𝑠2(𝛽𝐿−𝜖) , (5a) ⟨∆𝑝𝑠,𝐿 2 ⟩ = ℎ(𝐿) 𝑐𝑜𝑠 𝜖−𝑐𝑜𝑠(2𝛽𝐿+𝜖)−2𝑠𝑖𝑛2𝜖 𝑐𝑜𝑠(2𝛽𝐿−𝜖) 8 𝑐𝑜𝑠 𝜖𝑐𝑜𝑠2(𝛽𝐿−𝜖) , (5b) ⟨∆𝑞𝑠,𝐿 2 ⟩ = 2+𝑐𝑜𝑠2𝜑𝑒−2𝛾𝐿−𝑐𝑜𝑠 𝜑 𝑐𝑜𝑠(2𝜅𝐿+𝜑) 8 𝑐𝑜𝑠2(𝜅𝐿) , (5c) ⟨∆𝑝𝑖,0 2 ⟩ = (2+𝑐𝑜𝑠2𝜑)𝑒2𝛾𝐿−𝑐𝑜𝑠 𝜑 𝑐𝑜𝑠(2𝜅𝐿−𝜑) 8 𝑐𝑜𝑠2(𝜅𝐿) , (5d) where ℎ(𝐿) = 3 + 2𝛾𝐿 and 𝜑 = tan−1(𝛾/𝜅) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As expected, the PT-inherited variances (5a) and (5b) differentiate themselves from the rest two (5c) and (5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' A hallmark of such is the appearance of the argument 𝛽𝐿 in ⟨∆𝑞𝑖,0 2 ⟩ and ⟨∆𝑝𝑠,𝐿 2 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To have an intuitive picture, we exemplify these variances in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2(a)–(d) for some typical 𝛾/𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2(a) and (b), we observe a few extraordinary traits absent from all past studies on non- Hermitian physics as well as quantum squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' First, in the PT-phase intact region (𝛾 < 𝜅), different from the two-mode squeezed vacuum (TMSV) with an oscillation period of 2𝜋 , both log4⟨∆𝑞𝑖,0 2 ⟩ and log4⟨∆𝑝𝑠,𝐿 2 ⟩ generally display increased classical fluctuations with a period 𝑇 ≈ 2𝜋𝜅/𝛽, except that the former shows a little sub-vacuum-noise suppression at a very short distance range due to the insufficient competition between PT and squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In contrast, both cease to oscillate in the phase broken regime (𝛾 > 𝜅) and are upper bounded by their respective variance curves at the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Moreover, log4⟨∆𝑝𝑠,𝐿 2 ⟩ always grows monotonically above the vacuum-noise level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' while log4⟨∆𝑞𝑖,0 2 ⟩ invariably exhibits quantum squeezing, and the larger 𝛾/𝜅 the larger the squeezing and 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' What’s more, the completely incompatible nature, quantum versus classical, of the same physical observable 𝑞𝑖(0) before and after the PT phase transition renders our system a unique candidate to study the transition between these two different worlds, whose boundary is physically defined by the EP curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Contrarily, since {𝑝𝑖(0), 𝑞𝑠(𝐿)} are decoupled from {𝑞𝑖(0), 𝑝𝑠(𝐿)}, their variances fluctuate periodically, akin to the TMSV case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2(c) and (d), though notably affected by 𝑄𝑠 and 𝛾 , log4⟨∆𝑞𝑠,𝐿 2 ⟩ resembles the regular quadrature squeezing, but not log4⟨∆𝑝𝑖,0 2 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From the above analysis, we learned that for the same single-mode quadrature, PT results in a nontrivial fundamental transition from quantum to classical when the non-Hermitian parameter 𝛾 oversteps a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' One may wonder whether this exotic phenomenon can also take place in a two-mode quadrature measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The answer is affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To see how this works, we pay attention to 𝑑1 = [𝑞𝑖(0) + 𝑞𝑠(𝐿)]/√2 and 𝑑2 = [𝑝𝑖(0) + 𝑝𝑠(𝐿)]/√2, which satisfy [𝑑1, 𝑑2] = 𝑖/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For the vacuum input, it is easy to check that their variances are simply the sum of the single-mode ones (5a)–(5d), 〈∆𝑑1 2〉 = ⟨∆𝑞𝑖,0 2 ⟩+⟨∆𝑞𝑠,𝐿 2 ⟩ 2 , 〈∆𝑑2 2〉 = ⟨∆𝑝𝑖,0 2 ⟩+⟨∆𝑝𝑠,𝐿 2 ⟩ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (6) Based on Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2(b) and (d), 〈∆𝑑2 2〉 is expected to be distributed above the vacuum noise all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Moreover, because ⟨∆𝑝𝑠,𝐿 2 ⟩ and ⟨∆𝑝𝑖,0 2 ⟩ have different fluctuation periods before the phase transition, we envision that 〈∆𝑑2 2〉 will exhibit interleaved dual periodic oscillations but reduce to a single period after the phase breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Though the situation becomes somewhat subtle for 〈∆𝑑1 2〉 , its layout can be deduced similarly by compromising Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To be specific, in the PT-phase unbroken region, it is a double-cycle growth fluctuation staggered on top of the vacuum noise (except the very short distance case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' When PT symmetry spontaneously breaks down, counterintuitively, the single-period oscillating 〈∆𝑑1 2〉 will always return certain squeezing at some effective distances, and these distances will be extended for a bigger 𝛾/𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Same as 𝑞𝑖(0), 𝑑1 can serve as another physical probe to visualize the quantum-to-classical transition induced by quadrature PT symmetry, too, with the boundary defined by the EP curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' All these statements excellently agree with our numerical simulations given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 3(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' RISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—Other than homodyne detection, there is one additional means to explore quadrature PT, the so-called relative intensity squeezing measurement (RISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Traditionally, this method enables the shot-noise of one beam to be measured and subtracted from the other so as to attain lower-noise differential measurement of a signal of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To this end, we begin with our own relative- intensity operator, 𝑁𝑖,0 − 𝑁𝑠,𝐿 = 𝑎𝑖 †(0)𝑎𝑖(0) − 𝑎𝑠 †(𝐿)𝑎𝑠(𝐿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The degree of squeezing is then characterized by the noise figure (NF), which is determined by the relative-intensity variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Mathematically, it takes the form [38] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' PT-manifested log4⟨∆𝑞𝑖,0 2 ⟩ (a) and log4⟨∆𝑝𝑠,𝐿 2 ⟩ (b) in the presence of quantum Langevin noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Non-PT- symmetric but loss-noise-mediated log4⟨∆𝑞𝑠,𝐿 2 ⟩ (c) and log4⟨∆𝑝𝑖,0 2 ⟩ (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As the references, the black solid and dashed lines represent the regular TMSV (𝛾/𝜅 = 0) and vacuum noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' PT-symmetric log4⟨∆𝑑1 2⟩ (a) and log4⟨∆𝑑2 2⟩ (b) with account of quantum noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Again, as the references, the black solid and dashed curves are respectively the ideal TMSV (𝛾/𝜅 = 0) and vacuum noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 10 10 (a) (b) 8 8 人人人 人人人人人人 log4(Aqi,o) 4 2 2 0 0 2 0 5 10 15 20 25 0 5 10 15 20 25 2KL 2KL 8 25 (c) (d) 6 20 vacuumnoise Y/x=O y/x=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='5 y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='y/x=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='5 log4( 10 5 0 2 5 10 15 20 0 5 10 15 20 2KL 2kL10 25 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8 (b) 8 从人人 20 vacuum noise 6 y/x=0y/K=1 《pv) 15 人人人 Y/x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6—/x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4 5 10 15 20 25 0 5 10 15 20 2K 2KLNF = Var[𝑁𝑖,0−𝑁𝑠,𝐿] 〈𝑁𝑖(0)〉+〈𝑁𝑠(𝐿)〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (7) Here, the average photon numbers are computed by plugging Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (4a) and (4b) to 〈𝑁𝑖,0〉 = ⟨𝑞𝑖 2(0)⟩ + ⟨𝑝𝑖 2(0)⟩ − 1/2 and 〈𝑁𝑠,𝐿〉 = ⟨𝑞𝑠2(𝐿)⟩ + ⟨𝑝𝑠2(𝐿)⟩ − 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In stark contrast to the quadrature variances discussed earlier, the NF, while bringing about some alike characteristics, clearly reveals some quite opposite peculiarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For 𝛾/𝜅 ≥ 1, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 4(a), in addition to the incremental single-period fluctuation log10(NF≥0 + 1) grows along with the increment of 2𝜅𝐿, and the larger γ/κ is, the noisier it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From the plot, it is not difficult to conclude that in the PT-phase broken region, NF is essentially occupied by the noise anti- squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' However, NF behaves highly complex as 𝛾/𝜅 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Although it is still an interleaved double-period oscillation within this range, the EP curve is no longer the partition to separate the classical and quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In line with the numerical simulations, we find that quantum squeezing materializes when 𝛾/𝜅 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Some representative examples are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 4(b) by plotting −log10(NF<0 + 1) for different γ/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Their comparison suggests that the smaller the value of γ/κ, the more pronounced the achievable squeezing over a longer distance 2𝜅𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As a matter of fact, the RISM obviously supplies certain sharp signatures unreachable to the homodyne detection, regardless of the two highly unbalanced channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Before proceeding, a few remarks are ready here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' First, even in the presence of Langevin noise, utilizing PSA instead of PIA is practicable to accomplish quantum optical PT under fair sampling measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Second, contrary to PIA, PSA arouses the unusual quadrature PT and licenses the singular quantum-to-classical transition accompanied by the PT phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Last but not least, quadrature PT sheds new light on protecting continuous-variable (CV) qubits from decoherence in inevitable lossy transmission, a long-standing conundrum for various CV-based quantum technologies [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—Being a discipline of practical application, quantum sensing [40-42] exploits quantum properties, effects, or systems to fulfill high-resolution and super-sensitive measurements of physical parameters over the similar measurements performed within a classical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For this, quantum squeezing has long been recognized as one of the indispensable nonclassical resources for ultra-precision estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Among them, one far-reaching example is its recent adoption by the Laser Interferometer Gravitational-Wave Observatory (LIGO) for gravitational wave detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Nevertheless, the inevitable propagation loss often degrades the available squeezing and compromises the promised sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We note that in recent non- Hermitian studies, the abrupt change near EP has been capitalized for enhanced sensing in classical settings [43- 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Yet, its extension to the quantum level turns out to be problematic because of quantum noise [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To avoid such noise, one usually resorts to either ideal anti-PT systems or post-selection measurement [25,34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Unlike these studies, here we directly confront Langevin noise and explore the opportunity of quadrature PT in quantum sensing under fair sampling measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We are particularly interested to know whether the system could have any advantage in improving sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As shown below, the PT-quadrature observables can yield the best performance of classical sensing before the phase transition but departing far from the EP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' while the non- PT-quadrature observables are capable of optimal quantum sensing by noise-mediated squeezing for 𝛾/𝜅 less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This distinguishes our work from the previous anti-PT-, squeezing-, or EP-based proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Our analysis is carried out by estimating 𝜅 (or 𝛾) and comparing the achievable precision with the quantum Cramér-Rao bound (set by the quantum Fisher information of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (a) PT-regulated noise figure (NF) via relative intensity squeezing measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (b) Representative examples of quantum PT-symmetric NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 40 + (a) log10(NFzo+ 10 (b) 30 Y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8 /x=1 20 Y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 Y/x=2 10 +1) 0 Y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 + 07 107 Y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 AN INF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' /x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 10 log10l 5 l0g10() 0 10 15 20 25 30 35 0 5 10 15 20 25 30 35 2kL 2kLquantum state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Suppose that the two bosonic modes are initially prepared in a coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' An optimal homodyne measurement is then implemented on, say, 𝑞𝑖(0)after an evolution distance 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (4a), one can readily have its mean value and variance, ⟨𝑞𝑖(0)⟩ = 𝑐𝑜𝑠 𝜖⟨𝑞𝑖(𝐿)⟩−𝑠𝑖𝑛(𝛽𝐿)⟨𝑝𝑠(0)⟩ 𝑐𝑜𝑠(𝛽𝐿−𝜖) , (8) ⟨∆𝑞𝑖,0 2 ⟩ = 3+𝑤[2𝛾𝐿−tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖 8cos2(𝛽𝐿−𝜖) , (9) where 𝑤 = 2𝑛𝑡ℎ + 1 and 𝑛𝑡ℎ is the average thermal boson number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (8) and (9), it is clear that the Langevin noise shifts the peaks of ⟨∆𝑞𝑖,0 2 ⟩ away from the troughs of ⟨𝑞𝑖(0)⟩, so that they do not coincide at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To ease the subsequent derivations, hereafter we assume 𝛼𝑖 = 𝑖𝛼𝑠∗ = √2𝛼𝑒𝑖𝜋/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The estimation precision relies on measuring the change of ⟨𝑞𝑖(0)⟩ due to a tiny perturbation 𝛿𝜅 on a preset 𝜅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This alternatively suggests to examine the system response to ⟨𝑞𝑖(0)⟩ for a small variation 𝛿𝜅 around 𝜅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We thus define the susceptibility to capture such a response, 𝜒𝜅 𝑞𝑖(0) ≡ 𝜕⟨𝑞𝑖(0)⟩ 𝜕𝜅 = 𝛼{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]} 2𝛽cos2(𝛽𝐿−𝜖) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (10) When 𝜅 → 𝛾 or 𝛽 → 0 , 𝜒𝜅 𝑞𝑖(0) → 𝛼𝐿(3 + 𝜅2𝐿2)/ [3(1 + 𝜅𝐿)2] is a curvetureless constant, implying the loss of the sensing ability in the EP vicinity for the chosen observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' On the other hand, the κ-estimation is jointly determined by the variance and susceptibility, ∆𝜅,𝑞𝑖,0 2 = ⟨∆𝑞𝑖,0 2 ⟩/ [𝜒𝜅 𝑞𝑖(0)] 2 , whose inverse dictates the accuracy, ∆𝜅,𝑞𝑖,0 −2 = 2𝛼2{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]}2 𝛽2cos2(𝛽𝐿−𝜖){3+𝑤[2𝛾𝐿− tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (11) The sensing fulfillment is to compare ∆𝜅,𝑞𝑖,0 −2 ,0with the quantum Fisher information, 𝐹𝜅, which sets the ultimate precision, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', the lower quantum Cram´ er-Rao bound for any optimal measurement, 𝐹𝜅 ≥ ∆𝜅,𝑞𝑖,0 −2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Fκcan be accordingly derived from 𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (12) for the final state |𝜓𝑓⟩ of the system in the Schrödinger representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As detailed in Supplementary Information, one can perform similar sensing evaluations to the rest three quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Basing on the calculations shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 5(a) and (b), we note that in the current arrangement, 𝑞𝑖(0) and 𝑝𝑠(𝐿) can only permit optimal classical sensing for a moderate medium length in the phase-unbroken regime but away from the EP, as revealed by the ratios of ∆𝜅,𝑞𝑖,0 −2 /𝐹𝜅 and ∆𝜅,𝑝𝑠,𝐿 −2 /𝐹𝜅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' On the contrary, 𝑞𝑠(𝐿) and 𝑝𝑖(0) behave distinctively in the sense that, despite the impacts of the photon loss and Langevin noise, they are still able to offer optimal quantum sensing over a relatively longer distance for smaller 𝛾/𝜅, as sketched in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 5(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In short, fundamentally different from all previous research, the PSA-induced quadrature PT enables a unique way to observe a quantum-to-classical transition for a physical observable at the breakdown of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Such quadrature PT radically reshapes the dynamics of two-mode squeezing with a striking phase transition never seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Unfortunately, optical loss and Langevin noise hinder the PT quadrature pair to confer a quantum advantage in improving sensitivity, although the non-PT pair can still offer optimal quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Besides of nonlinear wave mixing, our model can be also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum sensing in quadrature PT symmetry characterized by the ratios of ∆𝜅,𝑞𝑖,0 −2 (a), ∆𝜅,𝑝𝑠,𝐿 −2 (b), ∆𝜅,𝑞𝑠,𝐿 −2 (c), and ∆𝜅,𝑝𝑖,0 −2 (d) to the quantum Fisher information 𝐹𝜅 for the parameters {𝛼 = 2, 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2, 𝜅 = 1 } (blue), {2,1,1} (red), and {2,2,1} (orange), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 X10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 ×10-3 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 Y/x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 6 Y/x=1 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 Y/x=2 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0 5 1015 202530 0 5 1015202530 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2KL 2KL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (c) (d) X10-3 X10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 6 6 2 4 AK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0 0 51015202530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0 5 1015202530 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2KL 2KLachieved in other platforms such as superconducting circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Most importantly, our work forges a new avenue to explore the long-sought, nontrivial quantum-to- classical transition utilizing non-Hermitian physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—This work was supported by NSF 1806519 and NSF EFMA-1741693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' acknowledges the support by the National Key R&D Program of China (2021YF A1400803).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' was supported by the Nature Science Foundation of Guangdong Province (2019A1515011401).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='—J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' conceived the theoretical scheme and supervised the whole project with the help of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', supervised by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', carried out the whole calculations with the assistance of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' All authors contributed to the discussions and writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' dmliu@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='cn † jxs@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='cn ‡ jianming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='wen@kennesaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu [1] Bender, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Boettcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Real spectra in non- Hermitian Hamiltonians having PT symmetry.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Khajavikhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Musslimani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Rotter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Christodoulides, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 51, 072001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' [10] R¨ oter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} 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N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Khajavikhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Enhanced sensitivity at higher- order exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Nature 548, 187-191 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' [48] Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Sweeney, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Hsu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Stone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum noise theory of exceptional point amplifying sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 123, 180501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 1 Supplementary Information for “Quantum-to-classical transition enabled by quadrature-PT symmetry” Wencong Wang, Yanhua Zhai, Dongmei Liu*, Xiaoshun Jiang*, Saeid Vashahri Ghamsari, and Jianming Wen* Emails: dmliu@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' jxs@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' jianming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='wen@kennesaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='edu I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Derivation of the Heisenberg-Langevin equations In our previous work [1], we have theoretically proved that a forward parametric optical process may lead to anti-PT symmetry while a backward parametric optical process can result in PT symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For this reason, as schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 1 in the main text [2], we are interested in a backward nonlinear parametric optical process such as backward four-wave mixing, where the two counter-propagating parametric modes, idler and signal, respectively experience balanced phase-sensitive linear quantum amplification (PSA) and attenuation in their own channels within the medium of length 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For such an open system, the evolution of the paired idler and signal field operators is effectively determined by a non-Hermitian Hamiltonian, 𝐻 = 𝑖 ℏ𝑔 2 (𝑎𝑖 †2 − 𝑎𝑖 2) − 𝑖ℏ𝛾𝑎𝑠 †𝑎𝑠 + ℏ𝜅(𝑎𝑖 †𝑎𝑠 † + 𝑎𝑖𝑎𝑠), (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1) where 𝑔 and 𝛾 respectively denote the PSA rate and loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1), one can readily obtain the Heisenberg equations of the idler-signal field operators, 𝑖ℏ 𝜕𝑎𝑖 𝜕(−𝑧) = [𝑎𝑖, 𝐻], (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2a) 𝑖ℏ 𝜕𝑎𝑠 𝜕𝑧 = [𝑎𝑠, 𝐻].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2b) Thanks to the noiseless amplification empowered by the PSA, the idler dynamics is not subject to the additive noise and the commutation relation can be always satisfied throughout the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' However, this is not true for the lossy signal propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To restore the commutation relation, one has to introduce the quantum Langevin noise in the Heisenberg equation of the signal field operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In this way, we arrive at the following coupled Heisenberg-Langevin equations for the system of interest, 𝜕𝑎𝑖 𝜕𝑧 = 𝑔𝑎𝑖 † + 𝑖𝜅𝑎𝑠 †, (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3a) 𝜕𝑎𝑠 𝜕𝑧 = −𝛾𝑎𝑠 − 𝑖𝜅𝑎𝑖 † + 𝑓𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3b) Though Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3a) and (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3b) seem to have nothing to do with PT symmetry at first glance, as pointed out in the main text, the hidden PT symmetry arises if transforming both equations into the dynamics of the corresponding quadrature operators, 𝑞𝑗 = (𝑎𝑗 † + 𝑎𝑗)/2 and 𝑝𝑗 = 𝑖(𝑎𝑗 † − 𝑎𝑗)/2 (𝑗 = 𝑖, 𝑠) with [𝑞𝑗, 𝑝𝑗] = 𝑖/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For simplicity, we concentrate on the case of the balanced PSA and loss, 𝑔 = 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' With these preparations, one can easily attain the following sets of the coupled-quadrature equations 𝑑 𝑑𝑧 [𝑞𝑖 𝑝𝑠] = [ 𝛾 𝜅 −𝜅 −𝛾] [𝑞𝑖 𝑝𝑠] + [0 𝑃𝑠], (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4a) 𝑑 𝑑𝑧 [𝑝𝑖 𝑞𝑠] = [−𝛾 𝜅 −𝜅 −𝛾] [𝑝𝑖 𝑞𝑠] + [ 0 𝑄𝑠], (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4b) with 𝑃𝑠 = 𝑖(𝑓𝑠 † − 𝑓𝑠)/2 and 𝑄𝑠 = (𝑓𝑠 † + 𝑓𝑠)/2 being the Langevin-noise quadrature operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4a), one can derive the effective Hamiltonian matrix for the quadrature pair (𝑞𝑖, 𝑝𝑠), which reads 𝐻(𝑞𝑖,𝑝𝑠) = [ 𝑖𝛾 𝑖𝜅 −𝑖𝜅 −𝑖𝛾] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='5) It is straightforward to show that 𝐻(𝑞𝑖,𝑝𝑠) is indeed PT-symmetric, because it satisfies 𝑃𝑇𝐻(𝑞𝑖,𝑝𝑠) = 𝐻(𝑞𝑖,𝑝𝑠)𝑃𝑇 for the combined PT operation with the parity operator being 𝑃 = [0 1 1 0] and the time- reversal operator assuming the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' For this reason, we call (𝑞𝑖, 𝑝𝑠) the PT- 2 quadrature pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This is in a sharp contrast to the effective Hamiltonian matrix 𝐻(𝑝𝑖,𝑞𝑠) = [−𝑖𝛾 𝑖𝜅 −𝑖𝜅 −𝑖𝛾] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4b) for the other conjugate quadrature pair (𝑝𝑖, 𝑞𝑠), which is apparently irrelevant to PT symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The two eigenvalues of 𝐻(𝑞𝑖,𝑝𝑠) are 𝛽± = ±√𝜅2 − 𝛾2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Akin to the classical PT symmetry, 𝛾 𝜅 < 1 corresponds to the quadrature PT-phase unbroken regime while for 𝛾 𝜅 > 1 , quadrature PT symmetry spontaneously breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The quadrature PT phase transition occurs at the singular or exceptional point (EP), 𝛾 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In terms of the initial boundary conditions, the general solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4a) and (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4b) are readily found to be [𝑞𝑖(0) 𝑝𝑠(𝐿)] = sec(𝛽𝐿 − 𝜖) [ cos 𝜖 −sin(𝛽𝐿) −sin(𝛽𝐿) cos 𝜖 ] [𝑞𝑖(𝐿) 𝑝𝑠(0)] + sec(𝛽𝐿 − 𝜖) ∫ 𝑑𝑧𝑃𝑠(𝑧) 𝐿 0 [−sin(𝛽(𝐿 − 𝑧)) cos(𝛽𝑧 − 𝜖) ], (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6a) [𝑝𝑖(0) 𝑞𝑠(𝐿)] = sec(𝜅𝐿) [ 𝑒𝛾𝐿 −sin(𝜅𝐿) −sin(𝜅𝐿) 𝑒−𝛾𝐿 ] [𝑝𝑖(𝐿) 𝑞𝑠(0)] + sec(𝜅𝐿) ∫ 𝑑𝑧𝑄𝑠(𝑧) [−𝑒𝛾𝑧sin(𝜅(𝐿 − 𝑧)) 𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧) ] 𝐿 0 , (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6b) with 𝜖 = arctan ( 𝛾 𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' It is not difficult to prove that the dynamical solutions (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6a) and (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6b) well maintain the commutation relations at all times, [𝑞𝑖(0), 𝑝𝑖(0)] = 𝑒𝛾𝐿cos 𝜖 cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑖(𝐿), 𝑝𝑖(𝐿)] + sin(𝛽𝐿)sin(𝜅𝐿) cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑠(0), 𝑞𝑠(0)] + ∫ 𝑑𝑧 𝐿 0 𝑒𝛾𝑧sin(𝛽(𝐿 − 𝑧))sin(𝜅(𝐿 − 𝑧)) cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑃𝑠, 𝑄𝑠] = 𝑖 2, (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='7a) [𝑞𝑠(𝐿), 𝑝𝑠(𝐿)] = sin(𝛽𝐿)sin(𝜅𝐿) cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑖(𝐿), 𝑞𝑖(𝐿)] + 𝑒−𝛾𝐿 cos 𝜖 cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑠(0), 𝑝𝑠(0)] + ∫ 𝑑𝑧 𝐿 0 𝑒𝛾(𝑧−𝐿)cos(𝛽𝑧 − 𝜖)cos(𝜅𝑧) cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑄𝑠, 𝑃𝑠] = 𝑖 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='7b) Note that the quantum Langevin noise of zero mean satisfies 〈𝑓𝑠(𝑧)𝑓𝑠 †(𝑧′)〉 = 2𝛾𝛿(𝑧 − 𝑧′) and 〈𝑓𝑠 †(𝑧)𝑓𝑠(𝑧′)〉 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The quantumness of quadrature PT symmetry can be further approached by analyzing the variances (or noise fluctuations) of 𝑞𝑗 and 𝑝𝑗 for the vacuum input state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' After some algebra, we have reached the following important results: ⟨∆𝑞𝑖,0 2 ⟩ = ⟨𝑞𝑖 2(0)⟩ − ⟨𝑞𝑖(0)⟩2 = ℎ(𝐿) − 2sin2𝜖 − sec 𝜖 cos(2𝛽𝐿 − 𝜖) 8cos2(𝛽𝐿 − 𝜖) , (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8a) ⟨∆𝑝𝑠,𝐿 2 ⟩ = ⟨𝑝𝑠 2(𝐿)⟩ − ⟨𝑝𝑠(𝐿)⟩2 = ℎ(𝐿) cos 𝜖 − cos(2𝛽𝐿 + 𝜖) − 2sin2𝜖 cos(2𝛽𝐿 − 𝜖) 8 cos 𝜖 cos2(𝛽𝐿 − 𝜖) , (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8b) ⟨∆𝑞𝑠,𝐿 2 ⟩ = ⟨𝑞𝑠 2(𝐿)⟩ − ⟨𝑞𝑠(𝐿)⟩2 = 2 + cos2𝜑𝑒−2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 + 𝜑) 8 cos2(𝜅𝐿) , (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8c) ⟨∆𝑝𝑖,0 2 ⟩ = ⟨𝑝𝑖 2(0)⟩ − ⟨𝑝𝑖(0)⟩2 = (2 + cos2𝜑)𝑒2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 − 𝜑) 8 cos2(𝜅𝐿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8d) where ℎ(𝐿) = 3 + 2𝛾𝐿 and 𝜑 = arctan ( 𝛾 𝜅) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We notice from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8a)—(S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8d) that although 3 these variances contain linear terms, they do not affect the periodic characteristics of the noise fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As revealed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 2 in the main text, when taking 2𝜅𝐿 as the dimensionless variable, we find that the oscillation period of the variances of the PT-quadrature pair (𝑞𝑖, 𝑝𝑠) is approximately to be 2𝜅𝜋 𝛽 while the fluctuation period of the variances of the non-PT quadrature pair (𝑝𝑖, 𝑞𝑠) simply assumes 2𝜋 for the parameter space in the PT-phase intact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As emphasized in the main text, the ultimate novelty of our work is not just to find a system capable of the observation of genuine quantum optical PT symmetry under fair sampling measurement, but to unearth an extraordinary phenomenon that has never been discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' That is, the PT-quadrature observable enables one to witness a compelling quantum-to-classical transition perfectly coinciding with the PT phase transition by varying the non-Hermitian parameter 𝛾, and the transition boundary is physically defined by the EP curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To the best of our knowledge, this is the first proposal on exploring the untrivial transition between two incompatible worlds, classical and quantum, with a well- defined physical boundary by measuring the same quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' We are aware that there exists a parallel way in the literature that exploits some massive quantum systems such as cooled cavity optomechanical structures to probe the quantum-to-classical transition by constantly checking the decoherence of a quantum state when manipulating some system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' However, even if these proposals are viable in the lab, they have to face an inescapable conundrum, that is, in these systems it becomes extremely challenging to determine the exact transition boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In other words, observing the sharp transition would be exceedingly difficult and even impossible for these protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In contrast, these difficulties do not appear in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Moreover, our method aims to measure the expectation of a quantum observable while the existing protocols concentrate on studying the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This difference fundamentally distinguishes our work from all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' All in all, our work for the first time presents a new way to explore the quantum-to-classical transition by taking advantage of non- Hermiticity and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Before ending this part of discussion, here we would like to add a couple of additional comments on the following important issues on quantum optical PT symmetry raised in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' One is in response to the quantum noncloning theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In fact, the amplification won’t violate the quantum noncloning theorem at all in our proposal, because the PSA here only acts on the single-mode idler field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This also concurs with the well-established knowledge in the field of quantum optics, especially in quantum squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The second question concerns the law of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Although the gain may lead to the fast-light or superluminal effect, since the quantum noise introduced by the transmission loss is inseparable from the actual signal of interest, the causality proves not to be a problem when considering PT symmetry at the quantum level (as we did here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Derivation of quantum sensing For the quantum sensor application, we consider the generation of the idler-signal bosonic modes from a seeding two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' If taking into account the thermal reservoir with an average thermal bosonic number 𝑛th, the quantum Langevin noise in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3b) obeys the following properties, 〈𝑓𝑠(𝑧)𝑓𝑠 †(𝑧′)〉 = 2𝛾(𝑛th + 1)𝛿(𝑧 − 𝑧′) and 〈𝑓𝑠 †(𝑧)𝑓𝑠(𝑧′)〉 = 2𝛾𝑛th𝛿(𝑧 − 𝑧′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' After an interaction length 𝐿, the mean values and variances of the quadrature measurements on 𝑞𝑗 and 𝑝𝑗 with respect to the final state |𝜓𝑓⟩ of the system can be obtained by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6a) and (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6b), ⟨𝑞𝑖(0)⟩ = cos 𝜖 cos(𝛽𝐿 − 𝜖) ⟨𝑞𝑖(𝐿)⟩ − sin(𝛽𝐿) cos(𝛽𝐿 − 𝜖) ⟨𝑝𝑠(0)⟩, (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1a) ⟨𝑝𝑠(𝐿)⟩ = − sin(𝛽𝐿) cos(𝛽𝐿−𝜖) ⟨𝑞𝑖(𝐿)⟩ + cos 𝜖 cos(𝛽𝐿−𝜖) ⟨𝑝𝑠(0)⟩, (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1b) ⟨𝑞𝑠(𝐿)⟩ = − sin(𝜅𝐿) cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ + 𝑒−𝛾𝐿 cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1c) 4 ⟨𝑝𝑖(0)⟩ = 𝑒𝛾𝐿 cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ − sin(𝜅𝐿) cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1d) and ⟨∆𝑞𝑖,0 2 ⟩ = 3 + 𝑤[2𝛾𝐿 − tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖 8cos2(𝛽𝐿 − 𝜖) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2a) ⟨∆𝑝𝑠,𝐿 2 ⟩ = 3 + 𝑤[2𝛾𝐿 + tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖 8cos2(𝛽𝐿 − 𝜖) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2b) ⟨∆𝑞𝑠,𝐿 2 ⟩ = 𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿) 8cos2(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2c) ⟨∆𝑝𝑖,0 2 ⟩ = 𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)] 8cos2(𝜅𝐿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2d) Here, 𝑤 = 2𝑛th + 1 with the thermal average boson number 𝑛th = [Exp ( ℎ𝑣𝜆 𝑘𝐵𝑇) − 1] −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' One can easily check that the thermal photon number becomes infinitesimal at the room temperature (~300 K) in the visible spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Alternatively, the above Langevin noise properties reduce to the simpler formats mentioned in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The ultimate precision of the parameter estimation of 𝜅 is essentially determined by the variance ∆𝜅 2 in terms of a targeted physical observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The performance of the proposed quadrature-PT sensing scheme can be however evaluated by comparing the inverse variances ∆𝜅,𝑞𝑗 −2 = (𝜒𝜅 𝑞𝑗) 2 〈∆𝑞𝑗 2〉 and ∆𝜅,𝑝𝑗 −2 = (𝜒𝜅 𝑝𝑗) 2 〈∆𝑝𝑗 2〉 with the quantum Fisher information 𝐹𝜅 at the system’s final state |𝜓𝑓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Here, we have introduced the susceptibilities 𝜒𝜅 𝑞𝑗 = 𝜕𝜅⟨𝑞𝑗⟩ and 𝜒𝜅 𝑝𝑗 = 𝜕𝜅⟨𝑝𝑗⟩ to capture the system response to ⟨𝑞𝑗⟩ and ⟨𝑝𝑗⟩ for a small perturbation 𝛿𝜅 about the preset 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' With the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1a)—(S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1d), after some labor one can show that 𝜒𝜅 𝑞𝑗 and 𝜒𝜅 𝑝𝑗 (𝑗 = 𝑖, 𝑠) take the form of, 𝜒𝜅 𝑞𝑖(0) = 𝛼{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]} 2𝛽cos2(𝛽𝐿 − 𝜖) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3a) 𝜒𝜅 𝑝𝑠(𝐿) = 𝜒𝜅 𝑞𝑖(0), (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3b) 𝜒𝜅 𝑞𝑠(𝐿) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1], (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3c) 𝜒𝜅 𝑝𝑖(0) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3d) By plugging Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2a)—(S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2d) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3a)—(S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='3d) into the inverse variances ∆𝜅,𝑞𝑗 −2 = (𝜒𝜅 𝑞𝑗) 2 〈∆𝑞𝑗 2〉 and ∆𝜅,𝑝𝑗 −2 = (𝜒𝜅 𝑝𝑗) 2 〈∆𝑝𝑗 2〉 , we arrive at the following key results: ⟨∆𝜅,𝑞𝑖,0 −2 ⟩ = 2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2 𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 − tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖} , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4a) ⟨∆𝜅,𝑝𝑠,𝐿 −2 ⟩ = 2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2 𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 + tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖}, (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4b) ⟨∆𝜅,𝑞𝑠,𝐿 −2 ⟩ = 8𝛼2𝐿2sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1]2 𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4c) 5 ⟨∆𝜅,𝑝𝑖,0 −2 ⟩ = 8𝛼2𝐿2sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1]2 𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4d) After having the inverse variances (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4a)—(S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='4d), now let us turn our attention to the quantum Fisher information 𝐹𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', the quantum Cramér-Rao bound, which demands the optimal measurement to satisfy the inequality ∆𝜅 −2≤ 𝐹𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In this sensing protocol, we start with the initial system state to be in a two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩ for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Then, the final state of the system evolves as |𝜓𝑓⟩ = 𝑈 √𝜇 |𝜓𝑖⟩, where 𝑈 = 𝑒−𝑖𝐻𝐿 is the evolution operator and 𝜇 = ⟨𝜓𝑓|𝜓𝑓⟩ is the normalization coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' By working in the Schrödinger picture and treating the idler-signal field operators 𝑎𝑖 = 𝑎𝑖(𝐿) and 𝑎𝑠 = 𝑎𝑠(0) as constant operators, the quantum Fisher information can be calculated by the definition of 𝐹𝜅 = 4 (⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ − |⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩| 2) for a parameter 𝜅 that controls the strength of the system’s Hamiltonian 𝐻 (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1) with respect to a known physical observable (in our case, it can be any of the four quadratures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To this end, let us give a detailed examination on the first term in 𝐹𝜅: ⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ = ⟨𝜓𝑖| √𝜇(𝜕𝜅𝑈†) − 𝜕𝜅𝜇 2√𝜇 𝑈† 𝜇 √𝜇(𝜕𝜅𝑈) − 𝜕𝜅𝜇 2√𝜇 𝑈 𝜇 |𝜓𝑖⟩ = 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿 𝜕𝜅𝜇 2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿 𝜕𝜅𝜇 2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ + (𝜕𝜅𝜇)2 4𝜇2 , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='5) where 𝜕𝜅𝑈 = −𝑖𝐿(𝜕𝜅𝐻)𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Note that 𝜕𝜅𝐻 commutes with 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In the same way, we can also obtain the exact expression for the second term as follows: |⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩| 2 = 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿 𝜕𝜅𝜇 2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿 𝜕𝜅𝜇 2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ + (𝜕𝜅𝜇)2 4𝜇2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6) Substituting these two results into 𝐹𝜅 yields the concise and intuitive expression of the quantum Fisher information, which is 𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='7) Since 𝜕𝜅𝐻 = 𝜕𝜅𝐻† = ℏ(𝑎𝑖 †𝑎𝑠 † + 𝑎𝑖𝑎𝑠) = 2ℏ(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖) in the Schrödinger picture and the expectation value of an operator does not change along with the picture transformation, we can transform the above formulae of the quantum Fisher information into the Heisenberg representation to ease the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' That is, 𝐹𝜅 = 16𝐿2 (⟨𝜓𝑓|(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖)2|𝜓𝑓⟩ − (⟨𝜓𝑓|𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖|𝜓𝑓⟩) 2) = 16𝐿2{⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2|𝜓𝑖⟩ − [⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]|𝜓𝑖⟩]2} = 16𝐿2{〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8), one can easily evaluate the term 𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6a) and (S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' After some lengthy derivations, we eventually get 𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) = 𝐴𝑞𝑖(𝐿)𝑝𝑖(𝐿) + 𝐵𝑝𝑠(0)𝑝𝑖(𝐿) + ∫ 𝑑𝑧𝐶𝑃𝑠𝑝𝑖(𝐿) 𝐿 0 + 𝐷𝑞𝑖(𝐿)𝑞𝑠(0) + 𝐸𝑝𝑠(0)𝑞𝑠(0) + ∫ 𝑑𝑧𝐹𝑃𝑠𝑞𝑠(0) 𝐿 0 + ∫ 𝑑𝑧𝐺𝑄𝑠𝑞𝑖(𝐿) 𝐿 0 + ∫ 𝑑𝑧𝐽𝑄𝑠𝑝𝑠(0) 𝐿 0 + ∫ 𝑑𝑧𝑅𝑃𝑠𝑄𝑠 𝐿 0 , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='9) where all the involved coefficients are 6 𝐴 = 𝑒𝛾𝐿sin(𝛽𝐿)−cos𝜖sin(𝜅𝐿) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10a) 𝐵 = sin(𝛽𝐿)sin(𝜅𝐿)−𝑒𝛾𝐿 cos𝜖 cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10b) 𝐶 = sin[𝛽(𝐿−𝑧)]sin(𝜅𝐿)−𝑒𝛾𝐿cos(𝛽𝑧−𝜖) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10c) 𝐷 = 𝑒−𝛾𝐿 cos 𝜖−sin(𝛽𝐿)sin(𝜅𝐿) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10d) 𝐸 = cos𝜖sin(𝜅𝐿)−𝑒−𝛾𝐿sin(𝛽𝐿) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10e) 𝐹 = sin (𝜅𝐿)cos(𝛽𝑧−𝜖)−𝑒−𝛾𝐿sin(𝛽(𝐿−𝑧)) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10f) 𝐺 = 𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧) cos𝜖−𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]sin(𝛽𝐿) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10g) 𝐽 = 𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos 𝜖−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin(𝛽𝐿) cos(𝛽𝐿−𝜖)cos(𝜅𝐿) , (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10h) 𝑅 = 𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos(𝛽𝑧−𝜖)−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin[𝛽(𝐿−𝑧)] cos(𝛽𝐿−𝜖)cos(𝜅𝐿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='10i) With these results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' we are ready to work out the following step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='= 𝐴2(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ − ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐵2(⟨𝑝𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2(𝐿)𝑝𝑠2(0)⟩ − ⟨𝑝𝑠(0)𝑝𝑖(𝐿)⟩2) + 𝐷2(⟨𝑞𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2(𝐿)𝑞𝑠2(0)⟩ − ⟨𝑞𝑠(0)𝑞𝑖(𝐿)⟩2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐸2(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ − ⟨𝑝𝑠(0)𝑞𝑠(0)⟩2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐴𝐵(⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑝𝑖(𝐿)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− 2⟨𝑝𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑝𝑠(0)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐴𝐷(⟨𝑞𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑞𝑠(0)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐵𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− 2⟨𝑝𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑝𝑖(𝐿)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐷𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑞𝑠(0)⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− 2⟨𝑞𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑞𝑖(𝐿)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ 𝐵𝐷(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑠(0)𝑞𝑠(0)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑠(0)⟩⟨𝑝𝑖(𝐿)⟩⟨𝑝𝑠(0)⟩) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ ∫ 𝑑𝑧[𝐶2⟨𝑝𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2(𝐿)⟩ + 𝐹2⟨𝑞𝑠2(0)⟩ + 2𝐶𝐹⟨𝑞𝑠(0)𝑝𝑖(𝐿)⟩]⟨𝑃𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ ∫ 𝑑𝑧[𝐺2⟨𝑞𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2(𝐿)⟩ + 𝐽2⟨𝑝𝑠2(0)⟩ + 2𝐺𝐽⟨𝑝𝑠(0)𝑞𝑖(𝐿)⟩]⟨𝑄𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ ∫ 𝑑𝑧𝐶𝐺[⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩⟨𝑃𝑠𝑄𝑠⟩] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ ∫ 𝑑𝑧𝐹𝐽[⟨𝑝𝑠(0)𝑞𝑠(0)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)⟩⟨𝑃𝑠𝑄𝑠⟩] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='+ ∫ 𝑑𝑧𝑅2⟨𝑃𝑠𝑄𝑠𝑃𝑠𝑄𝑠⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='− (∫ 𝑑𝑧𝑅⟨𝑃𝑠𝑄𝑠⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='11) 7 in terms of the quadrature operators at the initial boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' By further simplification, we finally approach the following resultant function for the quantum Fisher information, 𝐹𝜅 = 16𝐿2 [(𝐴2 + 𝐸2) ( 𝛼2 2 + 1 8) + (𝐴𝐵 + 𝐴𝐷 + 𝐵𝐸 + 𝐷𝐸) ( 𝛼2 2 ) + ( 1 16 + 𝛼2 2 ) (𝐵2 + 𝐷2) − 1 8 𝐵𝐷 + ( 1 4 + 𝛼2) 𝛾 2 (2𝑛th + 1) ∫ 𝑑𝑧(𝐶2 + 𝐹2 + 𝐺2 + 𝐽2) 𝐿 0 + 𝛼2𝛾(2𝑛th + 1) ∫ 𝑑𝑧(𝐶𝐹 + 𝐿 0 𝐽𝐺) + 𝛾 4 ∫ 𝑑𝑧(𝐽𝐹 − 𝐶𝐺) 𝐿 0 + 𝛾 2 (𝑛th + 𝛾 2) ∫ 𝑑𝑧𝑅2 𝐿 0 + 𝛾2 4 (∫ 𝑑𝑧𝑅 𝐿 0 ) 2 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='12) Obviously, the quantum Fisher information 𝐹𝜅 (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='12) will feature different manifestations in response to the contrasting PT domains of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As a representative example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S1 we accordingly present the quantum Fisher information log4𝐹𝜅 for three distinct scenarios: 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8 unbroken quadrature-PT phase), 𝛾 𝜅 = 1 (EP point), and 𝛾 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 (breaking quadrature-PT phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The quantum Fisher information at different quadrature PT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Blue, red, and orange lines are, respectively, corresponding to 𝛾 𝜅 Τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8 (unbroken quadrature-PT phase), 𝛾 𝜅 Τ = 1 (EP point), and 𝛾 𝜅 Τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 (broken quadrature-PT phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum sensing performance by comparing log4(Δ𝜅𝑞𝑖,0 −2 ) (a), log4(Δ𝜅𝑝𝑠,𝐿 −2 ) (b), log4(Δ𝜅𝑞𝑠,𝐿 −2 ) (c), and log4(Δ𝜅𝑝𝑖,0 −2 ) (d) with log4 𝐹𝑘 in the quadrature-PT phase unbroken region for the parameters (𝛼 = 2, 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2, 𝜅 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 60 50 y y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 K K t 40 30 20 10 0 10 0 5 10 15 20 25 30 35 40 45 50 2KL30 30 (a) (b) —log4((Fk) 25 25 log4((Ago) log4((Akpz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=')) 20 20 15 15 人儿从 10 10 in 0 0 5 5 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 30 30 (c) (d) 25 log4(Fk) 25 log4(Fk)) log4((Akgz)) 20 log4((Akpz0)) 20 15 15 10 10 5 5 0 0 5 5 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 308 Different from other existing (quantum) sensing protocols based on PT or EP enhancement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' we find that the PT-quadrature variables permit optimal classical sensor performance in the PT phase unbroken regime but far away from the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This observation is strongly supported by analyzing the quantum Fisher information with respect to the inverse variances across the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In the main text, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Evaluating quantum sensing performance (FIGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S3(a)—(d)) near the exceptional point by examining the ratios of ∆𝜅𝑞𝑖,0 −2 (a), ∆𝜅𝑝𝑠,𝐿 −2 (b), ∆𝜅𝑞𝑖,𝐿 −2 (c), and ∆𝜅𝑝𝑖,0 −2 (d) to 𝐹𝜅 for 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 (red curve) and 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 (black curve), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' The other parameters are 𝛼 = 2 and 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum sensing performed near the exceptional point by comparing log4(Δ𝜅𝑞𝑖,0 −2 ) (a), log4(Δ𝜅𝑝𝑠,𝐿 −2 ) (b), log4(Δ𝜅𝑞𝑠,𝐿 −2 ) (c), and log4(Δ𝜅𝑝𝑖,0 −2 ) (d) with log4 𝐹𝑘 for 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 and 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 for the parameters 𝛼 = 2 and 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (a) (b) 10-4 ×10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 8 Y/K= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 8 三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 6 —y/k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 6 25 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 2 4 9 8 0 0 0 2 4 6 8 0 2 4 6 8 2KL 2KL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (c) ×10-4 (d) ×10-4 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 8 8 6 K 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content="15 4 4 /T'S 2 2 2 20 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 0 6 7 8 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0 0 0 2 4 6 8 0 2 4 6 8 2kL 2KL60 60 (a) (b) 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95log4Fx 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95log4F K 40 Y =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 Y 40 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 30 30 20 20 10 10 0 0 10 10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 2KL 2KL 60 60 (c) (d) 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='951og4Fx 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='951og4Fx =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 K 40 Y =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 Y =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 log4Axqz 40 Y =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 Y =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 30 30 20 20 10 10 0 0 10 10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 2KL 2KL9 we have shown the precision of the 𝜅 -parameter estimation by looking at the ratio of the inverse variance ∆𝜅−2 to the quantum Fisher information 𝐹𝜅 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 5(a)—(d), which should be bounded in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S2(a)—(d), we have further given these inverse variances in comparison with 𝐹𝜅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To make the point more straightforward and convincing, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S3(a)—(d) we have particularly examined the measurement schemes implemented very close to the EP for 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quantum sensing implemented exactly at the exceptional point by comparing log4(Δ𝜅𝑞𝑖,0 −2 ) (a), log4(Δ𝜅𝑝𝑠,𝐿 −2 ) (b), log4(Δ𝜅𝑞𝑠,𝐿 −2 ) (c), and log4(Δ𝜅𝑝𝑖,0 −2 ) (d) with log4 𝐹𝑘 for 𝛼 = 2 and 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Evaluating quantum sensing performance (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S5(a)—(d)) at the exceptional point by examining the ratios of ∆𝜅𝑞𝑖,0 −2 (a), ∆𝜅𝑝𝑠,𝐿 −2 (b), ∆𝜅𝑞𝑖,𝐿 −2 (c), and ∆𝜅𝑝𝑖,0 −2 (d) to 𝐹𝜅 for the parameters 𝛼 = 2 and 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 40 40 (a) log4Fx (b) log4Fx 30 30 20 20 10 10 0 0 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2KL 2kL 40 40 (c) log4Fx (d) log4Fk 30 30 20 20 10 10 0 0 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2KL 2KL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (a) (q) ×10-4 × 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 8 K Y/K= 1 8 三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 6 6 4 2弘 4 AK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 2 4 6 8 0 0 0 2 4 6 8 0 2 4 6 8 2KL 2kL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='25 (c) × 10-4 (d) × 10-4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 8 8 K 6 E K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='15 4 4 2 2 /0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='1 2 0 AK 0 6 7 8 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='05 0 0 0 2 4 6 8 0 2 4 6 8 2kL 2KL10 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95 by plotting log4 Δ𝜅𝑞𝑖,0 −2 , log4 Δ𝜅𝑝𝑠,𝐿 −2 , log4 Δ𝜅𝑞𝑠,𝐿 −2 , and log4 Δ𝜅𝑝𝑖,0 −2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Similarly, the quantitative sensing performance offered by each quadrature can be well assessed by evaluating the corresponding ratio of ∆𝜅𝑞𝑖,0 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S4(a)), ∆𝜅𝑝𝑠,𝐿 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S4(b)), ∆𝜅𝑞𝑠,𝐿 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S4(c)), and ∆𝜅𝑝𝑖,0 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S4(d)) to 𝐹𝜅 for the same parameters used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S3(a)—(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' By comparing these figures with Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 5(a)—(d) in the main text, it is not difficult to conclude that indeed, the presence of gain and loss in gain-loss-coupled PT symmetry can substantially diminish the EP-based super-sensitivity promised in the classical settings and make it unavailable in the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Moreover, even if one still insists on performing any quantum sensing measurement in the vicinity of the EP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='94 and 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='95), it would become highly challenging due to the vast difference between the peak values of ∆𝜅 −2 and 𝐹𝜅 spanning over many orders of magnitude, regardless of whether the quadrature observables are associated with the characteristics of PT symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' This is especially true if comparing with the measurement carried out at 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' What happens if one attempts to fulfill the quantum sensing at the phase transition point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In such a case, unfortunately, the paired PT quadratures will cease to showcase any response to the parameter precision estimation, thereby making them fully unsuitable for quantum sensor applications when the symmetry spontaneously breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' As demonstrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S5(a) and (b), one can clearly see that log4 Δ𝜅𝑞𝑖,0 −2 and log4 Δ𝜅𝑝𝑠,𝐿 −2 for the PT-symmetric quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿)) become smooth and curvatureless, indicating that they are completely insensitive to any perturbation on an unknown parameter yet to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Alternatively, no gain on parameter estimation will be accessed at the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' On the other hand, we notice from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S5(c) and (d) that the non-PT-symmetric quadrature pair (𝑝𝑖(0), 𝑞𝑠(𝐿)) enables best sensing measurement only near the first peaks of the inverse variances log4 Δ𝜅𝑞𝑠,𝐿 −2 , and log4 Δ𝜅𝑝𝑖,0 −2 , in accordance with the quantum Fisher information log4 𝐹𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Obviously, this behaves differently from the cases of 𝛾 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content='2 , where the supersensitive measurements are available near the first two peaks of ∆𝜅 −2 and even more peaks (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S2(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' In fact, when PT symmetry disappears, the quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿)) lose to offer any sensing capabilities, despite (sub)optimal sensing may be accessible to the other non-PT-symmetric conjugate pair (𝑝𝑖(0), 𝑞𝑠(𝐿)), according to our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' To have a more intuitive evaluation on the quantum sensing performance exactly at the EP, it is better to look at the ratios of ∆𝜅𝑞𝑖,0 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6(a)), ∆𝜅𝑝𝑠,𝐿 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6(b)), ∆𝜅𝑞𝑖,𝐿 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6(c)), and ∆𝜅𝑝𝑖,0 −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6(d)) to 𝐹𝜅 in the same way as we did above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' S6(a)—(d), we can easily find that these ratios quickly approach zero for the longer medium length 𝐿, implying that the system loses its sensing ability at the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' References: [1] Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Mei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Zuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Zhai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Wen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Du, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Anti-parity-time symmetry optical four-wave mixing in cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' 123, 193604 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' [2] Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Zhai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=', Ghamsari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' & Wen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' Quadrature parity-time symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} +page_content=' (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE5T4oBgHgl3EQfQQ5g/content/2301.05511v1.pdf'} diff --git a/89AyT4oBgHgl3EQfqPhE/content/tmp_files/2301.00538v1.pdf.txt b/89AyT4oBgHgl3EQfqPhE/content/tmp_files/2301.00538v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4f4a2e474280218e2952dc24f4a55bdedf07b7b --- /dev/null +++ b/89AyT4oBgHgl3EQfqPhE/content/tmp_files/2301.00538v1.pdf.txt @@ -0,0 +1,2175 @@ +Topological Kondo Superconductors +Yung-Yeh Chang,1, 2 Khoe Van Nguyen,2 Kuang-Lung Chen,2 Yen-Wen Lu,3 Chung-Yu Mou,4 and Chung-Hou Chung2 +1Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Republic of China +2Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan Republic of China +3Department of Physics and Astronomy, University of California, Riverside, California 92511, U.S.A. +4Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan Republic of China +(Dated: January 3, 2023) +Spin-triplet p-wave superconductors are promising candidates for topological superconductors. They have +been proposed in various heterostructures where a material with strong spin-orbit interaction is coupled to a +conventional s-wave superconductor by proximity effect. However, topological superconductors existing in na- +ture and driven purely by strong electron correlations are yet to be studied. Here we propose a realization of +such a system in a class of Kondo lattice materials in the absence of spin-orbit coupling and proximity effect. +Therein, the odd-parity Kondo hybridization mediates ferromagnetic spin-spin coupling and leads to spin-triplet +resonant-valence-bond (t-RVB) pairing between local moments. Spin-triplet p ± ip′-wave topological super- +conductivity is reached when Kondo effect co-exists with t-RVB. We identify the topological nature by the +non-trivial topological invariant and the Majorana fermions at edges. Our results offer a comprehensive under- +standing of experimental observations on UTe2, a U-based ferromagnetic heavy-electron superconductor. +I. +INTRODUCTION +Searching for topological superconductors (TSc) and the +corresponding self-dual charge neutral Majorana zero modes +associated with their excitations at edges has become one of +the central problem in condensed matter physics [1, 2]. The- +oretical proposals and experimental realizations of TSc are +mostly heterostructure combining strong spin-orbit coupled +materials and conventional superconductors by proximity ef- +fect [3–5]. The emergence of the topological edge states in +such systems can be explained in terms of the single-particle +band structure without considering many-body electron corre- +lations. Recently, the search for topological phases of matter +has focused on a more intriguing class of materials that exist +in nature. Their topological properties are driven by strong +electron correlations instead of the proximity effect. Kondo +effect, describing the screening of a local spin moment by con- +duction electrons, is a well-known strong correlation between +electrons existing in heavy electron compounds. The Kondo- +mediated topological phases of matter have been studied in the +context of topological Kondo insulators [6–8] and topological +Kondo semi-metals [9], where the topological properties are +driven by either the odd-parity Kondo hybridization or by the +Kondo hybridization with strong spin-orbit coupling. +Spin-triplet p-wave superconductors are known to be the +prime candidates for TSc. However, they are scarce in na- +ture. While it is still debatable for SrRu2O4 [10–12], more +convincing evidence for p-wave triplet superconductivity was +observed in noncentrosymmetric superconductor BiPd from +phase-sensitive measurement [13]. More recently, signatures +of triplet chiral p-wave superconductivity were observed in +heavy-electron Kondo lattice compound UTe2 at the edge of +ferromagnetism, possibly marking the first example of topo- +logical superconductor induced by the strongly correlated +Kondo effect [14–17]. +Motivated by these discoveries, in this paper, we propose a +distinct class of triplet p-wave superconductors in the absence +of spin-orbit coupling or proximity effect/heterostructure [18] +in a two-dimensional Kondo lattice model driven by odd- +parity Kondo hybridization. We start from the Anderson lat- +tice model (ALM) with odd-parity hybridization, which oc- +curs between d- and f-orbital electrons in various heavy- +fermion compounds [6–8]. Via the Schrieffer-Wolff transfor- +mation [19, 20], we derive an effective Kondo lattice model +with odd-parity hybridization. +Furthermore, by integrating +out the conduction electron degrees of freedom, an effective +ferromagnetic RKKY interaction is generated. We explore +the mean-field phase diagram of this ferromagnetic Kondo- +Heisenberg model. In the fermionic mean-field approach, the +ferromagnetic RKKY coupling describes the p-wave (Sz = +±1) t-RVB spin-liquid state. A time-reversal invariant topo- +logical superconducting phase is reached when the Kondo ef- +fect co-exists with the p-wave t-RVB order parameter. The +topological nature of this superconducting phase is manifested +by the non-trivial Z2 topological Chern number of the bulk +band and by the existence of helical Majorana zero modes at +the edges of a finite-sized ribbon. Our results offer a qualita- +tive and some quantitative understanding of the spin-triplet su- +perconductivity recently observed in UTe2 (see Discussions). +II. +MODEL +A. +Anderson lattice model with odd-parity hybridization +We start with the odd-parity Anderson lattice model (ALM) +on a two-dimensional (2D) square lattice, which has been +shown to exhibit topologically non-trivial states [6–8]: +HP AM = Hc + Hf + Hcf, +(1) +where Hc = � +k,σ=↑,↓ εkc† +kσckσ describes the hopping of +electrons in the d orbits with orbital angular momentum l = 2 +and dispersion εk = −2t(cos kx + cos ky) − µ. The Hamil- +tonian Hf of the more localized electron in the f orbits with +arXiv:2301.00538v1 [cond-mat.str-el] 2 Jan 2023 + +2 +● +● +● +● +● +● +● +●●●● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●● +● +● +● +●●●●●●●●●●● +● +● +● +● +● +●●● +● +● +● +●●●● +● +● +● +●●●● +● +● +●●●●●●●● +●● +■ +■ +■ +■ +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +■ +■ +■ ■ ■ +■ +■ +■ ■ ■ ■ ■ +■ +■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ +● μ/t = -1.5 +■ μ/t = -2.0 +0 +2 +4 +6 +-0.1 +-0.05 +0 +0.05 +R/a +JH/JK +2 +FIG. 1. The effective RKKY coupling JH (normalized with J2 +K) as a +function of R/a for different chemical potentials µ. JH is computed +by Eq. (7) with Rij ∥ (1, 1) and a = 1 being chosen. +orbital angular momentum l = 3 is given by +Hf = +� +i,σ +� +εff † +iσfiσ + U +2 nf +iσnf +i,−σ +� +, +(2) +where εf denote the energy level of the f-electron, and U +is the repulsive on-site Coulomb potential (the Hubbard-U +term). Hybridization of the local and conduction electrons is +described by +Hcf = +� +⟨i,j⟩ +� +σ,σ′=↑↓ +V σσ′ +ij +c† +iσfjσ′ + H.c.. +(3) +To conserve the parity symmetry of hybridization between +electrons with their angular momentum quantum numbers dif- +fering by one, V σσ′ +ij +have to be odd under parity transforma- +tion. This restriction results in the hybridization having to +depend on sites and spins [6–8]: +V σσ′ +ij +≡ V σσ′ +ˆα += iV νˆασσσ′ +α +, +(4) +distinct from the well-known onsite and spin-conserving An- +derson hybridization. In Eq. (4), νij satisfies νij ≡ νˆα = +−νji with ˆα ≡ i − j ∈ ˆx, ˆy (α ∈ x, y) on a 2D square lattice, +and σα denotes the Pauli matrix of the α component. +B. +The effective odd-parity ferromagnetic Kondo lattice model +In this paper, we focus on the competition of the Kondo +and the magnetic interaction among impurities–the Doniach +scenario [21]. +We, therefore, derive the effective Kondo- +Heisenberg lattice Hamiltonian from ALM in the Kondo limit +where the vacant and doubly-occupied states are projected out +from the entire Hilbert space, namely 1 = � +σ f † +iσfiσ. The +low-energy effective Kondo term from the odd-parity ALM +of Eq. (1) can be derived by applying the Schrieffer-Wolff +transformation (SWT) [19, 20, 22], yielding +HK = (−JK) +� +i +� +σσ′ +� +σ′′σ′′′ +� +α,α′ +� +iνˆασσσ′ +α +c† +i+ˆα,σfiσ′ +� +× +� +iνˆα′σσ′′σ′′′ +α′ +f † +iσ′′ci−ˆα′,σ′′′ +� +(5) +with JK = +V 2 +U+εf −εF + +V 2 +εF −εf > 0 (see Appendix A). +The Kondo-like term of Eq. (5) describes the screening of +an impurity by its neighboring conduction electrons, distinct +from the conventional (on-site) Kondo term. +Here, we go beyond the topological Kondo insulating phase +by further deriving the magnetic RKKY interaction among +the local f-fermions. By perturbatively expanding the Kondo +term to second order [22–24], we obtain the effective RKKY- +like interaction between the local f fermions fiσ, +HJ = +� +i,j +� +σ,σ′ +Jijf † +iσf † +jσ′fjσfiσ′ += +� +⟨i,j⟩ +Jij +� +f † +i↑f † +j↑fj↑fi↑ + f † +i↓f † +j↓fj↓fi↓ +� ++ +� +⟨i,j⟩ +Jij +2 +� +f † +i↑f † +j↓ + f † +i↓f † +j↑ +� +(fj↓fi↑ + fj↑fi↓) +− +� +⟨i,j⟩ +Jij +2 +� +f † +i↑f † +j↓ − f † +i↓f † +j↑ +� +(fj↓fi↑ − fj↑fi↓) , (6) +where +Jij ≡ JH(R) = 16J2 +K +N 2s +� +εk<µ +� +εk′′>µ +ei(k−k′′)·Rij +εk − εk′′ +× +� +sin2 kx + sin2 ky +� � +sin2 k′′ +x + sin2 k′′ +y +� +(7) +denotes the effective coupling of the spinons of sites i and +j with R ≡ |Rij| ≡ |ri − rj|. +The HJ term of Eq. +(6) can be re-expressed as a linear combination of a spinon +pair wave function with total spin S += 0 (spin-singlet) +and S = 1 (spin-triplet). +Note that the associated effec- +tive spinon coupling of the spin-triplet channel is opposite +to that of the spin-singlet. When HJ is expressed in terms +of fermion pair with different spins, Eq. +(6) is reminis- +cent of the conventional Heisenberg interaction Si · Sj = +− 1 +2 +� +f † +i↑f † +j↓ − f † +i↓f † +j↑ +� +(fi↓fj↑ − fi↑fj↓)+ 1 +4nf +i nf +j , except for +the difference in the constant coefficients of the pair opera- +tors. As expected, the RKKY coupling Jij in Eq. (7) shows +an oscillatory behavior in R, accompanied by a decrease in +its magnitude with increasing R, similar to the behavior of +the conventional RKKY coupling. Due to the rapid attenua- +tion of Jij, we only consider the dominated nearest-neighbor +interaction and assume Jij to be spatially homogeneous, i.e. +Jij → J(R = a) ≡ JH. Furthermore, when R = a, we find +the effective RKKY coupling is attractive (or of the ferromag- +netic type), i.e., JH < 0 (see Fig. 1), which energetically fa- +vors the spin-triplet pairing of spinons. On the other hand, the +effective RKKY coupling in the spin-singlet channel shows +repulsive interaction and can be neglected here since it is not + +3 +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●● +◆ x +● Δt +0.5 +1 +1.5 +2. +2.5 +0 +0.2 +0.4 +JH +Mean-field parameters +JK = 0.3, δ = -0.3 +FIG. 2. The zero-temperature mean-field solutions of t-RVB order +parameter ∆t (brown) and the Kondo correlation x (black) as a func- +tion of JH. We fix JK = 0.3 and doping of the conduction band +δ = −0.3 (30 percent hole doping). Without loss of generality, we +set t = 1. This plot reveals a (co-existing) superconducting ground +state with x ̸= 0, ∆t ̸= 0 for 0 < JH ≲ 2.5 and a pure t-RVB +phase where x = 0, ∆t ̸= 0 when JH ≳ 2.52. A pure Kondo phase +(x ̸= 0, ∆t = 0) only exists at JH = 0. +energetically favorable. Lastly, on a two-dimensional lattice, +the triplet spin state | ↑↓⟩ + | ↓↑⟩ does not exist since the cor- +responding structure factor is proportional to kz, and kz = 0 +is fixed here. Therefore, based on the above arguments, only +the equal-spin states, | ↑↑⟩ and | ↓↓⟩, survive, and the HJ term +is reduced to +HJ ≈ − |JH| +� +⟨i,j⟩ +� +f † +i↑f † +j↑fj↑fi↑ + f † +i↓f † +j↓fj↓fi↓ +� +. +(8) +Combining HK and HJ of Eqs. (5), (6) and (8), the effec- +tive Kondo-Heisenberg lattice model with odd-parity Kondo +hybridization reads HF KH = H0 + Hλ + HK + HJ. Here, +Hλ = − � +i iλi +�� +σ(f † +iσfiσ) − 1 +� +enforces the singly occu- +pied local f-spinons with λi being the Lagrange multiplier. +The Hamiltonian HF KH offers a platform for discovering a +distinct class of topological superconducting states induced +by electron correlations via collaboration between the ferro- +magnetic RKKY coupling and the Kondo effect. To facilitate +our numerical calculations of the mean-field phase diagram, +we treat JK and JH as independent couplings here since it is +more convenient to explore the phase diagram by tuning the +ratio of JK/JH [25, 26]. In experiments, varying the non- +thermal parameter can be expected to follow a certain trajec- +tory of JK/JH in the phase diagram. +III. +MEAN-FIELD TREATMENT OF THE EFFECTIVE +KONDO-HEISENBERG-LIKE MODEL +We now employ a mean-field analysis on the above ef- +fective Kondo-Heisenberg-like Hamiltonian with an effective +ferromagnetic RKKY interaction and odd-parity Kondo hy- +bridization. +Via performing Hubbard-Stratonovich transformation, HK +and HJ of Eqs. (5) and (6) can be factorized as +HK → +� +i,α +� +σσ′ +� +χ† +i +� +iνˆασσσ′ +α +f † +iσci−ˆα,σ′ +� ++ H.c. +� ++ +� +i +|χi|2 +JK +, +HJ → +� +⟨i,j⟩ +� +∆↑ +t (i, j)f † +i↑f † +j↑ + ∆↓ +t (i, j)f † +i↓f † +j↓ + H.c. +� ++ +� +⟨i,j⟩ +���∆↑ +t (i, j) +��� +2 ++ +���∆↓ +t (i, j) +��� +2 +JH +(9) +where the mean-field values of the bosonic Hubbard- +Stratonovich fields, χi and ∆σ +t (i, j) (σ =↑, ↓), represent the +order parameters of the Kondo correlation and the Sz = ±1 +spin-triplet RVB bonds between two adjacent up/down spins, +respectively. +To describe the Kondo-screened Fermi-liquid state, we al- +low the χi field to acquire uniformly Bose condensation over +the real space; hence, χi can be expressed as χi → x + ˆχi +with x = (−JK/Ns) � +iσσ′α⟨iνˆασσσ′ +α +f † +iσci−ˆα,σ′⟩ being the +Bose-condensed stiffness of χi while ˆχi represents its fluctua- +tions. The mean-field order parameter of the tRVB is given by +∆σ +t = (−JH/4Ns) � +⟨i,j⟩⟨fjσfiσ⟩. Since the ferromagnetic +coupling is expected to favor spin-triplet p-wave pairing sim- +ilar to superfluid helium-3 [27], we restrict ourselves to the +p-wave pairing, i.e., ∆σ +t (i, j) here is taken the p-wave form, +see Eqs. (11) and (12) below. We further fix the Lagrange +multiplier at the mean-field level via iλi → λ and neglect the +fluctuations of λi, χi, and ∆σ +t , leading to the following mean- +field Kondo-Heisenberg-like Hamiltonian: +HMF = +� +k,σ +εkc† +kσckσ + +� +kσ +λf † +kσfkσ ++ +� +k +� +V1kf ∗ +k↑ck↓ + V2kf ∗ +k↓ck↑ + H.c. +� ++ +� +k +� +∆↑ +kf † +k↑f † +−k↑ + ∆↓ +kf † +k↓f † +−k↓ + H.c. +� ++ 8Ns∆2 +t +JH ++ Nsx2 +JK +− Nsλ, +(10) +where +V1k += +2x (sin kx − i sin ky) +and +V2k += +2x (sin kx + i sin ky). +The Fourier transformation for +the +second-quantized +operator +is +defined +as +ψiσ += +1 +√Ns +� +k e−ik·riψkσ. Note that the mean-field Kondo term +of Eq. +(10) is reminiscent of the topological Kondo insu- +lator shown in Ref. [28]. In Eq. (10), ∆σ +t (k) represents +the gap structure of the spin-triplet p-wave RVB pairing in +the momentum space for the spin-σ sector, defined as ∆↑ +k = +∆t (− sin ky − i sin kx) and +∆↓ +k = ∆t (sin ky − i sin kx) +with ∆t being denoted the mean-field pairing potential (see +Appendix, Section II). This momentum-dependent gap struc- + +4 +ture for the up- and down-spin sectors correspond to the fol- +lowing real-space patterns of ∆↑ +t (i, j) and ∆↓ +t (i, j) of Eq. (9): +∆↑ +t (i, j) → ∆↑ +t (i, i + ˆx) = −∆↑ +t (i, i − ˆx) = −∆t, +∆↑ +t (i, i + ˆy) = −∆↑ +t (i, i − ˆy) = i∆t, +(11) +and +∆↓ +t (i, j) →∆↓ +t (i, i + ˆx) = −∆↓ +t (i, i − ˆx) = −∆t, +∆↓ +t (i, i + ˆy) = −∆↓ +t (i, i − ˆy) = −i∆t. +(12) +Choosing Ψk = (φAk, φBk)T with the Nambu spinors +defined by φAk = +� +ck↑, c† +−k↑, fk↓, f † +−k↓ +�T +and φBk = +� +ck↓, c† +−k↓, fk↑, f † +−k↑ +�T +, +the +mean-field +Hamiltonian +HMF = � +k Ψ † +kHkΨk + C can be expressed as a summation +of two decoupled 4 × 4 matrices as follows +HMF = HA + HB + C, +HA(B) = +� +k +φ† +A(B)kHA(B) +k +φA(B)k +(13) +with C ≡ � +k εk + 8Ns∆2 +t +JH ++ Nsx2 +JK , and +HA +k = +� +� +� +� +� +εk +2 +0 +V ∗ +2k +2 +0 +0 +− εk +2 +0 +V2k +2 +V2k +2 +0 +λ +2 +∆↓ +k +0 +V ∗ +2k +2 +∆↓∗ +k +− λ +2 +� +� +� +� +� , +(14) +HB +k = +� +� +� +� +� +εk +2 +0 +V ∗ +1k +2 +0 +0 +− εk +2 +0 +V1k +2 +V1k +2 +0 +λ +2 +∆↑ +k +0 +V ∗ +1k +2 +∆↑∗ +k +− λ +2 . +� +� +� +� +� +(15) +The Hamiltonian Eq. (13) possesses time-reversal symme- +try: HA and HB constitute the time-reversal partner of each +other, i.e. ΘHA(B)Θ−1 = HB(A) where the time-reversal +operator Θ = ρ0 ⊗ (−iσy)K with σy being the y-component +Pauli matrix on the spin subspace, ρ0 being a 2 × 2 identity +matrix on the orbital subspace while K being the complex- +conjugate operator. Under time-reversal transformation, the +spin and quasi-momentum of conduction (c) and pseud- +ofermion (f) operators are flipped: (ck↑, ck↓, fk↑, fk↓) +Θ +−→ +(c−k↓, −c−k↑, f−k↓, −f−k↑). +Meanwhile, our Hamilto- +nian respects charge-conjugation (particle-hole) symmetry: +PHkP−1 = −H−k where P ≡ τ xK is the particle-hole op- +erator with τx being the x-component of the Pauli matrices on +the particle-hole basis. Due to the odd-parity p±ip′ RVB pair- +ing of our model, the parity symmetry is broken here. Thus, +our model Eq. (13) belongs to the DIII class of topological +symmetry [29]. +IV. +RESULTS +A. +Mean-field phase diagram +The mean-field ground states are determined by minimiz- +ing the mean-field free energy per site FMF += +C +Ns − +kBT +Ns +� +nk ln +� +1 + exp +� +− Enk +kBT +�� +with respect to the mean- +field variables q = (λ, x, ∆t), i.e. ∂FMF /∂qi = 0. +Here, +Enk < 0 is the n-th band of Hk. The chemical potential µ is +determined by the relation ∂FMF /∂µ = −(1 + δ) with δ be- +ing the chemical doping of the c-electrons for which δ ⪋ 0 is +for p/un/n− doped (half-filling corresponds to δ = 0). This +leads to the following saddle-point equations at zero tempera- +ture, +1 +Ns +� +nk +∂Enk +∂x ++ 2x +JK += 0, +1 +Ns +� +nk +∂Enk +∂∆t ++ 16∆t +JH += 0, +1 +Ns +� +nk +∂Enk +∂λ += 0, +1 +Ns +� +nk +∂Enk +∂µ ++ δ = 0. +(16) +The ground-state phase diagram (Fig. 2) of our model is ob- +tained by solving the saddle-point equations self-consistently. +The phase diagram contains three distinct mean-field phases: +a pure Kondo phase is found at JH = 0 where x ̸= 0, ∆t = 0. +At the opposite limit where the RKKY interaction dominates, +the ground state shows short-range magnetic correlation with +p-wave spin-triplet RVB pairing (∆t ̸= 0, x = 0). In the +intermediate range of 0 < JH/JK < (JH/JK)c, we find a +Kondo-tRVB co-existing (superconducting) phase with x ̸= 0 +and ∆t ̸= 0, which can be explained via the mechanism of +Kondo-stabilized spin liquid [26, 30]. The development of +superconductivity in this co-existing phase requires higher- +order processes involving both the Kondo and t-RVB terms: +the mean-field t-RVB pairings of the local f fermions pro- +vide preformed Cooper pairs. When the Kondo hybridization +field χ gets Bose-condensed (x ̸= 0), the local fermions de- +localize into the conduction band and make the preformed t- +RVB Cooper pairs superconduct [31]. These processes can +be described by the effective mean-field Hamiltonian Hsc = +� +k +� +¯ +∆↓∗ +k c−k↓ck↓ + ¯ +∆↑∗ +k c−k↑ck↑ + H.c. +� +, where the effec- +tive gap functions take the form ¯ +∆↓∗ +k += V1kV1,−k∆↑∗ +k +∼ +x2∆t(sin2 kx + sin2 ky)(sin kx − i sin ky) and +¯ +∆↑∗ +k += +V2kV2,−k∆↓∗ +k ∼ x2∆t(sin2 kx + sin2 ky)(sin kx + i sin ky) +with the size of the superconducting gap being proportional +to x2∆t. The superconducting gap function ¯ +∆↑ +k we obtained +here shows a f-wave-like pairing symmetry on a generic +anisotropic (non-circular) 2D Fermi surface. +Nevertheless, +as we are taking the continuous limit of the conduction band +here, ¯ +∆↑ +k can be expressed as a product of s and p±ip′ pairing + +5 +(a) +Γ +X +M +(b) +FIG. 3. Figures (a) (red curves) and (b) show the bulk energy spec- +trum of the co-existing superconducting state near the Fermi level µ. +The Fermi level locates at E(k) = 0. The coupling constants are +JK = 0.3 and JH = 1.0. Inset of (a) displays the First Brillouin +zone of a square lattice with indications of high-symmetry points +Γ, X, M. +orders, i.e., ¯ +∆↑/↓∗ +k +∼ k2(kx±iky) with k2 ≡ k2 +x+k2 +y on a cir- +cular Fermi surface but only the p±ip′ component plays a role +here. Note that we find the co-existing superconducting state +persists for an arbitrary small value of JH/JK → 0+. This is +likely due to the overestimation of the co-existing phase at the +mean-field level. Upon including fluctuations of the Kondo +and t-RVB order parameters beyond the mean-field level, we +expect a narrower co-existing superconducting phase. A first- +order transition similar to the results found in Refs. [26, 32] +is observed at the transition of the t-RVB and the co-existing +superconducting phases (see Fig. 2). The bulk band structure +in the co-existing superconducting state is shown in Fig. 3. +B. +Topological invariance +We now address the topological properties of the coexisting +superconducting state. Since this system is invariant under +time-reversal transformation, the bulk topological properties +of the coexisting Kondo-RVB superconducting state with p ± +ip′ spin-triplet RVB pairing can be thus characterized by the +Z2 Chern number cT (or time-reversal polarization) [33–35], +kx +E(kx) +FIG. 4. The left figure displays the electronic band structure of the +coexisting superconductor state for a strip with Ny = 81 described +by HA at JK/t = 0.3 and JH/t = 1.0. Three pairs of edge states +with Dirac spectra are observed near kx = 0 (the pink curves). The +edge states at zero energy correspond to the Majorana zero modes. +Due to the time-reversal symmetry of the model, the band structure +for a strip for HB is identical to that of HA. The close-up band +structures near three pairs of edge states (pink curves) on the top, +middle and bottom bounded by the red squares are shown on the +right figures. +given by +cT = cA − cB +2 +(17) +with cI (I ∈ A, B) being the Thouless-Kohmoto-Nightingal- +den Nijs (TKNN) number [36] of HI, defined as +cI = 1 +2π +� +k∈FBZ +dSk · +� +∇k × AI +k +� +. +(18) +The Berry’s connection AI +k for HI is given by AI +k ≡ +i � +n∈I⟨uI +nk|∇k|uI +nk⟩ with |uI +nk⟩ being the normalized +Bloch state of the n-th filled band for HI +k. We numerically +calculate the TKNN numbers [37], cA and cB, and find that +cA = −cB = 1 in the co-existing phase, indicating a topolog- +ically non-trivial Z2 Chern number cT = 1. By the bulk-edge +correspondence, we expect this co-existing superconducting +state to support a pair of counter-propagating Majorana zero +modes at the edges of a finite-sized strip. Further band struc- +ture calculations of our model on a strip in the following sub- +section confirm our expectation. +C. +Edge states of the coexisting Kondo-RVB spin-triplet +p ± ip′-wave superconducting state +We now check whether our model would support helical +Majorana zero modes at the edge of a finite-sized system. +We shall examine our model’s band structures and edge-state +wave functions on a finite-sized strip that extends infinitely +along the x direction but contains a finite number of lattice +sites in y. The results are shown in Figs. 4 to 6. As shown + +0.5 +E(k) +-0.5 +π +一π +0 +0 +π一 +2 +2 +Ky +kx +π－ +2 +26 +E +E +x +y +yi = 1 +yi = Ny +Ribbon +γRA,k +γL +B,k +γR +B,k +γL +A,k +(a) +(b) +(c) +(d) +(e) +(f) +(g) +FIG. 5. Figures (a) and (d) show the Bogoliubov excitation spectra of HA and HB, respectively, near the chemical potential on a nano-strip with +Ny = 81 chains. Figures (b), (c) and (e), (f) demonstrate the probability density of the Majorana edge state wave functions of HA and HB as +a function of atom position yi, +��γΓ +I,kx(yi) +��2 with I = A, B and Γ = R, L (pink curves in (a) and (d)), at a fixed energy E ≡ E(kx = ±0.03). +The probability density is described by +��γΓ +I,kx(yi) +��2 = +���uΓ +I,kx +��2 , +��¯uΓ +I,kx +��2 , +��vΓ +I,kx +��2 , +��¯vΓ +I,kx +��2� +(yi). The parameters are JK/t = 0.3, +JH/t = 1.0, and doping δ = −0.3. The edge states are of the helical type, as schematically represented in (g). +E +E +1 +2 +4 +3 +1 +3 +2 +4 +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 6. The finite-energy (E(kx) > 0) Bogoliubov excitation spectra of (a) HA (shown on top right in Fig. 5) and (d) HB. A pair of “helical” +edge states is found to exist at finite energy [pink curve in (a) and (d)], and their probability densities are shown in (b) and (c), (e) and (f), +respectively, at a fixed energy E(kx = ±0.22). +in Fig. 4, gapless Dirac spectra of the Bogoliubov excitations +around kx = 0 near zero energy are observed, exhibiting one +of the typical features of topological edge states. The exci- +tations can be effectively described by the linear-dispersed +Hamiltonian ˜HI += � +kx vx|kx| +� +γR † +I,kxγR +I,kx − γL † +I,kxγL +I,kx +� +with +γΓ +I,kx = +� +yi +� +uΓ +I,kx(yi)ckx,yi,↑ + ¯uΓ +I,kx(yi)c† +−kx,yi,↑ ++vΓ +I,kx(yi)fkx,yi,↓ + ¯vΓ +I,kx(yi)f † +−kx,yi,↓ +� +(19) +with u, ¯u and v, ¯v being the coherent factors. In Eq. (19), +I ∈ A, B, Γ ∈ R, L, and γR/L +A/B,kx represents the right/left- +moving Bogoliubov quasiparticle of ˜HA/B. Here, vx in ˜HI +denotes the velocity. +Due to time-reversal symmetry, HA +is the time-reversal partner of HB, and thus their spectra +are identical. The low-energy eigenstates with Dirac spectra +near kx = 0 for both HA and HB exhibit the typical prop- +erty of edge states, as their probability densities accumulate +mostly at the edges of strip, as shown in Fig. 5. Combin- +ing the directions of propagation inferred from the velocity +vx ∼ ∂E(kx)/∂kx, we can classify these edge states into two + +7 +Tc +Tonset +Tc +FIG. 7. +Plot of the temperature-dependent mean-field order pa- +rameters x(T)/t and ∆t(T)/t with kB = 1, JK/t = 0.3 and +JH/t = 1.0 fixed. +Inset shows the enlarged plot of ∆t(T). +The single-impurity Kondo temperature occurs at Tonset/t ≈ 0.16 +while the transition of superconductivity takes places at temperature +Tc/t ≈ 0.015. +groups, each of them constitutes a pair of counter-propagating +edge states (see Fig. 5), revealing the nature of helical Ma- +jorana zero modes. The helical type of the Majorana zero +modes is the consequence of time-reversal symmetry of our +model, reminiscent of the well-known Kane-Mele model on a +single-layered graphene [38, 39]. Remarkably, in addition to +the Majorana fermions at zero energy, two pairs of counter- +propagating edge-states are observed at finite energy, see Fig. +6. The two pairs of edge states correspond to the edge states +of the topological Kondo insulator, where the spin-triplet RVB +order parameter is absent (∆t = 0) [6–8]. +V. +DISCUSSIONS AND CONCLUSIONS +We now discuss the application of our results for heavy- +electron superconductors, particularly the Kondo lattice com- +pound UTe2. Experimental evidence indicates that this com- +pound does not show long-range magnetic order and is in +the vicinity of the ferromagnetic quantum critical point, ex- +hibiting both strong ferromagnetic fluctuations, possibly due +to magnetic frustrations induced by sub-leading antiferromag- +netic fluctuations [40, 41], and Kondo screening [14, 17, 42]. +The DFT+U calculations indicate that the dynamics of elec- +tron bands and the physical properties of UTe2 are dominated +by the electrons near the quasi-two-dimensional (cylindrical) +Fermi surface with weak kz dependence despite its 3D crystal +structure [40]. Superconductivity is reached at Tc = 1.6K, +while the resistivity maximum observed at T ⋆ ≈ 15 ∼ 75 K +reveals signature of coherent Kondo scattering [14, 43], indi- +cating T ⋆/Tc ≈ 10 ∼ 50. The superconductivity can, in gen- +eral, co-exist and compete with the Kondo effect [17]. When +a magnetic field is applied along the hard-magnetic axis b of +UTe2 and before entering the superconducting phase, a corre- +lated paramagnetic phase is observed below the temperature +at which the magnetic susceptibility shows a broad maximum +[44]. Similar spin-liquid behavior has been observed in the +magnetic susceptibility of another heavy fermion compound +CePdAl [45]. This similarity suggests this correlated para- +magnetic phase may feature short-range magnetic order. Our +theoretical framework based on competition and collaboration +between a Kondo-screened and a ferromagnetic t-RVB spin- +liquid states on a two-dimensional Kondo lattice is consistent +with the above observations in UTe2. It, therefore, consti- +tutes a promising approach to account for its exotic phenom- +ena. On the other hand, the chiral in-gap state, a signature of +chiral topological superconductor, has been observed by scan- +ning tunneling spectroscopy in the superconducting phase of +UTe2 [17]. Combining with the ferromagnetic fluctuations +that are known to induce spin-triplet pairing, people believe +UTe2 is a promising candidate for the spin-triplet chiral topo- +logical superconductor [14, 17]. Furthermore, the supercon- +ducting phase co-existing with Kondo coherence in this ma- +terial strongly suggests the role played by the Kondo effect +in this possible topological superconductor. +The topologi- +cal Kondo superconducting state with equal-spin spin-triplet +p-wave pairings we proposed here bears striking similarities +to and strong relevance for the experimental observations on +UTe2: (i) the d- and f-orbitals electrons with their angular +momentum quantum number differing by 1 in the uranium +atoms of UTe2 likely give rise to the odd-parity Kondo effect +[6–8], (ii) the t-RVB state in our theory may be considered +as one possible realization of the short-ranged ferromagnetic +fluctuations in UTe2, (iii) the Kondo-t-RVB co-existing su- +perconducting state we find here qualitatively agrees with the +co-existence between superconductivity and Kondo effect ob- +served in UTe2, (iv) the high upper critical field exceeding the +Pauli limit [14, 46] implies that the superconducting state of +UTe2 may have equal-spin Cooper pairs, and (v) the effective +pairing ∆ +σ +k formed in the conduction band mentioned in Sec- +tion IV A shows characteristics of spin-triplet point-node gap +structure [47]. Various characteristic temperature scales esti- +mated from our mean-field calculations with JH/t = 1.0 and +JK/t = 0.3 at finite temperatures agree reasonably well with +experimental observations (see Fig. 7): The superconducting +transition temperature Tc, theoretically determined from our +mean-field analysis Tc = Min[T(x = 0), T(∆t = 0)], shows +Tc ≈ 0.015t ≈ 2.3 K by taking estimated values of t = 150 K +and half-bandwidth D = 1.25t [42]. The Kondo coherent +scale can be obtained by T ⋆ = x2(T = 0)/D ≈ 17.4 K +[48]. The ratio T ⋆/Tc ≈ 8 is in reasonable agreement with +experimental observations. The onset temperature Tonset of +Kondo hybridization, which occurs at x(T = Tonset) = 0, +displays Tonset ≈ 0.16t ≈ 24 K, within the theoretically es- +timated range 10K < Tonset < 100K by DMFT calculation +[42]. Meanwhile, there have been evidences of TRS breaking +in UTe2 from the observed two superconducting transitions +and a finite polar Kerr effect at T < Tc [49], likely due to +proximity to the ferromagnetic ordered phase. A number of +theoretical attempts were proposed based on these observa- +tions [50, 51]. However, the observed single superconducting +transition near ambient pressure and zero field [44, 52, 53] as +well as the theoretically proposed unitary triplet pairing [40] + +8 +suggest TRS may be preserved in UTe2. Though our results +shown above are obtained in the presence of TRS, the chi- +ral p-wave superconducting state with chiral Majorana zero +mode at edges is expected to occur here once a time-reversal +breaking magnetic field is applied [54]. Our distinct predic- +tions with and without fields serve as theoretical guidance for +future experiments to distinguish the time-reversal breaking +from time-reversal preserving triplet pairing states in UTe2. +Since the Kondo correlations stabilize the t-RVB spin liquid +in the co-existing superconducting phase, it is expected to be +robust against gauge-field fluctuations beyond the mean field. +Our approach and results are distinct from the spin-triplet non- +topological superconducting state recently proposed based on +the Hund’s-Kondo coupling and Sz = 0 t-RVB state to ac- +count for UTe2 [51]. +In conclusion, we propose a first realization of the topo- +logical superconductivity in the Kondo lattice model, a dis- +tinct class of topological superconductors due to purely strong +electron correlations without employing spin-orbit coupling +or proximity effect. +A topological Kondo superconductor +essentially constitutes of 1) itinerant c and localized f bands +with different orbital quantum numbers, 2) strong Hubbard in- +teraction of the f electrons, 3) odd-parity Kondo hybridization +of the c and f bands, and 4) the attractive exchange interac- +tion of the f electrons with spin-triplet correlations. Start- +ing from the odd-parity Anderson lattice model, we obtain +the unconventional type of Kondo hybridization and ferro- +magnetic RKKY-like interaction via perturbation theory, lead- +ing to spin-triplet resonating-valence-bond (RVB) pairing be- +tween f-electrons with time-reversal invariant p ± ip′-wave +gap symmetry. Via the mean-field approach, we find a Kondo +triplet-RVB coexisting phase in the intermediate range of the +Kondo to RKKY coupling ratio. This phase is shown as a +time-reversal invariant topological superconducting state with +a spin-triplet p ± ip′-wave RVB pairing gap. It exhibits non- +trivial topology in the bulk band structure, and supports heli- +cal Majorana zero modes at edges. Our prediction in the pres- +ence of a time-reversal breaking field leads to chiral p-wave +spin-triplet topological Kondo superconductor. Our results on +the superconducting transition temperature, Kondo coherent +scale, and onset temperature of Kondo hybridization not only +qualitatively but also quantitatively agree with the observa- +tions for UTe2. The theoretical framework we propose here +opens up the search for topological superconductors induced +by strongly electronic correlations on the Kondo lattice com- +pounds. +VI. +ACKNOWLEDGEMENTS +This work is supported by the Ministry of Science +and Technology Grants 104-2112-M-009-004-MY3 and 107- +2112-M-009-010-MY3, the National Center for Theoretical +Sciences of Taiwan, Republic of China (to C.-H. C.). +Appendix A: The Schrieffer-Wolff transformation (SWT) +In this section, we provide derivations of the Kondo term +via using the SWT. We first perform the SWT on an odd-parity +single-impurity Anderson model where an impurity at an ar- +bitrary site i hybridizes with the conduction electrons on the +four nearest-neighbor sites of i. This result will be succes- +sively generalized to the lattice version. +The single-impurity Anderson model takes the following +form +H = +� +kσ +εkc† +kσckσ + +� +σ +εff † +iσfiσ + Unf +i↑nf +i↓ ++ +� +σσ′ +� +α=x,y +� +iV νˆασσσ′ +α +c† +i+ˆα,σfiσ′ + H.c. +� +, +(A1) +where ˆα ≡ ±ˆx, ±ˆy denotes the nearest-neighbor vectors of a +square lattice, and νˆα satisfies νˆα = −ν−ˆα and νˆx = νˆy = 1. +The SWT aims at projecting out the empty and doubly oc- +cupied states to generate the effective Hamiltonian Heff in +the Kondo (singly-occupied) limit. Following Ref. [20], we +first use the states of impurity occupation as the basis set, +{|f 0⟩, |f 1⟩, |f 2⟩} with the superscripts being denoted as the +occupation of the localized electrons, to expand the Hamilto- +nian of Eq. (A1) in the following matrix form, +H = +� +� +H00 H01 H02 +H10 H11 H12 +H20 H21 H22 +� +� . +(A2) +The matrix elements of Eq. +(A2), denoted as Hij +≡ +⟨f i|H|f j⟩ with i, j = 0, 1, 2, are +H10 = +� +σσ′ +� +α=±x,±y +iV νˆασσσ′ +α +f † +iσci−ˆα,σ′ = H21, +H01 = H† +10 = +� +σσ′ +� +α=±x,±y +iV νˆασσσ′ +α +c† +i+ˆα,σfiσ′ = H12, +H11 = +� +kσ +εkc† +kσckσ + +� +σ +εff † +iσfiσ, H00 = +� +kσ +εkc† +kσckσ, +H22 = +� +kσ +εkc† +kσckσ + +� +σ +εff † +jσfjσ + Unf +i↑nf +i↓. +(A3) +We then project out |f 0⟩ and |f 2⟩ from the Hilbert space to +obtain the effective Hamiltonian Heff at the Kondo limit sat- +isfying Heff|f 1⟩ = E|f 1⟩ with E being the eigenenergy. Via +Eq. (A2), Heff can be expressed as Heff = H11 + H′ with + +9 +H′ =H10(E − H00)−1H01 + H12(E − H22)−1H21 += +� +α,α′=x,y +� +σσ′ +� +σ′′σ′′′ +� +V 2 +εF − εf − U +� +iνˆασσσ′ +α +c† +i+ˆα,σfiσ′ +� � +iνˆα′σσ′′σ′′′ +α′ +f † +iσ′′ci−ˆα′,σ′′′ +� +(A4) ++ +V 2 +εf − εF +� +iνˆασσσ′ +α +f † +iσci−ˆα,σ′ +� � +iνˆα′σσ′′σ′′′ +α′ +c† +i+ˆα′,σ′′fiσ′′′ +�� +(A5) +Here, we skip the derivations of H10(E − H00)−1H01 and +H12(E − H22)−1H21 in Eq. (A5) as those are standard and +can be found in a number of references. See, for example, +Refs. [19, 20]. H′ can be further cast into the form simi- +lar to the conventional single-impurity Kondo term, with the +following antiferromagnetic Kondo coupling +JK = +V 2 +U + εf − εF ++ +V 2 +εF − εf +> 0, +(A6) +plus a potential scattering term. Eq. (A5) can be generalized +to the lattice version by summing over all lattice sites, as de- +scribed by Eq. (5). +Appendix B: Derivation of the effective ferromagnetic +RKKY-like interaction +In the section, we derive the RKKY-like interaction by per- +turbatively expanding HK of Eq. (5) to second order. +The unperturbed state is described as +|0, f⟩ = |k1m1, k2m2, · · · , kNmN⟩ |f⟩ , +(B1) +where conduction electrons do not interact with the impuri- +ties. In Eq. (B1), |k1m1, k2m2, · · · , kNmN⟩ represents the +Fermi sea with all wave vectors lying below the Fermi wave +vector, namely ki < kF . After imposing perturbation, the un- +perturbed state acquires correction and the corrected eigenen- +ergy is expressed in powers of JK, E = E0 + ∆E(1) + +∆E(2) + O(J3 +K) with E0 being the eigenenergy of the un- +perturbed state. +The first and second order energy corrections take the form +∆E(1) = ⟨0, f| HK |0, f⟩ , +∆E(2) = +� +(0,f)̸=(A,f ′) +|⟨0, f| HK |A, f ′⟩|2 +E0 − EA +, +(B2) +where |A, f ′⟩ denotes the excited state which can be expressed +as a direct product of the building blocks |k′′ +i , m′′ +i ⟩, with part +of wave vectors lying above the Fermi surface, i.e. k′′ +i > kF . +Here, we first derive the effective interaction of the f +fermions for a simpler two-impurity model and generalize the +results to the lattice version. +∆E(1) can be evaluated by summing over the subspace of +the conduction electron, yielding +∆E(1) = ⟨0, f| HK |0, f⟩ += 4nfJK +Ns +� +kµ +� +m,τ=± +� +β,β′ +� +J2 +K +εk − εk′′ +� � +iνˆαiνˆα′iν ˆβiν ˆβ′ +� +× eik·(ri+ˆα)−ik′′·(ri−ˆα′)eik′′·(rj+ ˆβ)−ik·(rj− ˆβ′)σm,−m +α +σ−τ,τ +α′ +στ,−τ +β +σ−m,m +β′ +⟨f| fi,−mf † +i,−τfj,−τf † +j,−m |f⟩ +(B10) + +11 +The effective interacting term among the f fermions can be +obtained by removing the bracket ⟨f| · · · |f⟩. This result can +be simply generalized to the lattice version by extending the +summation of i and j over the entire lattice, as shown in Eqs. +(6) and (7). +Appendix C: The mean-field Kondo-Heisenberg Hamiltonian on +a strip +In this section, we provide the details of the matrix elements +of the Kondo-Heisenberg Hamiltonian on a nano-strip with +Ny chains along y-axis. We choose the basis of the Kondo- +Heisenberg strip as +φA,k = +� +ck1↑, ck2↑, · · · , ckNy↑, c† +−k1↑, c† +−k2↑, · · · , c† +−kNy↑, fk1↓, fk2↓, · · · , fkNy↓, f † +−k1↓, f † +−k2↓, · · · , f † +−kNy↓ +�T +, +φB,k = +� +ck1↓, ck2↓, · · · , ckNy↓, c† +−k1↓, c† +−k2↓, · · · , c† +−kNy↓, fk1↑, fk2↑, · · · , fkNy↑, f † +−k1↑, f † +−k2↑, · · · , f † +−kNy↑ +�T +, +(C1) +where we take kx → k. The total Hamiltonian H is repre- +sented as a summation of two decoupled Hamiltonians, HA +and HB, each of which is 4Ny × 4Ny in size, given by +H = +� +k +φ† +A,kHA(k)φA,k + +� +k +φ† +B,kHB(k)φB,k. +(C2) +Below, we provides the matrix elements of HA(k) and HB, +respectively: +1. +HA +The matrix elements of the hopping term for HA are +HA(yi, yi) = −t cos k − µ +2 , +HA(yi + Ny, yi + Ny) = t cos k + µ +2 +(C3) +for yi = 1, 2, · · · , Ny while +HA(yi, yi + 1) = − t +2, +HA(yi + 1, yi) = − t +2, +HA(Ny + yi + 1, Ny + yi) = t +2, +HA(Ny + yi, Ny + yi + 1) = t +2 +(C4) +for yi = 1, 2, · · · , Ny − 1. +For Hf, we have for yi = 1, 2, · · · , Ny +HA(2Ny + yi, 2Ny + yi) = λ/2, +HA(3Ny + yi, 3Ny + yi) = −λ/2. +(C5) +The Kondo term HK for HA describes the Kondo interaction +with the following matrix form: the Kondo hybridization of c +and f with the same y chain are +HA(2Ny + yi, yi) = x sin k, +HA(Ny + yi, 3Ny + yi) = x sin k, +HA(yi, yi + 2Ny) = x sin k, +HA(3Ny + yi, Ny + yi) = x sin k +(C6) +for yi = 1, · · · , Ny. The matrix elements of the Kondo term +for yi = 1, · · · , Ny − 1 are +HA(2Ny + yi + 1, yi) = −x +2 , +HA(Ny + yi, 3Ny + yi + 1) = x +2 , +HA(2Ny + yi, yi + 1) = x +2 , +HA(Ny + yi + 1, 3Ny + yi) = −x +2 , +HA(yi, 2Ny + yi + 1) = −x +2 , +HA(3Ny + yi + 1, Ny + yi) = x +2 , +HA(yi + 1, 2Ny + yi) = x +2 , +HA(3Ny + yi, Ny + yi + 1) = −x +2 , +(C7) +which corresponds to the hybridization of c and f with the +nearest-neighboring y chains. +The RVB pairing term HJ on a nano-strip is described by +the following matrix elements: for yi = 1, · · · , Ny, +HA +∆(2Ny + i, 3Ny + i) = −i∆t sin k, +HA +∆(3Ny + i, 2Ny + i) = i∆t sin k +(C8) +are the matrix elements for the pairing of spinons with the +same yi. For yi = 1, · · · , Ny − 1, we have +HA(2Ny + yi, 3Ny + yi + 1) = − i +2∆t, +HA(2Ny + yi + 1, 3Ny + yi) = i +2∆t. +HA(3Ny + yi + 1, 2Ny + yi) = i +2∆t, +HA(3Ny + yi, 2Ny + yi + 1) = − i +2∆t. +(C9) + +12 +2. +HB +The matrix elements for the hopping term in HB are +HB(yi, yi) = −t cos k − µ +2 , +HB(yi + Ny, yi + Ny) = t cos k + µ +2 +(C10) +for yi = 1, 2, · · · , Ny. While, for for yi = 1, 2, · · · , Ny − 1, +we obtain +HB(yi, yi + 1) = − t +2, +HB(yi + 1, yi) = − t +2, +HB(Ny + yi + 1, Ny + yi) = t +2, +HB(Ny + yi, Ny + yi + 1) = t +2. +(C11) +The matrix elements for Hf are +HB(2Ny + yi, 2Ny + yi) = λ/2, +HB(3Ny + yi, 3Ny + yi) = −λ/2 +(C12) +with yi = 1, 2, · · · , Ny. +The matrix elements of the Kondo term for c and f lying +on the same chain yi are +HB(2Ny + yi, yi) = x sin k, +HB(Ny + yi, 3Ny + yi) = x sin k, +HB(yi, 2Ny + yi) = x sin k, +HB(3Ny + yi, Ny + yi) = x sin k, +(C13) +where yi = 1, · · · , Ny. 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B 79, 094504 (2009). + diff --git a/89AyT4oBgHgl3EQfqPhE/content/tmp_files/load_file.txt b/89AyT4oBgHgl3EQfqPhE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1752654cf4a03c9d2b5b7b274a1a5df13078a193 --- /dev/null +++ b/89AyT4oBgHgl3EQfqPhE/content/tmp_files/load_file.txt @@ -0,0 +1,988 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf,len=987 +page_content='Topological Kondo Superconductors Yung-Yeh Chang,1, 2 Khoe Van Nguyen,2 Kuang-Lung Chen,2 Yen-Wen Lu,3 Chung-Yu Mou,4 and Chung-Hou Chung2 1Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Republic of China 2Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan Republic of China 3Department of Physics and Astronomy, University of California, Riverside, California 92511, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 4Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan Republic of China (Dated: January 3, 2023) Spin-triplet p-wave superconductors are promising candidates for topological superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' They have been proposed in various heterostructures where a material with strong spin-orbit interaction is coupled to a conventional s-wave superconductor by proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' However, topological superconductors existing in na- ture and driven purely by strong electron correlations are yet to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here we propose a realization of such a system in a class of Kondo lattice materials in the absence of spin-orbit coupling and proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Therein, the odd-parity Kondo hybridization mediates ferromagnetic spin-spin coupling and leads to spin-triplet resonant-valence-bond (t-RVB) pairing between local moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Spin-triplet p ± ip′-wave topological super- conductivity is reached when Kondo effect co-exists with t-RVB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We identify the topological nature by the non-trivial topological invariant and the Majorana fermions at edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our results offer a comprehensive under- standing of experimental observations on UTe2, a U-based ferromagnetic heavy-electron superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' INTRODUCTION Searching for topological superconductors (TSc) and the corresponding self-dual charge neutral Majorana zero modes associated with their excitations at edges has become one of the central problem in condensed matter physics [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The- oretical proposals and experimental realizations of TSc are mostly heterostructure combining strong spin-orbit coupled materials and conventional superconductors by proximity ef- fect [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The emergence of the topological edge states in such systems can be explained in terms of the single-particle band structure without considering many-body electron corre- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Recently, the search for topological phases of matter has focused on a more intriguing class of materials that exist in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Their topological properties are driven by strong electron correlations instead of the proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Kondo effect, describing the screening of a local spin moment by con- duction electrons, is a well-known strong correlation between electrons existing in heavy electron compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Kondo- mediated topological phases of matter have been studied in the context of topological Kondo insulators [6–8] and topological Kondo semi-metals [9], where the topological properties are driven by either the odd-parity Kondo hybridization or by the Kondo hybridization with strong spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Spin-triplet p-wave superconductors are known to be the prime candidates for TSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' However, they are scarce in na- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' While it is still debatable for SrRu2O4 [10–12], more convincing evidence for p-wave triplet superconductivity was observed in noncentrosymmetric superconductor BiPd from phase-sensitive measurement [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' More recently, signatures of triplet chiral p-wave superconductivity were observed in heavy-electron Kondo lattice compound UTe2 at the edge of ferromagnetism, possibly marking the first example of topo- logical superconductor induced by the strongly correlated Kondo effect [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Motivated by these discoveries, in this paper, we propose a distinct class of triplet p-wave superconductors in the absence of spin-orbit coupling or proximity effect/heterostructure [18] in a two-dimensional Kondo lattice model driven by odd- parity Kondo hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We start from the Anderson lat- tice model (ALM) with odd-parity hybridization, which oc- curs between d- and f-orbital electrons in various heavy- fermion compounds [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Via the Schrieffer-Wolff transfor- mation [19, 20], we derive an effective Kondo lattice model with odd-parity hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Furthermore, by integrating out the conduction electron degrees of freedom, an effective ferromagnetic RKKY interaction is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We explore the mean-field phase diagram of this ferromagnetic Kondo- Heisenberg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In the fermionic mean-field approach, the ferromagnetic RKKY coupling describes the p-wave (Sz = ±1) t-RVB spin-liquid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A time-reversal invariant topo- logical superconducting phase is reached when the Kondo ef- fect co-exists with the p-wave t-RVB order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The topological nature of this superconducting phase is manifested by the non-trivial Z2 topological Chern number of the bulk band and by the existence of helical Majorana zero modes at the edges of a finite-sized ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our results offer a qualita- tive and some quantitative understanding of the spin-triplet su- perconductivity recently observed in UTe2 (see Discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Anderson lattice model with odd-parity hybridization We start with the odd-parity Anderson lattice model (ALM) on a two-dimensional (2D) square lattice, which has been shown to exhibit topologically non-trivial states [6–8]: HP AM = Hc + Hf + Hcf, (1) where Hc = � k,σ=↑,↓ εkc† kσckσ describes the hopping of electrons in the d orbits with orbital angular momentum l = 2 and dispersion εk = −2t(cos kx + cos ky) − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Hamil- tonian Hf of the more localized electron in the f orbits with arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='00538v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='str-el] 2 Jan 2023 2 ●●●● ●●●●●●●●●●●●● ●●●●●●●●●●● ●●● ●●●● ●●●● ●●●●●●●● ●● ■ ■ ■ ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ μ/t = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 ■ μ/t = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='05 R/a JH/JK 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The effective RKKY coupling JH (normalized with J2 K) as a function of R/a for different chemical potentials µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' JH is computed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (7) with Rij ∥ (1, 1) and a = 1 being chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' orbital angular momentum l = 3 is given by Hf = � i,σ � εff † iσfiσ + U 2 nf iσnf i,−σ � , (2) where εf denote the energy level of the f-electron, and U is the repulsive on-site Coulomb potential (the Hubbard-U term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Hybridization of the local and conduction electrons is described by Hcf = � ⟨i,j⟩ � σ,σ′=↑↓ V σσ′ ij c† iσfjσ′ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='. (3) To conserve the parity symmetry of hybridization between electrons with their angular momentum quantum numbers dif- fering by one, V σσ′ ij have to be odd under parity transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This restriction results in the hybridization having to depend on sites and spins [6–8]: V σσ′ ij ≡ V σσ′ ˆα = iV νˆασσσ′ α , (4) distinct from the well-known onsite and spin-conserving An- derson hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (4), νij satisfies νij ≡ νˆα = −νji with ˆα ≡ i − j ∈ ˆx, ˆy (α ∈ x, y) on a 2D square lattice, and σα denotes the Pauli matrix of the α component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The effective odd-parity ferromagnetic Kondo lattice model In this paper, we focus on the competition of the Kondo and the magnetic interaction among impurities–the Doniach scenario [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We, therefore, derive the effective Kondo- Heisenberg lattice Hamiltonian from ALM in the Kondo limit where the vacant and doubly-occupied states are projected out from the entire Hilbert space, namely 1 = � σ f † iσfiσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The low-energy effective Kondo term from the odd-parity ALM of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (1) can be derived by applying the Schrieffer-Wolff transformation (SWT) [19, 20, 22], yielding HK = (−JK) � i � σσ′ � σ′′σ′′′ � α,α′ � iνˆασσσ′ α c† i+ˆα,σfiσ′ � × � iνˆα′σσ′′σ′′′ α′ f † iσ′′ci−ˆα′,σ′′′ � (5) with JK = V 2 U+εf −εF + V 2 εF −εf > 0 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Kondo-like term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (5) describes the screening of an impurity by its neighboring conduction electrons, distinct from the conventional (on-site) Kondo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here, we go beyond the topological Kondo insulating phase by further deriving the magnetic RKKY interaction among the local f-fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' By perturbatively expanding the Kondo term to second order [22–24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' we obtain the effective RKKY- like interaction between the local f fermions fiσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' HJ = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='j � σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='σ′ Jijf † iσf † jσ′fjσfiσ′ = � ⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='j⟩ Jij � f † i↑f † j↑fj↑fi↑ + f † i↓f † j↓fj↓fi↓ � + � ⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='j⟩ Jij 2 � f † i↑f † j↓ + f † i↓f † j↑ � (fj↓fi↑ + fj↑fi↓) − � ⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='j⟩ Jij 2 � f † i↑f † j↓ − f † i↓f † j↑ � (fj↓fi↑ − fj↑fi↓) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (6) where Jij ≡ JH(R) = 16J2 K N 2s � εk<µ � εk′′>µ ei(k−k′′)·Rij εk − εk′′ × � sin2 kx + sin2 ky � � sin2 k′′ x + sin2 k′′ y � (7) denotes the effective coupling of the spinons of sites i and j with R ≡ |Rij| ≡ |ri − rj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The HJ term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (6) can be re-expressed as a linear combination of a spinon pair wave function with total spin S = 0 (spin-singlet) and S = 1 (spin-triplet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Note that the associated effec- tive spinon coupling of the spin-triplet channel is opposite to that of the spin-singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' When HJ is expressed in terms of fermion pair with different spins, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (6) is reminis- cent of the conventional Heisenberg interaction Si · Sj = − 1 2 � f † i↑f † j↓ − f † i↓f † j↑ � (fi↓fj↑ − fi↑fj↓)+ 1 4nf i nf j , except for the difference in the constant coefficients of the pair opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' As expected, the RKKY coupling Jij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (7) shows an oscillatory behavior in R, accompanied by a decrease in its magnitude with increasing R, similar to the behavior of the conventional RKKY coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Due to the rapid attenua- tion of Jij, we only consider the dominated nearest-neighbor interaction and assume Jij to be spatially homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Jij → J(R = a) ≡ JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Furthermore, when R = a, we find the effective RKKY coupling is attractive (or of the ferromag- netic type), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=', JH < 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 1), which energetically fa- vors the spin-triplet pairing of spinons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='effective RKKY coupling in the spin-singlet channel shows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='repulsive interaction and can be neglected here since it is not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='◆ x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='Δt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='4 JH Mean-field parameters JK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3, δ = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The zero-temperature mean-field solutions of t-RVB order parameter ∆t (brown) and the Kondo correlation x (black) as a func- tion of JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We fix JK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 and doping of the conduction band δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 (30 percent hole doping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Without loss of generality, we set t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This plot reveals a (co-existing) superconducting ground state with x ̸= 0, ∆t ̸= 0 for 0 < JH ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 and a pure t-RVB phase where x = 0, ∆t ̸= 0 when JH ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A pure Kondo phase (x ̸= 0, ∆t = 0) only exists at JH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Lastly, on a two-dimensional lattice, the triplet spin state | ↑↓⟩ + | ↓↑⟩ does not exist since the cor- responding structure factor is proportional to kz, and kz = 0 is fixed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Therefore, based on the above arguments, only the equal-spin states, | ↑↑⟩ and | ↓↓⟩, survive, and the HJ term is reduced to HJ ≈ − |JH| � ⟨i,j⟩ � f † i↑f † j↑fj↑fi↑ + f † i↓f † j↓fj↓fi↓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (8) Combining HK and HJ of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (5), (6) and (8), the effec- tive Kondo-Heisenberg lattice model with odd-parity Kondo hybridization reads HF KH = H0 + Hλ + HK + HJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here, Hλ = − � i iλi �� σ(f † iσfiσ) − 1 � enforces the singly occu- pied local f-spinons with λi being the Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Hamiltonian HF KH offers a platform for discovering a distinct class of topological superconducting states induced by electron correlations via collaboration between the ferro- magnetic RKKY coupling and the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' To facilitate our numerical calculations of the mean-field phase diagram, we treat JK and JH as independent couplings here since it is more convenient to explore the phase diagram by tuning the ratio of JK/JH [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In experiments, varying the non- thermal parameter can be expected to follow a certain trajec- tory of JK/JH in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' MEAN-FIELD TREATMENT OF THE EFFECTIVE KONDO-HEISENBERG-LIKE MODEL We now employ a mean-field analysis on the above ef- fective Kondo-Heisenberg-like Hamiltonian with an effective ferromagnetic RKKY interaction and odd-parity Kondo hy- bridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Via performing Hubbard-Stratonovich transformation, HK and HJ of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (5) and (6) can be factorized as HK → � i,α � σσ′ � χ† i � iνˆασσσ′ α f † iσci−ˆα,σ′ � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � + � i |χi|2 JK , HJ → � ⟨i,j⟩ � ∆↑ t (i, j)f † i↑f † j↑ + ∆↓ t (i, j)f † i↓f † j↓ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � + � ⟨i,j⟩ ���∆↑ t (i, j) ��� 2 + ���∆↓ t (i, j) ��� 2 JH (9) where the mean-field values of the bosonic Hubbard- Stratonovich fields, χi and ∆σ t (i, j) (σ =↑, ↓), represent the order parameters of the Kondo correlation and the Sz = ±1 spin-triplet RVB bonds between two adjacent up/down spins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' To describe the Kondo-screened Fermi-liquid state, we al- low the χi field to acquire uniformly Bose condensation over the real space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' hence, χi can be expressed as χi → x + ˆχi with x = (−JK/Ns) � iσσ′α⟨iνˆασσσ′ α f † iσci−ˆα,σ′⟩ being the Bose-condensed stiffness of χi while ˆχi represents its fluctua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The mean-field order parameter of the tRVB is given by ∆σ t = (−JH/4Ns) � ⟨i,j⟩⟨fjσfiσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Since the ferromagnetic coupling is expected to favor spin-triplet p-wave pairing sim- ilar to superfluid helium-3 [27], we restrict ourselves to the p-wave pairing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=', ∆σ t (i, j) here is taken the p-wave form, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (11) and (12) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We further fix the Lagrange multiplier at the mean-field level via iλi → λ and neglect the fluctuations of λi, χi, and ∆σ t , leading to the following mean- field Kondo-Heisenberg-like Hamiltonian: HMF = � k,σ εkc† kσckσ + � kσ λf † kσfkσ + � k � V1kf ∗ k↑ck↓ + V2kf ∗ k↓ck↑ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � + � k � ∆↑ kf † k↑f † −k↑ + ∆↓ kf † k↓f † −k↓ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � + 8Ns∆2 t JH + Nsx2 JK − Nsλ, (10) where V1k = 2x (sin kx − i sin ky) and V2k = 2x (sin kx + i sin ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Fourier transformation for the second-quantized operator is defined as ψiσ = 1 √Ns � k e−ik·riψkσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Note that the mean-field Kondo term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (10) is reminiscent of the topological Kondo insu- lator shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (10), ∆σ t (k) represents the gap structure of the spin-triplet p-wave RVB pairing in the momentum space for the spin-σ sector, defined as ∆↑ k = ∆t (− sin ky − i sin kx) and ∆↓ k = ∆t (sin ky − i sin kx) with ∆t being denoted the mean-field pairing potential (see Appendix, Section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This momentum-dependent gap struc- 4 ture for the up- and down-spin sectors correspond to the fol- lowing real-space patterns of ∆↑ t (i, j) and ∆↓ t (i, j) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (9): ∆↑ t (i, j) → ∆↑ t (i, i + ˆx) = −∆↑ t (i, i − ˆx) = −∆t, ∆↑ t (i, i + ˆy) = −∆↑ t (i, i − ˆy) = i∆t, (11) and ∆↓ t (i, j) →∆↓ t (i, i + ˆx) = −∆↓ t (i, i − ˆx) = −∆t, ∆↓ t (i, i + ˆy) = −∆↓ t (i, i − ˆy) = −i∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (12) Choosing Ψk = (φAk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' φBk)T with the Nambu spinors defined by φAk = � ck↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' c† −k↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' fk↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' f † −k↓ �T and φBk = � ck↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' c† −k↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' fk↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' f † −k↑ �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' the mean-field Hamiltonian HMF = � k Ψ † kHkΨk + C can be expressed as a summation of two decoupled 4 × 4 matrices as follows HMF = HA + HB + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' HA(B) = � k φ† A(B)kHA(B) k φA(B)k (13) with C ≡ � k εk + 8Ns∆2 t JH + Nsx2 JK ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' and HA k = � � � � � εk 2 0 V ∗ 2k 2 0 0 − εk 2 0 V2k 2 V2k 2 0 λ 2 ∆↓ k 0 V ∗ 2k 2 ∆↓∗ k − λ 2 � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (14) HB k = � � � � � εk 2 0 V ∗ 1k 2 0 0 − εk 2 0 V1k 2 V1k 2 0 λ 2 ∆↑ k 0 V ∗ 1k 2 ∆↑∗ k − λ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � � � � � (15) The Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (13) possesses time-reversal symme- try: HA and HB constitute the time-reversal partner of each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' ΘHA(B)Θ−1 = HB(A) where the time-reversal operator Θ = ρ0 ⊗ (−iσy)K with σy being the y-component Pauli matrix on the spin subspace, ρ0 being a 2 × 2 identity matrix on the orbital subspace while K being the complex- conjugate operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Under time-reversal transformation, the spin and quasi-momentum of conduction (c) and pseud- ofermion (f) operators are flipped: (ck↑, ck↓, fk↑, fk↓) Θ −→ (c−k↓, −c−k↑, f−k↓, −f−k↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Meanwhile, our Hamilto- nian respects charge-conjugation (particle-hole) symmetry: PHkP−1 = −H−k where P ≡ τ xK is the particle-hole op- erator with τx being the x-component of the Pauli matrices on the particle-hole basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Due to the odd-parity p±ip′ RVB pair- ing of our model, the parity symmetry is broken here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Thus, our model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (13) belongs to the DIII class of topological symmetry [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Mean-field phase diagram The mean-field ground states are determined by minimiz- ing the mean-field free energy per site FMF = C Ns − kBT Ns � nk ln � 1 + exp � − Enk kBT �� with respect to the mean- field variables q = (λ, x, ∆t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' ∂FMF /∂qi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here, Enk < 0 is the n-th band of Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The chemical potential µ is determined by the relation ∂FMF /∂µ = −(1 + δ) with δ be- ing the chemical doping of the c-electrons for which δ ⪋ 0 is for p/un/n− doped (half-filling corresponds to δ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This leads to the following saddle-point equations at zero tempera- ture, 1 Ns � nk ∂Enk ∂x + 2x JK = 0, 1 Ns � nk ∂Enk ∂∆t + 16∆t JH = 0, 1 Ns � nk ∂Enk ∂λ = 0, 1 Ns � nk ∂Enk ∂µ + δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (16) The ground-state phase diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 2) of our model is ob- tained by solving the saddle-point equations self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The phase diagram contains three distinct mean-field phases: a pure Kondo phase is found at JH = 0 where x ̸= 0, ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' At the opposite limit where the RKKY interaction dominates, the ground state shows short-range magnetic correlation with p-wave spin-triplet RVB pairing (∆t ̸= 0, x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In the intermediate range of 0 < JH/JK < (JH/JK)c, we find a Kondo-tRVB co-existing (superconducting) phase with x ̸= 0 and ∆t ̸= 0, which can be explained via the mechanism of Kondo-stabilized spin liquid [26, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The development of superconductivity in this co-existing phase requires higher- order processes involving both the Kondo and t-RVB terms: the mean-field t-RVB pairings of the local f fermions pro- vide preformed Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' When the Kondo hybridization field χ gets Bose-condensed (x ̸= 0), the local fermions de- localize into the conduction band and make the preformed t- RVB Cooper pairs superconduct [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' These processes can be described by the effective mean-field Hamiltonian Hsc = � k � ¯ ∆↓∗ k c−k↓ck↓ + ¯ ∆↑∗ k c−k↑ck↑ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � , where the effec- tive gap functions take the form ¯ ∆↓∗ k = V1kV1,−k∆↑∗ k ∼ x2∆t(sin2 kx + sin2 ky)(sin kx − i sin ky) and ¯ ∆↑∗ k = V2kV2,−k∆↓∗ k ∼ x2∆t(sin2 kx + sin2 ky)(sin kx + i sin ky) with the size of the superconducting gap being proportional to x2∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The superconducting gap function ¯ ∆↑ k we obtained here shows a f-wave-like pairing symmetry on a generic anisotropic (non-circular) 2D Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Nevertheless, as we are taking the continuous limit of the conduction band here, ¯ ∆↑ k can be expressed as a product of s and p±ip′ pairing 5 (a) Γ X M (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Figures (a) (red curves) and (b) show the bulk energy spec- trum of the co-existing superconducting state near the Fermi level µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Fermi level locates at E(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The coupling constants are JK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 and JH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Inset of (a) displays the First Brillouin zone of a square lattice with indications of high-symmetry points Γ, X, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' orders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=', ¯ ∆↑/↓∗ k ∼ k2(kx±iky) with k2 ≡ k2 x+k2 y on a cir- cular Fermi surface but only the p±ip′ component plays a role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Note that we find the co-existing superconducting state persists for an arbitrary small value of JH/JK → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This is likely due to the overestimation of the co-existing phase at the mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Upon including fluctuations of the Kondo and t-RVB order parameters beyond the mean-field level, we expect a narrower co-existing superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A first- order transition similar to the results found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' [26, 32] is observed at the transition of the t-RVB and the co-existing superconducting phases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The bulk band structure in the co-existing superconducting state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Topological invariance We now address the topological properties of the coexisting superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Since this system is invariant under time-reversal transformation, the bulk topological properties of the coexisting Kondo-RVB superconducting state with p ± ip′ spin-triplet RVB pairing can be thus characterized by the Z2 Chern number cT (or time-reversal polarization) [33–35], kx E(kx) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The left figure displays the electronic band structure of the coexisting superconductor state for a strip with Ny = 81 described by HA at JK/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 and JH/t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Three pairs of edge states with Dirac spectra are observed near kx = 0 (the pink curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The edge states at zero energy correspond to the Majorana zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Due to the time-reversal symmetry of the model, the band structure for a strip for HB is identical to that of HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The close-up band structures near three pairs of edge states (pink curves) on the top, middle and bottom bounded by the red squares are shown on the right figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' given by cT = cA − cB 2 (17) with cI (I ∈ A, B) being the Thouless-Kohmoto-Nightingal- den Nijs (TKNN) number [36] of HI, defined as cI = 1 2π � k∈FBZ dSk · � ∇k × AI k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (18) The Berry’s connection AI k for HI is given by AI k ≡ i � n∈I⟨uI nk|∇k|uI nk⟩ with |uI nk⟩ being the normalized Bloch state of the n-th filled band for HI k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We numerically calculate the TKNN numbers [37], cA and cB, and find that cA = −cB = 1 in the co-existing phase, indicating a topolog- ically non-trivial Z2 Chern number cT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' By the bulk-edge correspondence, we expect this co-existing superconducting state to support a pair of counter-propagating Majorana zero modes at the edges of a finite-sized strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Further band struc- ture calculations of our model on a strip in the following sub- section confirm our expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Edge states of the coexisting Kondo-RVB spin-triplet p ± ip′-wave superconducting state We now check whether our model would support helical Majorana zero modes at the edge of a finite-sized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We shall examine our model’s band structures and edge-state wave functions on a finite-sized strip that extends infinitely along the x direction but contains a finite number of lattice sites in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 4 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' As shown 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 E(k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='5 π 一π 0 0 π一 2 2 Ky kx π－ 2 26 E E x y yi = 1 yi = Ny Ribbon γRA,k γL B,k γR B,k γL A,k (a) (b) (c) (d) (e) (f) (g) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Figures (a) and (d) show the Bogoliubov excitation spectra of HA and HB, respectively, near the chemical potential on a nano-strip with Ny = 81 chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Figures (b), (c) and (e), (f) demonstrate the probability density of the Majorana edge state wave functions of HA and HB as a function of atom position yi, ��γΓ I,kx(yi) ��2 with I = A, B and Γ = R, L (pink curves in (a) and (d)), at a fixed energy E ≡ E(kx = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The probability density is described by ��γΓ I,kx(yi) ��2 = ���uΓ I,kx ��2 , ��¯uΓ I,kx ��2 , ��vΓ I,kx ��2 , ��¯vΓ I,kx ��2� (yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The parameters are JK/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3, JH/t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0, and doping δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The edge states are of the helical type, as schematically represented in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' E E 1 2 4 3 1 3 2 4 (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The finite-energy (E(kx) > 0) Bogoliubov excitation spectra of (a) HA (shown on top right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 5) and (d) HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A pair of “helical” edge states is found to exist at finite energy [pink curve in (a) and (d)], and their probability densities are shown in (b) and (c), (e) and (f), respectively, at a fixed energy E(kx = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 4, gapless Dirac spectra of the Bogoliubov excitations around kx = 0 near zero energy are observed, exhibiting one of the typical features of topological edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The exci- tations can be effectively described by the linear-dispersed Hamiltonian ˜HI = � kx vx|kx| � γR † I,kxγR I,kx − γL † I,kxγL I,kx � with γΓ I,kx = � yi � uΓ I,kx(yi)ckx,yi,↑ + ¯uΓ I,kx(yi)c† −kx,yi,↑ +vΓ I,kx(yi)fkx,yi,↓ + ¯vΓ I,kx(yi)f † −kx,yi,↓ � (19) with u, ¯u and v, ¯v being the coherent factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (19), I ∈ A, B, Γ ∈ R, L, and γR/L A/B,kx represents the right/left- moving Bogoliubov quasiparticle of ˜HA/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here, vx in ˜HI denotes the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Due to time-reversal symmetry, HA is the time-reversal partner of HB, and thus their spectra are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The low-energy eigenstates with Dirac spectra near kx = 0 for both HA and HB exhibit the typical prop- erty of edge states, as their probability densities accumulate mostly at the edges of strip, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Combin- ing the directions of propagation inferred from the velocity vx ∼ ∂E(kx)/∂kx, we can classify these edge states into two 7 Tc Tonset Tc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Plot of the temperature-dependent mean-field order pa- rameters x(T)/t and ∆t(T)/t with kB = 1, JK/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 and JH/t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Inset shows the enlarged plot of ∆t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The single-impurity Kondo temperature occurs at Tonset/t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='16 while the transition of superconductivity takes places at temperature Tc/t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' groups, each of them constitutes a pair of counter-propagating edge states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 5), revealing the nature of helical Ma- jorana zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The helical type of the Majorana zero modes is the consequence of time-reversal symmetry of our model, reminiscent of the well-known Kane-Mele model on a single-layered graphene [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Remarkably, in addition to the Majorana fermions at zero energy, two pairs of counter- propagating edge-states are observed at finite energy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The two pairs of edge states correspond to the edge states of the topological Kondo insulator, where the spin-triplet RVB order parameter is absent (∆t = 0) [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSIONS We now discuss the application of our results for heavy- electron superconductors, particularly the Kondo lattice com- pound UTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Experimental evidence indicates that this com- pound does not show long-range magnetic order and is in the vicinity of the ferromagnetic quantum critical point, ex- hibiting both strong ferromagnetic fluctuations, possibly due to magnetic frustrations induced by sub-leading antiferromag- netic fluctuations [40, 41], and Kondo screening [14, 17, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The DFT+U calculations indicate that the dynamics of elec- tron bands and the physical properties of UTe2 are dominated by the electrons near the quasi-two-dimensional (cylindrical) Fermi surface with weak kz dependence despite its 3D crystal structure [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Superconductivity is reached at Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='6K, while the resistivity maximum observed at T ⋆ ≈ 15 ∼ 75 K reveals signature of coherent Kondo scattering [14, 43], indi- cating T ⋆/Tc ≈ 10 ∼ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The superconductivity can, in gen- eral, co-exist and compete with the Kondo effect [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' When a magnetic field is applied along the hard-magnetic axis b of UTe2 and before entering the superconducting phase, a corre- lated paramagnetic phase is observed below the temperature at which the magnetic susceptibility shows a broad maximum [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Similar spin-liquid behavior has been observed in the magnetic susceptibility of another heavy fermion compound CePdAl [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This similarity suggests this correlated para- magnetic phase may feature short-range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our theoretical framework based on competition and collaboration between a Kondo-screened and a ferromagnetic t-RVB spin- liquid states on a two-dimensional Kondo lattice is consistent with the above observations in UTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' It, therefore, consti- tutes a promising approach to account for its exotic phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' On the other hand, the chiral in-gap state, a signature of chiral topological superconductor, has been observed by scan- ning tunneling spectroscopy in the superconducting phase of UTe2 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Combining with the ferromagnetic fluctuations that are known to induce spin-triplet pairing, people believe UTe2 is a promising candidate for the spin-triplet chiral topo- logical superconductor [14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Furthermore, the supercon- ducting phase co-existing with Kondo coherence in this ma- terial strongly suggests the role played by the Kondo effect in this possible topological superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The topologi- cal Kondo superconducting state with equal-spin spin-triplet p-wave pairings we proposed here bears striking similarities to and strong relevance for the experimental observations on UTe2: (i) the d- and f-orbitals electrons with their angular momentum quantum number differing by 1 in the uranium atoms of UTe2 likely give rise to the odd-parity Kondo effect [6–8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (ii) the t-RVB state in our theory may be considered as one possible realization of the short-ranged ferromagnetic fluctuations in UTe2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (iii) the Kondo-t-RVB co-existing su- perconducting state we find here qualitatively agrees with the co-existence between superconductivity and Kondo effect ob- served in UTe2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (iv) the high upper critical field exceeding the Pauli limit [14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 46] implies that the superconducting state of UTe2 may have equal-spin Cooper pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' and (v) the effective pairing ∆ σ k formed in the conduction band mentioned in Sec- tion IV A shows characteristics of spin-triplet point-node gap structure [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Various characteristic temperature scales esti- mated from our mean-field calculations with JH/t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='0 and JK/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 at finite temperatures agree reasonably well with experimental observations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' 7): The superconducting transition temperature Tc, theoretically determined from our mean-field analysis Tc = Min[T(x = 0), T(∆t = 0)], shows Tc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='015t ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='3 K by taking estimated values of t = 150 K and half-bandwidth D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='25t [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The Kondo coherent scale can be obtained by T ⋆ = x2(T = 0)/D ≈ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='4 K [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The ratio T ⋆/Tc ≈ 8 is in reasonable agreement with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The onset temperature Tonset of Kondo hybridization, which occurs at x(T = Tonset) = 0, displays Tonset ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='16t ≈ 24 K, within the theoretically es- timated range 10K < Tonset < 100K by DMFT calculation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Meanwhile, there have been evidences of TRS breaking in UTe2 from the observed two superconducting transitions and a finite polar Kerr effect at T < Tc [49], likely due to proximity to the ferromagnetic ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A number of theoretical attempts were proposed based on these observa- tions [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' However, the observed single superconducting transition near ambient pressure and zero field [44, 52, 53] as well as the theoretically proposed unitary triplet pairing [40] 8 suggest TRS may be preserved in UTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Though our results shown above are obtained in the presence of TRS, the chi- ral p-wave superconducting state with chiral Majorana zero mode at edges is expected to occur here once a time-reversal breaking magnetic field is applied [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our distinct predic- tions with and without fields serve as theoretical guidance for future experiments to distinguish the time-reversal breaking from time-reversal preserving triplet pairing states in UTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Since the Kondo correlations stabilize the t-RVB spin liquid in the co-existing superconducting phase, it is expected to be robust against gauge-field fluctuations beyond the mean field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our approach and results are distinct from the spin-triplet non- topological superconducting state recently proposed based on the Hund’s-Kondo coupling and Sz = 0 t-RVB state to ac- count for UTe2 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In conclusion, we propose a first realization of the topo- logical superconductivity in the Kondo lattice model, a dis- tinct class of topological superconductors due to purely strong electron correlations without employing spin-orbit coupling or proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' A topological Kondo superconductor essentially constitutes of 1) itinerant c and localized f bands with different orbital quantum numbers, 2) strong Hubbard in- teraction of the f electrons, 3) odd-parity Kondo hybridization of the c and f bands, and 4) the attractive exchange interac- tion of the f electrons with spin-triplet correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Start- ing from the odd-parity Anderson lattice model, we obtain the unconventional type of Kondo hybridization and ferro- magnetic RKKY-like interaction via perturbation theory, lead- ing to spin-triplet resonating-valence-bond (RVB) pairing be- tween f-electrons with time-reversal invariant p ± ip′-wave gap symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Via the mean-field approach, we find a Kondo triplet-RVB coexisting phase in the intermediate range of the Kondo to RKKY coupling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This phase is shown as a time-reversal invariant topological superconducting state with a spin-triplet p ± ip′-wave RVB pairing gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' It exhibits non- trivial topology in the bulk band structure, and supports heli- cal Majorana zero modes at edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our prediction in the pres- ence of a time-reversal breaking field leads to chiral p-wave spin-triplet topological Kondo superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Our results on the superconducting transition temperature, Kondo coherent scale, and onset temperature of Kondo hybridization not only qualitatively but also quantitatively agree with the observa- tions for UTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The theoretical framework we propose here opens up the search for topological superconductors induced by strongly electronic correlations on the Kondo lattice com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported by the Ministry of Science and Technology Grants 104-2112-M-009-004-MY3 and 107- 2112-M-009-010-MY3, the National Center for Theoretical Sciences of Taiwan, Republic of China (to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Appendix A: The Schrieffer-Wolff transformation (SWT) In this section, we provide derivations of the Kondo term via using the SWT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' We first perform the SWT on an odd-parity single-impurity Anderson model where an impurity at an ar- bitrary site i hybridizes with the conduction electrons on the four nearest-neighbor sites of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' This result will be succes- sively generalized to the lattice version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The single-impurity Anderson model takes the following form H = � kσ εkc† kσckσ + � σ εff † iσfiσ + Unf i↑nf i↓ + � σσ′ � α=x,y � iV νˆασσσ′ α c† i+ˆα,σfiσ′ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' � , (A1) where ˆα ≡ ±ˆx, ±ˆy denotes the nearest-neighbor vectors of a square lattice, and νˆα satisfies νˆα = −ν−ˆα and νˆx = νˆy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The SWT aims at projecting out the empty and doubly oc- cupied states to generate the effective Hamiltonian Heff in the Kondo (singly-occupied) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' [20], we first use the states of impurity occupation as the basis set, {|f 0⟩, |f 1⟩, |f 2⟩} with the superscripts being denoted as the occupation of the localized electrons, to expand the Hamilto- nian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A1) in the following matrix form, H = � � H00 H01 H02 H10 H11 H12 H20 H21 H22 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A2) The matrix elements of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A2), denoted as Hij ≡ ⟨f i|H|f j⟩ with i, j = 0, 1, 2, are H10 = � σσ′ � α=±x,±y iV νˆασσσ′ α f † iσci−ˆα,σ′ = H21, H01 = H† 10 = � σσ′ � α=±x,±y iV νˆασσσ′ α c† i+ˆα,σfiσ′ = H12, H11 = � kσ εkc† kσckσ + � σ εff † iσfiσ, H00 = � kσ εkc† kσckσ, H22 = � kσ εkc† kσckσ + � σ εff † jσfjσ + Unf i↑nf i↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A3) We then project out |f 0⟩ and |f 2⟩ from the Hilbert space to obtain the effective Hamiltonian Heff at the Kondo limit sat- isfying Heff|f 1⟩ = E|f 1⟩ with E being the eigenenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A2), Heff can be expressed as Heff = H11 + H′ with 9 H′ =H10(E − H00)−1H01 + H12(E − H22)−1H21 = � α,α′=x,y � σσ′ � σ′′σ′′′ � V 2 εF − εf − U � iνˆασσσ′ α c† i+ˆα,σfiσ′ � � iνˆα′σσ′′σ′′′ α′ f † iσ′′ci−ˆα′,σ′′′ � (A4) + V 2 εf − εF � iνˆασσσ′ α f † iσci−ˆα,σ′ � � iνˆα′σσ′′σ′′′ α′ c† i+ˆα′,σ′′fiσ′′′ �� (A5) Here, we skip the derivations of H10(E − H00)−1H01 and H12(E − H22)−1H21 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A5) as those are standard and can be found in a number of references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' See, for example, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' H′ can be further cast into the form simi- lar to the conventional single-impurity Kondo term, with the following antiferromagnetic Kondo coupling JK = V 2 U + εf − εF + V 2 εF − εf > 0, (A6) plus a potential scattering term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (A5) can be generalized to the lattice version by summing over all lattice sites, as de- scribed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Appendix B: Derivation of the effective ferromagnetic RKKY-like interaction In the section, we derive the RKKY-like interaction by per- turbatively expanding HK of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (5) to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The unperturbed state is described as |0, f⟩ = |k1m1, k2m2, · · · , kNmN⟩ |f⟩ , (B1) where conduction electrons do not interact with the impuri- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' (B1), |k1m1, k2m2, · · · , kNmN⟩ represents the Fermi sea with all wave vectors lying below the Fermi wave vector, namely ki < kF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' After imposing perturbation, the un- perturbed state acquires correction and the corrected eigenen- ergy is expressed in powers of JK, E = E0 + ∆E(1) + ∆E(2) + O(J3 K) with E0 being the eigenenergy of the un- perturbed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' The first and second order energy corrections take the form ∆E(1) = ⟨0, f| HK |0, f⟩ , ∆E(2) = � (0,f)̸=(A,f ′) |⟨0, f| HK |A, f ′⟩|2 E0 − EA , (B2) where |A, f ′⟩ denotes the excited state which can be expressed as a direct product of the building blocks |k′′ i , m′′ i ⟩, with part of wave vectors lying above the Fermi surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' k′′ i > kF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' Here, we first derive the effective interaction of the f fermions for a simpler two-impurity model and generalize the results to the lattice version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfqPhE/content/2301.00538v1.pdf'} +page_content=' ∆E(1) can be evaluated by summing over the subspace of the conduction electron, yielding ∆E(1) = ⟨0, f| HK |0, f⟩ = 4nfJK Ns � k1 +eV/H2) with the host atoms. Another type of adsorption process observed is similar to the +physisorption, in which the inter-atomic H-H bond in the H2 molecule is elongated but not +dissociated and adsorbed by Kubas-type orbital interactions[2]. It enhances the H2 adsorption +energy and makes most of the H2 storage capacities that fulfil the target of the US department of +energy (DOE-US) [5, 6]. +Since last few years, researchers are engaged extensively to study various materials, including +carbon nanostructures [7, 8], metal hydrides [9, 10], graphene [11, 12], metal alloys[13, 14], +metal-organic frameworks (MOF)[15, 16], and covalent-organic frameworks [17], etc. for the +reversible hydrogen storage at ambient condition. However, it has been reported that these +materials often have several limitations, including poor storage capacity, instability at significantly +high temperatures, and low reversibility at normal temperatures. For example, Mg-based metal +hydrides showed a high storage capacity of up to 7.6 wt% under ambient condition, however; it + +could be used only for 2-3 cycles [18].Similarly, metal alloys have very poor reversibility when +used as hydrogen storage materials [19]. Using MOFs as H2 storage materials, researchers could +attain up to 15 wt% of storage capacity at temperatures and pressures of 77 K and 80 bar. However, +under normal environmental condition its gravimetric and volumetric storage capacity remained +very low [20]. To address the aforesaid issue and to develop commercially effective hydrogen +storage materials, the experimentally synthesized organic compounds functionalized with +transition metals (TMs), such as TM-doped organometallic buckyballs, TM-ethylene, etc., were +introduced and investigated extensively [21, 22]. Early reports show that the TM atoms form a +strong bond with the π -electron delocalized compounds through the Dewar mechanism and adsorb +hydrogen molecules via Kubas interaction[23, 24]. For example, Chakraborty et al. studied the +hydrogen storage in Ti-doped Ψ-graphene and reported an H2 uptake capacity of up to 13.1 wt% +with an average adsorption energy of -0.30 eV/H2 [25]. Dewangan et al. predicted up to 10.52 wt% +of H2 adsorption in Ti-functionalized holey graphyne via the Kubas mechanism with adsorption +energy and desorption temperature of 0.38 eV/H2 and 486 K, respectively[26]. +Numerous theoretical and experimental studies revealed that metal-adorned small organic +molecules like CnHn could capture a large number of H2 molecules. For example, Zhou et al. +estimated hydrogen uptake capacity up to 12 wt% in TiC2H4 with H2 binding energy of 0.24 +eV/H2[27, 28]. High capacities of H2 storage in TMC2H4 (M = Ti, Sc, V, Ni, Ce, Nb) complexes +was reported by Chaudhari et al.[29, 30, 31]. At low benzene pressure (35 millitorrs) and ambient +temperature, TiC6H6 was experimentally shown to absorb up to 6 wt% hydrogens [32]. Phillips et +al. obtained an H2 uptake of up to 14 wt% and quick kinetics at room temperature on TiC6H6 by +laser ablation; however, the experiments did not discuss the desorption process[33]. Recently, +Ma et al. theoretically studied an interesting combination of chemisorption and physisorption in +Ti-doped C6H6 and reported an uptake capacity of 6.02 wt % with complete desorption at 935 +K [34]. Mahamiya et al. revealed the H2 storage capacities of 11.9 wt % in K and Ca decorated +biphenylene with an average adsorption energy of 0.24-0.33 eV [35]. Y atom doped zeolite showed +high capacity adsorption of H2 with binding energy 0.35 eV/H2 and the desorption energy of 437K +for fuel cells[36]. +Macrocyclic compounds, like paracyclophane (PCP), a subgroup derivative of cyclophanes, +comprises aromatic benzene rings with number of -CH2- moieties linking the subsequent benzene + +rings [37]. The PCPs are easier to synthesize in the laboratory; they can be functionalized with +metal atoms due to the presence of aromatic benzene rings in the geometry, making them a feasible +alternative for hydrogen storage prospects. For instance, Sathe et al. studied the Sc and Li +decorated PCP and reported the molecular H2 physisorbed via Kubas-Niu-Jena interaction +resulting in up to 10.3 wt% H2 uptake capacity [38]. The hydrogen storage transition metal (Sc, +Y) functionalized [1,1]paracyclophane was investigated by Sahoo et al. and reported a storage +capacity of 6.33-8.22 wt%, with an average adsorption energy of 0.36 eV/H2 and desorption +temperature of 412 K - 439 K[39]. The H2 storage on Li and Sc functionalized [4,4]paracyclophane +shows an uptake capacity of 11.8 wt% and 13.7 wt%, as estimated by Sathe et al. [40]. Kumar et +al. revealed the combination of physisorption and chemisorption of hydrogen on Sc and Ti +functionalized BN-analogous [2.2]PCP[41]. They showed the first hydrogen molecule +chemisorbed on the host material followed by physisorption of other H2, resulting in a storage of +~8.9 wt% via Kubas interaction. Numerous other metal-decorated macrocyclic compounds have +been explored as hydrogen storage possibilities, with storage capacities above the DOE +requirement; however, only a few have shown practical H2 capacity at varied thermodynamic +conditions. Though few PCP-based hydrogen storage systems are available in the literature, the +[2,2,2]paracyclophane, which is experimentally synthesized by Tabushi et al.[42] is yet to be +explored as a hydrogen storage material. +In the present work, we investigated the chemisorption and physisorption properties of hydrogen +molecules on [2,2,2]paracyclophane (PCP222) functionalized with Ti atoms and estimated their +hydrogen uptake capacity at varied thermodynamics. In paracyclophane, there are many molecules +in the group and are named after their pattern of arene substitution. The preceding square bracket +number, “[2,2,2]” in [2,2,2]paracyclophane, indicates that the consecutive benzene rings (3 +benzene rings) in paracyclophane are linked with two (-CH2-) moieties. The linking bridges are +relatively short; thus, the separation between consecutive benzene rings is small, which develops +a strain in the aromatic rings. This strain in the rings can be utilized for Ti functionalization over +the aromatic benzene ring. Due to the strain and metal functionalization, the aromatic benzene +rings lose their inherent planarity. We choose to functionalize Ti metal atoms over the PCP222, as +the d- block transition metal elements are well known for reversible hydrogen adsorption and could +bind the H2 molecules via Kubas interaction[25, 26]. Though there are few reports available based +on hydrogen storage in macrocyclic organic compounds and other Ti-doped nanostructures, our + +work is the first to investigate the efficiency of Ti-functionalized PCP222 using the atomistic MD +simulation, practical storage capacity, and diffusion energy barrier estimation +2 Theory and Computation +We have performed the theoretical calculations on [2.2.2] paracyclophane (PCP222) and their +hydrogenated structures within the framework of density functional theory (DFT)[43]. In the +computation, the advanced hybrid ωB97Xd functional is used, and molecular orbitals (MO) are +expressed as the linear combination of atom-centered basis function for which the valence diffuse +and polarization function 6-311+G(d,p) basis set is used for all atoms. ωB97Xd includes the long- +range and Grimme’s D2 dispersion correction which is a range-separated version of Becke’s 97 +functional[44, 45]. It is important to note that the ωB97Xd technique is a trustworthy method for +studying the non-covalent interactions, Organometallic complexes, and their thermochemistry. To +ensure the studied structures are in true ground state on the potential surface, the harmonic +frequencies of all the systems are determined and are found to be positive. All the theoretical +computations are performed with the computational program Gaussian 09[43]. +In order to investigate the binding strength of titanium (Ti) atoms on the PCP222, we have +calculated the average binding energy of decorated Ti atoms by using the following equation. +𝐸𝑏 = +1 +𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝑇𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖] + + + (1) +Where EPCP222, ETi, and EPCP222+mTi is the total energy of PCP222, Ti atom and Ti-decorated +PCP222 respectively. m is the number of Ti atoms added PCP222. +The average adsorption energy of molecular hydrogen with metal atoms is calculated as[46]. +𝐸𝑎𝑑𝑠 = +1 +𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖+𝑛𝐻2] + (2) +Where EPCP222+mTi, EH2, and EPCP222+mTi+nH2 is the total energy of host material, hydrogen molecule +and hydrogen trapped complexes respectively. n is the number of H2 molecules adsorbed in each +complex. +The global reactivity descriptors such as hardness (η), electronegativity (χ), and electrophilicity +(ω) were estimated and used to study the stability and reactivity of Ti functionalized PCP222 and +their hydrogen adsorbed derivatives [47, 48]. The energy gap between the highest occupied + +molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is computed to +assure the kinetic stability of the studied systems. Further, to understand the electronic charge +transfer properties, the Hirshfeld charge and electrostatic potential map (ESP) were explored. +Moreover, partial density of states (PDOS) investigation was also carried out to further understand +the process of hydrogen interaction. The topological parameters were studied using Bader’s theory +of atoms in molecules (AIM) to analyze more about the nature of the interaction between metal on +PCP222 and adsorbed hydrogen molecules. +To obtained the hydrogen uptake capacity, gravimetric density (wt%) of hydrogen is calculated +using the following equation[49]: +𝐻2(𝑤𝑡%) = +𝑀𝐻2 +𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 + + + + (3) +Here MH2represent the mass of the total number of H2 molecules adsorbed and MHost represent the +mass of metal-doped PCP222. +3 Results and Discussion +3.1 Structural properties of PCP222 +The optimized geometrical structure of PCP222 is depicted in Figure 1(a). PCP222 has three +benzene rings connected by two -CH2- moiety as a bridge between the consecutive rings. The +distance between the two consecutive -CH2- moiety and the -CH2- across the benzene ring are +found to be 1.54 Å and 5.84 Å respectively, which is consistent with the earlier experimentally +reported value by Cohen-Addad et al. [50]. To validate the π aromaticity of the optimized +molecule, we computed the Nucleus Independent Chemical Shift (NICS) of PCP222 before +functionalizing by any metal atom. The NICS values are determined with 1 Å increment from the +center to 3 Å above the three benzene rings. NICS(1) is found to be negative maximum (-10.1 +ppm), suggesting the aromatic nature of PCP222. This indicates that the benzene rings of PCP222 +are π electron-rich and can bind a metal atom outside the benzene rings. +3.2 Functionalization of Ti atom on PCP222 + + +Figure 1: (a) Optimized structure of PCP222 with all possible marked adsorption site marked, (b) Ti +functionalized PCP222 + + +Next, we explore different possible adsorption sites of pristine PCP222, such as C-C bridge of +benzene ring (B1), CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the +center of benzene (Rc) which are depicted in Figure 1(a). To design the host material for hydrogen +adsorption, a single Ti atom is positioned about 2 Å above at the regioselective sites of PCP222, +and the resulting structure is re-optimized. The binding energy between Ti and PCP222 calculated +using Equation 1 at different adsorption sites shows that the Ti atom is stable at two positions, B3 +and Rc sites of PCP222 with binding energies of 0.37 eV and 2.20 eV, respectively which fairly +agree with the previously reported value of Ti on CNT by Yildirim et al. [51]. Hence, the most +favourable site for Ti atom functionalization is at the Rc site above the benzene ring of PCP222. +3.2.1 Bonding mechanism of Ti on PCP222 +To understand the binding mechanism of Ti on PCP222, we analyzed the partial density of state +(PDOS), electrostatic potential map (ESP), Hirshfeld charge, and Bader’s topological parameters +of the Ti functionalized PCP222 system as discussed below. +Density of states + +5.84 +B2 +5.87 +1.543 +(a) +(b) +Figure 2: Density of states plot on Ti and C atom on PCP222 + +The Ti atom is functionalized on PCP222 via the Dewar mechanism in which π-electron gets +transferred from the highest occupied molecular orbitals (HOMO) of the substrates to the vacant +d-orbital of Ti followed by the back-donation of charges from the partially filled d-orbital of Ti to +empty π*-anti-bonding of the benzene ring of PCP222[26]. To understand the orbital interaction +between the Ti and C atom of PCP222, we have performed the partial density of states (PDOS) +calculation of PCP222-Ti and the result is plotted in Figure 2. Figure 2 clearly shows that the +electronic states of the Ti atom and the C atom of PCP222 overlap below and above the Fermi +level (E = 0). The transferred electrons partially fill the unoccupied states of PCP222, as seen by +the intense peaks near the Fermi level. This infers an orbital interaction between Ti and C atom of +PCP222 mediated by charge transfer. The fact is also obvious because Ti has the relatively lower +ionization potential than the C atom. +ESP and Hirshfeld charges +To get a picture of electronic charge distribution over the PCP222 during Ti functionalization, we +plotted the electrostatic potential (ESP) map over the total electron density, as shown in Figure.S1. +The variation of electron density in the ESP map is shown by using different colour codes, which +follows the pattern of accumulation and reduction of electron density as; red (maximum electron +density) >orange > yellow > green > blue (minimum electron density). In the ESP plot (Figure.S1), +the red region over the benzene ring of PCP222 implies the aggregation of electron density. After + +22 +c +PCP222-Ti +20. +Ti +18- +Total +16- +14. +12 - +DOS +10- +8. +6. +4. +2 +0 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +2 +0 +4 +Energy (eV)the functionalization of the Ti atom, the region changed to dark blue, indicating the deficiency of +electron density around the metal making it susceptible to bind with the guest molecules. +Moreover, region around the carbon atoms of PCP222 turns from red to green supporting the +charge transfer as discussed above. The estimated Hirshfeld charge on C and Ti atoms is computed +to be -0.121 e.u and +0.511 e.u, respectively, which makes the Ti atom nearly ionic, opening the +possibility for H2 adsorption. +3.2.2 Diffusion energy barrier calculation + +Figure 3: Ti diffusion energy barrier over the PCP222 + +According to earlier reports, the aggregation of transition metal atoms on the substrate may lower +the ability of the host material for hydrogen adsorption. So, before hydrogen adsorption on the +surface of PCP222, it is necessary to study the possibility of metal clustering on the substrate. If +the Ti atom is displaced from its stable adsorption position on PCP222 due to an increase in +temperature, there is a strong possibility of metal clustering. Since the Ti binding energy on +PCP222 (2.2 eV/Ti) is lower than the cohesive energy of an isolated single Ti atom (4.85 eV), we +evaluated whether or not there is an energy barrier for Ti atom diffusion on PCP222. The diffusion +energy barrier is calculated by displacing Ti to a finite neighbourhood (δr) over the adsorption site +of PCP222. as shown in Figure 3. The difference in energy calculated between the initial and that +of the close neighbourhood is then plotted with the diffusion coordinates as shown in Figure 3. +The figure illustrates the diffusion energy barrier to be 5.97 eV, which is sufficient to prevent the +diffusion of the Ti atom over PCP222 and therefore avoid Ti-Ti clustering which is also supported + +AE= 5.97 eV +6 +5 +4 +ev +4 +2 +1 +0 +2 +3 +5 +0 +1 +4 +Diffusion coordinatesby the works of Dewangan et al. [26] and Chakraborty et al. [25]. Therefore, the present Ti +functionalized PCP222 can be considered a suitable candidate for hydrogen adsorption. +3.3 Adsorption of H2 molecules on PCP222-Ti + + + + + +Figure 4: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-1H2, (b) +PCP222-Ti-2H2, (c) PCP222-Ti-3H2, (d) PCP222-Ti-4H2, (e) PCP222-Ti-5H2, (f) PCP222-Ti-6H2 + +Figure 5: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-2H, (b) +PCP222-Ti-2H-1H2, (c) PCP222-Ti-2H-2H2, (d) PCP222-Ti-2H-3H2, (e) PCP222-Ti-2H-4H2. + +(a) +(b) +(c) +d) +e +f)a +b +(c)Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) +and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen Hydrogen +(H-H) in Å. Average adsorption energy of H2 on PCP222-Ti. + +Name of complex +Bridge C-C +Rc -Ti +Ti-H +H-H +Eads (eV) +PCP222-Ti +1.542 +1.566 + + + +PCP222-Ti-2H +1.540 +1.800 +1.750 +2.796 +1.797 +PCP222-Ti-2H2 +1.540 +1.765 +1.770 +0.884 +0.953 +PCP222-Ti-3H2 +1.540 +1.798 +1.830 +0.852 +0.784 +PCP222-Ti-4H2 +1.540 +1.818 +1.905 +0.806 +0.672 +PCP222-Ti-5H2 +1.540 +1.842 +2.332 +0.816 +0.554 +PCP222-Ti-6H2 +1.540 +1.842 +2.633 +0.804 +0.467 + + + + + + +PCP222-Ti-2H-1H2 +1.540 +1.822 +1.926 +0.800 +0.480 +PCP222-Ti-2H-2H2 +1.540 +1.837 +1.868 +0.803 +0.474 +PCP222-Ti-2H-3H2 +1.540 +1.851 +1.899 +0.801 +0.406 +PCP222-Ti-2H-4H2 +1.540 +1.837 +2.840 +0.774 +0.256 + +To investigate the hydrogen adsorption on the surface of Ti functionalized PCP222, we added the +H2 molecules sequentially to PCP222-Ti. First, we added a single H2 molecule at about 2 Å above +the Ti atom functionalized on PCP222 and allowed the system to relax. It is observed that the +H2 molecule dissociates into two fragments of H atoms and forms chemical bond with the Ti atom. +The Ti-H bond length is found to be 1.75 Å which is close to the experimental result for titanium +monohydride [52]. The H-H bond distance is noted to be about 2.8 Å (Figure 4(a)). The binding +energy between Ti and H is calculated to be 1.79 eV which lies in the range of chemisorption +mechanized by Kubas’s interaction [2, 38]. Similar result was also reported by Ciraci et al. for the +adsorption of a single H2 molecule on Ti-decorated SWNT8 ( and SWBNT ) where the +H2 molecules dissociate into individual H atoms with a binding energy of 0.83 eV/H (0.93 eV/H) +and H-H- distance of 2.71Å (3.38 Å)[51, 53]. However, when two H2 molecules are +simultaneously added to the sorption center, the calculated average adsorption energy is reduced +to 0.95 eV/H2, with the average H-H bond length stretching from 0.74 Å to 0.8 Å. This result + +clearly indicates the adsorption process to be physisorptive. This is because of reduced interaction +strength between Ti atoms and H2 molecules caused due to screening effect. From the ESP analysis +(7) it is obvious that simultaneous presence of two H2 molecules reduces the charge densities of +Ti and H2 thereby inducing a weak charge polarization which causes the physisorption of hydrogen +on the surface of Ti functionalized PCP222. Another way of generating similar isomeric +configuration is chemisorption induced physisorption of H2 molecules on Ti functionalized +PCP222 in which one H2 molecule is adsorbed over n PCP222-Ti-2H (Figure 5(b)). Interestingly, +this configuration is 0.37 eV lower in energy than that of PCP222-Ti-2H2, and the H2 adsorbed +with lower adsorption energy (0.48 eV). Therefore, we proceed with both configurations for +further hydrogen adsorption. Sequential adsorption of H2 molecules on PCP222-Ti results in the +maximum adsorption up to 6H2 molecules. The adsorption of 3rd, 4th, 5th, and 6th H2 molecules +to PCP222-Ti reduces the average H2 adsorption energy to 0.784, 0.68, 0.554, and 0.467 eV/H2, +respectively. On the other hand, successive addition of H2 molecules to PCP222-Ti-2H leads to +maximum adsorption of four hydrogen molecules. More addition of H2 molecules beyond maxima +in both the cases causes them to fly away from the sorption center. It is observed that the average +adsorption energy decreases with an increase in the number of H2 molecules in the system which +is due to the steric hindrance among the adsorbed H2 crowed and the increase in distances between +the H2 and sorption centers. The estimated data of adsorption energy and geometrical parameters +of all the bare hydrogenated systems and presented in Table 1. +3.3.1 Partial density of states + + +Figure 6: Partial density of state on Ti and H atoms of (a) PCP222-Ti-2H, (b) PCP222-Ti-2H-1H2, (c) +PCP222-Ti-2H2, and (d) PCP222-Ti-6H2 + +The partial density of states (PDOS) of Ti and H atoms of the hydrogen adsorbed PCP222-Ti with +the chemisorbed, and physisorbed hydrogen is plotted in Figure 6. The adsorption of 1H2 to the +host resulting in chemisorption is contributed from the strong overlapping of H and Ti orbital near +-9 eV. Upon adsorption of another H2 molecule over PCP222-Ti-2H, the peaks of σ-orbital +(HOMO) of hydrogen and Ti orbital appears at around -15.7 eV below the Fermi level +and σ* (LUMO) of hydrogen interacts with the orbital of Ti and chemisorbed H above the Fermi +level (figure 6(b)) which can be explained by the Kubas mechanism in which a small charge +transfer occurs from the σ(HOMO) orbital of H2 to the vacant 3d orbital of the Ti atom, followed +by a back-donation of charges in the other direction from the partially filled 3d orbitals of Ti +to σ* (LUMO) of H2 molecules. When two H2 molecules are introduced simultaneously to the +PCP222-Ti, similar DOS peaks are observed, suggesting the H2 adsorption via the Kubas + +2.5 +Ti +(a) PCP222-Ti-2H +2.0 +H (chemisorbed) +1.5 +1.0 +0.5 - +V +0.0 +2 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +4 +- +2.4 +Ti +(b) PCP222-Ti-2H-1H2 +2.0 +1.6 +H (chemisorbed) +1.2 +H (physisorbed) +0.8 +0.4 +人 +PDOS +0.0 +1 +. +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +4 +2.5 +Ti +(c) PCP222-Ti-2H, +2.0 +H (physisorbed) +1.5 +1.0 - +人 +0.5 - +0.0 +-16 +-14 +-12 +-10 +-8 +-6 +- +-18 +-4 +-2 +2 +4 +2.5 +Ti +(d) PCP222-Ti-6H2 +2.0 +H (physisorbed) +1.5 +1.0- +0.5 +0.0 +1 +T2 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +4 +Energy (eV)mechanism. However, here the σ orbital of H2 splits into several peaks in the range of -15.2 to - +6.2 eV and moves closer to the Fermi level inferring lower in the interaction strength. On +adsorption of 6H2 molecules to Ti functionalized PCP222, the σ orbitals split into numerous peaks +in a broad range of -16.3 eV to -6.1 eV with enhanced intensity. This signifies that the adsorption +strength gets weaker with an increase in the quantity of H2 molecules in the host systems. +3.3.2 Electrostatics potential and Hirshfeld charges + +Figure 7: Electrostatics potential map of (a) PCP222-Ti, (b) PCP222-Ti-2H, (c) PCP222-Ti-2H2, (d) +PCP222-Ti-3H2, (e) PCP222-Ti-4H2, (f) PCP222-Ti-5H2, (f) PCP222-Ti-6H2. + +To obtain a qualitative depiction of electronic charge distribution over the bare and hydrogenated +PCP222-Ti, we generated and plotted the electrostatic potential (ESP) map on the total electron +density as shown in Figure 7 The charge distribution is used to determine the active adsorption +region for the guest hydrogen molecules. The dark blue zone above the Ti atom on PCP222-Ti +(Figure 7(a)) and the dark red region over the first adsorbed hydrogen atom indicates a strong +interaction between them leading to chemisorption of hydrogen atom. Upon adsorption of two +H2 molecules simultaneously, the region over Ti turns from dark blue to light blue, suggesting the +fact that, positive charge get transferred from the Ti atom to the adsorbed H2 and C atom of +PCCP222 thereby inducing charge polarization which causes physisorption of the second +H2 molecule. Further addition of H2 molecules to PCP222-Ti, the region over Ti atom turns to +bluish-green and then to green inferring further charge transfer (depletion of electron density near +Ti ) and the yellow region over the adsorbed H2 represents a little accumulation of electron density +at hydrogen molecules[26]. + +4.000e-2 ++ 4.000 e-2 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +Sideview +Topview +Figure 8: Hirshfeld charges before and after hydrogen adsorption on PCP222-Ti + +Figure 8 shows the average Hirshfeld charges on the Ti atom, the adsorbed H2 molecules, and the +C atoms of the benzene ring (Ti functionalized site) as a function of the number of H2 adsorbed on +the host. The average charges on the C atom of the benzene ring are initially computed to be - +0.031 e which then raises to -0.121 e with the functionalization of the Ti atom. The charge on the +Ti atom of PCP222-Ti is found to be +0.511 e, indicating the transfer of electronic charges from +the Ti atom to the C atom of the benzene ring. On chemisorption of the first hydrogen on PCP222- +Ti, the electronic charges on the Ti and H atoms are +0.41 a.u and -0.24 a.u implying a strong +attractive interaction between them as discussed above. Adding more H2 molecules gradually +lessen the Hirshfeld charges over the Ti and H atoms implying polarization induced weak +interaction between them. (Figure 8). +3.3.3 Bader’s topological analysis +The topological analysis at the bond critical point (BCP) is used to investigate the nature of +interactions between the Ti-functionalized PCP222 and the adsorbed H2 molecules employing +Bader’s quantum theory of atoms in molecules (QTAIM). The topological descriptors associated +with the electronic distribution, such as electron density (), Laplacian (2), and total energy +density (ℌ) (calculated as the sum of local kinetic G() potential energy density V() ), at BCPs + +0.8 +- Ring C before Ti decoration +0.7 + Ring C after Ti decoration +0.6 + Ti atom +0.5. +H atom +Hirshfeld Charges (eu) +0.4 +0.3 +0.2 +0.1 +0.0 +-0.2 +0.3 +-0.4 +-0.5 +0 +2 +5 +6 +Number of H, molecules, nare presented in Table S1. Kumar et al. reported that the positive value of the Laplacian of electron +density (2>0) at BCP indicates a decrease in  at the bonding region, suggesting an interaction +of closed-shell (non-covalent) type [56]. For PCP222-Ti-6H2, the value of  and 2 at BCP of Ti +and adsorbed H2 are found to be 0.057 a.u and 0.208 a.u, respectively which infers a closed-shell +interaction between Ti and H2. Moreover, the negative value of ℌBCP and − +G() +V() > 1 at BCP of +Ti and H2 confirm the closed-shell interaction among sorption center and H2 as proposed by +Koch et al. (Table S1) [57]. For C–C and C-Ti bond, the average  value shows very nominal +changes after the hydrogen adsorption which suggests the post-adsorption chemical stability of the +host material. Additionally, the average  on BCP of the H-H bond in PCP222-Ti-6H2 is 0.231 a.u +which is almost the same as on isolated bare H2 molecule (0.263 a.u). This implies that the +adsorbed hydrogens are in quasi-molecular form during the adsorption which also reflected in H- +H bond elongation by 0.06-0.14 Å. +3.4 Thermodynamically usable H2 capacity +3.4.1 Storage capacity + +Figure 9: Optimized geometry of hydrogen saturated 3Ti functionalized PCP222 + +To examine the maximum H2 gravimetric storage capacity of the system, we have +functionalized the Ti atom on each benzene ring of PCP222 resulting in the structure of +PCP222-3Ti as shown in Figure 9 and S3. Further, we added H2 molecules to each Ti + +atom functionalized on PCP222 sequentially as discussed in previous section (3.3). The +calculated average H2 adsorption energy and the change in geometrical parameters are +presented in Table 2. The adsorption of H2 on PCP222-3Ti is observed to behave similar +to that of on single Ti atom on PCP222. On saturation of the H2 uptake capacity of +PCP222-3Ti, each sorption center is found holding a maximum of 6H2 molecules with +a gravimetric storage capacity of 7.37 wt%. Since the first H2 molecule on each Ti atom +dissociate into two H atom and bonded strongly with Ti atoms, 1.31 wt% of hydrogen +adsorbed via the chemisorption process is difficult to desorb. However, the concurrent +addition of two or more H2 molecules to each Ti atom over PCP222, results in +physisorption kind of adsorption. Further, to confirm the stability of maximum +hydrogenated systems, the energy gap (Eg) (gap between HOMO-LUMO) and global +reactivity parameters such as η, χ, and ω were estimated using the Koopmans +theorem[58]. Notwithstanding, the studied system follow the “maximum hardness and +minimum electrophilicity principle,” ensuring their chemical stability (Figure S4)[59]. + + Table 2: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) +and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen-hydrogen +(H-H) in Å. Average adsorption energy and successive desorption energy of PCP222-3Ti- +nH2 (n=3,6,9,12,15,18) +Name of complex +Bridge C-C +Rc-Ti +Ti-H +H-H +Eads (eV) +Edes (eV) +PCP222_3Ti +1.543 +1.590 +- +- +- +- +PCP222_3Ti-3H2 +1.537 +1.799 +1.747 +2.824 +1.824 +1.824 +PCP222_3Ti-6H2 +1.537 +1.756 +1.776 +0.880 +0.988 +0.152 +PCP222_3Ti-9H2 +1.537 +1.790 +1.832 +0.849 +0.813 +0.464 +PCP222_3Ti-12H2 +1.536 +1.824 +1.801 +0.821 +0.700 +0.360 +PCP222_3Ti-15H2 +1.535 +1.825 +2.332 +0.806 +0.570 +0.050 +PCP222_3Ti-18H2 +1.536 +1.838 +2.622 +0.803 +0.482 +0.043 + + + + +Figure 10: Hydrogen occupation number for PCP222-3Ti at various T and P. + +For a practically usable hydrogen medium, a substantial amount of H2 molecules should be +adsorbed by the host material at attainable adsorption conditions and the adsorbed H2 molecules +should be desorbed effectively at a suitable temperature (T) and pressure (P). Thus, it is essential +to estimate the number of hydrogen molecules usable at a wide variety of T and P. We have +estimated the usable hydrogen gravimetric density of the studied system by calculating the number +of H2 molecules stored in PCP222-3Ti at different T and P using the empirical value of H2 gas +chemical potential (μ). The H2 gravimetric density is estimated from the occupation number (N) +by the following equation and plotted with various T and P in Figure 10[60]. +𝑁 = +∑ +𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] +𝑁𝑚𝑎𝑥 +𝑛=0 +∑ +𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] +𝑛𝑚𝑎𝑥 +𝑛=0 + + + + +(4) +Here Nmax is the maximum number of H2 molecules adsorbed on each Ti atom on +PCP222, n and gn represents the number of H2 molecules adsorbed and configurational +degeneracy for a n respectively. kB is the Boltzmann constant and -Eads (>0) indicates the average +adsorption energy of H2 molecules over PCP222-3Ti. μ is the empirical value of chemical potential +of H2 gas at specific T and P, obtained by using the following expression [61]. +𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln ( +𝑃 +𝑃0) + + (5) +Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar). + +7.380 +6.772 +7 +6.164 +6 +5.556 +4.948 +5- +4.340 +wt% +3.732 +4 +工 +3.124 +3- +2.516 +1.908 +2 +1.300 +50 +100 +150 +30 +e(bar) +200 +250 +Pressure +300 +400From the Figure 10 it is clear that, the PCP222-3Ti can store 18H2 molecules at temperatures up +to 80 K and 10-60 bar pressure. Up-to these thermodynamic conditions, the maximum H2 storage +capacity of the studied system is estimated as 7.37 wt%, which is consistent the experimentally +reported value for Pd functionalized carbon nanotubes [62] and is fairly above the target set by +US-DOE (5.5 wt% by 2025). On raising the temperature above 80 K, the H2 molecules start to +desorb from the PCP222-3Ti and retain >5.5 wt% of H2 till the temperature of 120 K under 30-60 +bar. Further, rise in temperature, the system maintains an H2 gravimetric density of 5 wt% (close +to the target of US-DOE) throughout a temperature range of 120-300 K and a pressure range of 3- +60. This thermodynamic condition may be treated as an ideal storage condition for H2 on PCP222- +3Ti. At the temperature of 400 K and pressure of 1-10 bar, the system retains 1.31 wt% of +hydrogen, that are adsorbed via the chemisorption process and may be desorbed at very high +temperatures. Thus, a total gravimetric density of 6.06 wt% (difference in G.D at 80 K and 400 K) +H2 molecules are usable under ambient conditions, which is fairly higher than the US-DOE target. +This result justifies that the Ti functionalization over PCP222 can be used as a potential reversible +hydrogen storage material. +3.5 Molecular dynamics simulations + +Figure 11: (a) Potential energy trajectories of hydrogenated PCP222-3Ti and (b) Time evolution +trajectory of average bond length between the Ti atom and C atoms of PCP222 at 300K and 500K. + + +3498.14 +300K +(Hartree) +-3498.16 +500K +-3498.18 +-3498.20 +Potentialenergy +3498.22 +3498.24 +3498.26 +-3498.28 +3498.30 +3498.32 +-3498.34 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +Time (fs) +2.8 +2.7 +C-Tidistance@300K +2.6 +C-Tidistance@500K +2.5 +2.4 +2.3 +2.2 +2.1 +2.0 +1.9 +1.8 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +Time (fs)We have performed molecular dynamic (MD) simulations using the atom-centered density matrix +propagation (ADMP) to check the desorption of hydrogen from the PCP222-3Ti-nH2and the +structural integrity of the host. During the simulations, the temperature was maintained by the +velocity scaling method, and the temperature was checked and corrected at every time step of 10 +fs. Figure 11(a) and S5, show the time variation potential energy trajectories and system snapshots, +respectively. The MD simulations at 300K and 1 ps reveal that 2H2 molecules from each Ti atom +fly away, and each Ti continues to hold three physisorbed H2 molecules and two chemisorbed +hydrogen atoms. When the temperature is elevated to 500 K, almost all the H2 molecules get +desorbed and each sorption center hold one physisorbed H2 and two chemisorbed H atoms. Since +the first physisorbed H2 is bound strongly with the host material, it may desorb at a higher +temperature and time scale. This indicates that the system PCP222-3Ti is not complete reversible +at normal temperatures and may show 100% desorption at a higher temperature. +For a practical hydrogen storage material, it is necessary that the host material must keep the +structural integrity above the average desorption temperature. To examine the structural integrity +of the host material (PCP222-3Ti), we carried out the MD simulations with the host material at +300 K and significantly above the room temperature (500 K) using ADMP. With a time step of 1 +fs, the ADMP-MD simulations are carried out for 1 ps. Figure 11(b) depicts the time variation +trajectory of the average distance between the Ti atom and the carbon atoms of PCP222 benzene +rings. We observe that the PCP222-3Ti maintains its structural stability at 500 K, and the distances +between the C-C and C-H bonds essentially remain unchanged. The time evolution trajectories of +the average distance between the Ti and C atom of PCP222 were noticed to oscillate about the +mean value (2.32 Å) with little variance. This illustrates that the host material’s structural stability +is maintained significantly above room temperature. In light of this, we believe that PCP222-3Ti +can be a viable option for hydrogen storage material. +4 Conclusion +In this study, we investigated the thermodynamical stability and hydrogen storage properties of +Ti-functionalized [2,2,2]paracyclophane, using the density functional theory. The Ti atoms are +strongly bonded to the PCP222 via Dewar mechanism, and no clustering of Ti atoms over PCP222 +was noticed. The first H2 molecule is chemisorbed with binding energy of 1.797 eV, while the + +remaining H2 molecules are physisorbed with an average H2 adsorption energy in the range of +0.467 - 0.953 eV/H2. On saturation with the H2, the Ti atom on PCP222 could adsorb up to +6H2 molecules, while the Ti-2H on PCP222 could adsorb up to 4H2. The average H-H bond +distance elongated by 0.06-0.14 Å during the adsorption process which implied that the adsorbed +H2 molecules were in quasi-molecular form and the fact is supported by the Hirshfeld charge +distribution analysis. . When three Ti atoms were functionalized on PCP222, the H2 gravimetric +capacity of the system was up to 7.37 wt%, which was fairly above the US-DOE requirements for +practical hydrogen applications. During saturation of H2 adsorption, the host material displayed no +significant change in geometry. The thermodynamic usable hydrogen capacity was found to be up +to 5 wt% throughout a temperature range of 120-300 K and a pressure range of 3-60 bar. At the +temperature of 400 K and pressure of 1-10 bar, the system retains 1.31 wt% of hydrogen which +could be desorbed at very high temperatures. 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Hydrogen storage in multi-walled +carbon nanotubes decorated with palladium nanoparticles using laser ablation/chemical +reduction methods. Materials Research Express, 2017, 4(9), 095030. + + diff --git a/89E2T4oBgHgl3EQflgfM/content/tmp_files/load_file.txt b/89E2T4oBgHgl3EQflgfM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d3335cbebd74d042d2027ac1119403128c672e1 --- /dev/null +++ b/89E2T4oBgHgl3EQflgfM/content/tmp_files/load_file.txt @@ -0,0 +1,1079 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf,len=1078 +page_content='Transition from chemisorption to physisorption of H2 on Ti functionalized [2,2,2]paracyclophane: A computational search for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Rakesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Sahoo, Sridhar Sahu* Computational Materials Research Lab, Department of Physics, Indian Institute of Technology (Indian School of Mines) Dhanbad, India Abstract In this work, we studied the hydrogen adsorption-desorption properties and storage capacities of Ti functionalized [2,2,2]paracyclophane (PCP222) using density functional theory and molecular dynamic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The Ti atom was bonded strongly with the benzene ring of PCP222 via Dewar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Subsequently, the calculation of the diffusion energy barrier revealed a significantly high energy barrier of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='97 eV preventing the Ti clustering over PCP222 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On adsorption of hydrogen, the first H2 molecule was chemisorbed over PCP222 with a binding energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='79 eV with the Ti metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further addition of H2 molecules, however, exhibited their physisorption over PCP222-Ti through the Kubas-type H2 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Charge transfer mechanism during the hydrogen adsorption was explored by the Hirshfeld charge analysis and electrostatic potential map, and the PDOS, Bader’s topological analysis revealed the nature of the interaction between Ti and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The PCP222 functionalized with three Ti atoms showed a maximum hydrogen uptake capacity of up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 wt%, which was fairly above the US-DOE criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The practical H2 storage estimation revealed that at ambient conditions, the gravimetric density of up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='06 wt% H2 molecules could be usable, and up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='31 wt% of adsorbed H2 molecules were retained with the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The ADMP molecular dynamics simulations assured the reversibility by desorption of adsorbed H2 and the structural integrity of the host material at sufficiently above the desorption temperature (300K and 500K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Therefore, the Ti-functionalized PCP222 can be considered as a thermodynamically viable and potentially reversible H2 storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Keywords: Hydrogen storage, DFT, ADMP, [2,2,2]paracyclophane, PCP222, ESP, Chemisorption, Physisorption 1 Introduction Extensive use of fossil fuels not only results in the depletion of those energy resources but also leads the world towards an alarming environmental catastrophe in terms of pollution and global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' These consequences have motivated researchers across the globe to search for alternative sustainable and environment-friendly energy resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Therefore, hydrogen drew the attention because it is considered as an ideal, pollution-free, and sustainable energy carrier, which can replace fossil fuels by fulfilling the energy need of the world, and thus can resolve the pollution due to fossil fuels[1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, the major difficulty in hydrogen energy as fuel for domestic and vehicular application is its efficient storage and delivery at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Hydrogen can be stored mainly in two ways: system-based and material-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' System-based storage methods which is being adopted by few industries require huge volume vessels which should be made of composite material to withstand high pressure (~70 MPa) making the process quite expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, compressed hydrogen storage systems are reported to have low volumetric densities, even at high pressure [3], and hydrogen storage in liquid state requires a very low temperature (~ 253oC) under high pressure (~ 250-350 atm) which is highly prone to safety concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On the other hand, the solid-state material-based hydrogen storage method is substantiated as efficient alternative to use hydrogen energy provided it adsorbs and desorb a desirable amount of H2 at ambient conditions [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In solid-state materials, hydrogen is usually adsorbed by the physisorption or chemisorption process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In the physisorption process, the adsorbed hydrogen binds in molecular to the surface of host materials through weak interaction (adsorption energy ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 eV/H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, in the chemisorption process, the H2 molecules dissociate into individual H atoms and migrate to the host materials by producing a strong chemical bond (with a binding energy of >1 eV/H2) with the host atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Another type of adsorption process observed is similar to the physisorption, in which the inter-atomic H-H bond in the H2 molecule is elongated but not dissociated and adsorbed by Kubas-type orbital interactions[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' It enhances the H2 adsorption energy and makes most of the H2 storage capacities that fulfil the target of the US department of energy (DOE-US) [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Since last few years, researchers are engaged extensively to study various materials, including carbon nanostructures [7, 8], metal hydrides [9, 10], graphene [11, 12], metal alloys[13, 14], metal-organic frameworks (MOF)[15, 16], and covalent-organic frameworks [17], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' for the reversible hydrogen storage at ambient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, it has been reported that these materials often have several limitations, including poor storage capacity, instability at significantly high temperatures, and low reversibility at normal temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For example, Mg-based metal hydrides showed a high storage capacity of up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='6 wt% under ambient condition, however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' it could be used only for 2-3 cycles [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='Similarly, metal alloys have very poor reversibility when used as hydrogen storage materials [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Using MOFs as H2 storage materials, researchers could attain up to 15 wt% of storage capacity at temperatures and pressures of 77 K and 80 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, under normal environmental condition its gravimetric and volumetric storage capacity remained very low [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To address the aforesaid issue and to develop commercially effective hydrogen storage materials, the experimentally synthesized organic compounds functionalized with transition metals (TMs), such as TM-doped organometallic buckyballs, TM-ethylene, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=', were introduced and investigated extensively [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Early reports show that the TM atoms form a strong bond with the π -electron delocalized compounds through the Dewar mechanism and adsorb hydrogen molecules via Kubas interaction[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For example, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' studied the hydrogen storage in Ti-doped Ψ-graphene and reported an H2 uptake capacity of up to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 wt% with an average adsorption energy of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='30 eV/H2 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Dewangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' predicted up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='52 wt% of H2 adsorption in Ti-functionalized holey graphyne via the Kubas mechanism with adsorption energy and desorption temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='38 eV/H2 and 486 K, respectively[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Numerous theoretical and experimental studies revealed that metal-adorned small organic molecules like CnHn could capture a large number of H2 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For example, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' estimated hydrogen uptake capacity up to 12 wt% in TiC2H4 with H2 binding energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='24 eV/H2[27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' High capacities of H2 storage in TMC2H4 (M = Ti, Sc, V, Ni, Ce, Nb) complexes was reported by Chaudhari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' At low benzene pressure (35 millitorrs) and ambient temperature, TiC6H6 was experimentally shown to absorb up to 6 wt% hydrogens [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' obtained an H2 uptake of up to 14 wt% and quick kinetics at room temperature on TiC6H6 by laser ablation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' however, the experiments did not discuss the desorption process[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Recently, Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' theoretically studied an interesting combination of chemisorption and physisorption in Ti-doped C6H6 and reported an uptake capacity of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='02 wt % with complete desorption at 935 K [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Mahamiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' revealed the H2 storage capacities of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='9 wt % in K and Ca decorated biphenylene with an average adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='24-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='33 eV [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Y atom doped zeolite showed high capacity adsorption of H2 with binding energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='35 eV/H2 and the desorption energy of 437K for fuel cells[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Macrocyclic compounds, like paracyclophane (PCP), a subgroup derivative of cyclophanes, comprises aromatic benzene rings with number of -CH2- moieties linking the subsequent benzene rings [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The PCPs are easier to synthesize in the laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' they can be functionalized with metal atoms due to the presence of aromatic benzene rings in the geometry, making them a feasible alternative for hydrogen storage prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For instance, Sathe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' studied the Sc and Li decorated PCP and reported the molecular H2 physisorbed via Kubas-Niu-Jena interaction resulting in up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 wt% H2 uptake capacity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The hydrogen storage transition metal (Sc, Y) functionalized [1,1]paracyclophane was investigated by Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' and reported a storage capacity of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='33-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='22 wt%, with an average adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='36 eV/H2 and desorption temperature of 412 K - 439 K[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The H2 storage on Li and Sc functionalized [4,4]paracyclophane shows an uptake capacity of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 wt% and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='7 wt%, as estimated by Sathe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' revealed the combination of physisorption and chemisorption of hydrogen on Sc and Ti functionalized BN-analogous [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2]PCP[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' They showed the first hydrogen molecule chemisorbed on the host material followed by physisorption of other H2, resulting in a storage of ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='9 wt% via Kubas interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Numerous other metal-decorated macrocyclic compounds have been explored as hydrogen storage possibilities, with storage capacities above the DOE requirement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' however, only a few have shown practical H2 capacity at varied thermodynamic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Though few PCP-based hydrogen storage systems are available in the literature, the [2,2,2]paracyclophane, which is experimentally synthesized by Tabushi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [42] is yet to be explored as a hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In the present work, we investigated the chemisorption and physisorption properties of hydrogen molecules on [2,2,2]paracyclophane (PCP222) functionalized with Ti atoms and estimated their hydrogen uptake capacity at varied thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In paracyclophane, there are many molecules in the group and are named after their pattern of arene substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The preceding square bracket number, “[2,2,2]” in [2,2,2]paracyclophane, indicates that the consecutive benzene rings (3 benzene rings) in paracyclophane are linked with two (-CH2-) moieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The linking bridges are relatively short;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' thus, the separation between consecutive benzene rings is small, which develops a strain in the aromatic rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This strain in the rings can be utilized for Ti functionalization over the aromatic benzene ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Due to the strain and metal functionalization, the aromatic benzene rings lose their inherent planarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' We choose to functionalize Ti metal atoms over the PCP222, as the d- block transition metal elements are well known for reversible hydrogen adsorption and could bind the H2 molecules via Kubas interaction[25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Though there are few reports available based on hydrogen storage in macrocyclic organic compounds and other Ti-doped nanostructures, our work is the first to investigate the efficiency of Ti-functionalized PCP222 using the atomistic MD simulation, practical storage capacity, and diffusion energy barrier estimation 2 Theory and Computation We have performed the theoretical calculations on [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2] paracyclophane (PCP222) and their hydrogenated structures within the framework of density functional theory (DFT)[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In the computation, the advanced hybrid ωB97Xd functional is used, and molecular orbitals (MO) are expressed as the linear combination of atom-centered basis function for which the valence diffuse and polarization function 6-311+G(d,p) basis set is used for all atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' ωB97Xd includes the long- range and Grimme’s D2 dispersion correction which is a range-separated version of Becke’s 97 functional[44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' It is important to note that the ωB97Xd technique is a trustworthy method for studying the non-covalent interactions, Organometallic complexes, and their thermochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To ensure the studied structures are in true ground state on the potential surface, the harmonic frequencies of all the systems are determined and are found to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' All the theoretical computations are performed with the computational program Gaussian 09[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In order to investigate the binding strength of titanium (Ti) atoms on the PCP222, we have calculated the average binding energy of decorated Ti atoms by using the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 𝐸𝑏 = 1 𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝑇𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖] (1) Where EPCP222, ETi, and EPCP222+mTi is the total energy of PCP222, Ti atom and Ti-decorated PCP222 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' m is the number of Ti atoms added PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The average adsorption energy of molecular hydrogen with metal atoms is calculated as[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 𝐸𝑎𝑑𝑠 = 1 𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖+𝑛𝐻2] (2) Where EPCP222+mTi, EH2, and EPCP222+mTi+nH2 is the total energy of host material, hydrogen molecule and hydrogen trapped complexes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' n is the number of H2 molecules adsorbed in each complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The global reactivity descriptors such as hardness (η), electronegativity (χ), and electrophilicity (ω) were estimated and used to study the stability and reactivity of Ti functionalized PCP222 and their hydrogen adsorbed derivatives [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is computed to assure the kinetic stability of the studied systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further, to understand the electronic charge transfer properties, the Hirshfeld charge and electrostatic potential map (ESP) were explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Moreover, partial density of states (PDOS) investigation was also carried out to further understand the process of hydrogen interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The topological parameters were studied using Bader’s theory of atoms in molecules (AIM) to analyze more about the nature of the interaction between metal on PCP222 and adsorbed hydrogen molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To obtained the hydrogen uptake capacity, gravimetric density (wt%) of hydrogen is calculated using the following equation[49]: 𝐻2(𝑤𝑡%) = 𝑀𝐻2 𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 (3) Here MH2represent the mass of the total number of H2 molecules adsorbed and MHost represent the mass of metal-doped PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 Structural properties of PCP222 The optimized geometrical structure of PCP222 is depicted in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' PCP222 has three benzene rings connected by two -CH2- moiety as a bridge between the consecutive rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The distance between the two consecutive -CH2- moiety and the -CH2- across the benzene ring are found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='54 Å and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='84 Å respectively, which is consistent with the earlier experimentally reported value by Cohen-Addad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To validate the π aromaticity of the optimized molecule, we computed the Nucleus Independent Chemical Shift (NICS) of PCP222 before functionalizing by any metal atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The NICS values are determined with 1 Å increment from the center to 3 Å above the three benzene rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' NICS(1) is found to be negative maximum (-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 ppm), suggesting the aromatic nature of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This indicates that the benzene rings of PCP222 are π electron-rich and can bind a metal atom outside the benzene rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 Functionalization of Ti atom on PCP222 Figure 1: (a) Optimized structure of PCP222 with all possible marked adsorption site marked, (b) Ti functionalized PCP222 Next, we explore different possible adsorption sites of pristine PCP222, such as C-C bridge of benzene ring (B1), CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the center of benzene (Rc) which are depicted in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To design the host material for hydrogen adsorption, a single Ti atom is positioned about 2 Å above at the regioselective sites of PCP222, and the resulting structure is re-optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The binding energy between Ti and PCP222 calculated using Equation 1 at different adsorption sites shows that the Ti atom is stable at two positions, B3 and Rc sites of PCP222 with binding energies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='20 eV, respectively which fairly agree with the previously reported value of Ti on CNT by Yildirim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Hence, the most favourable site for Ti atom functionalization is at the Rc site above the benzene ring of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 Bonding mechanism of Ti on PCP222 To understand the binding mechanism of Ti on PCP222, we analyzed the partial density of state (PDOS), electrostatic potential map (ESP), Hirshfeld charge, and Bader’s topological parameters of the Ti functionalized PCP222 system as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Density of states 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='84 B2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='543 (a) (b) Figure 2: Density of states plot on Ti and C atom on PCP222 The Ti atom is functionalized on PCP222 via the Dewar mechanism in which π-electron gets transferred from the highest occupied molecular orbitals (HOMO) of the substrates to the vacant d-orbital of Ti followed by the back-donation of charges from the partially filled d-orbital of Ti to empty π*-anti-bonding of the benzene ring of PCP222[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To understand the orbital interaction between the Ti and C atom of PCP222, we have performed the partial density of states (PDOS) calculation of PCP222-Ti and the result is plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Figure 2 clearly shows that the electronic states of the Ti atom and the C atom of PCP222 overlap below and above the Fermi level (E = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The transferred electrons partially fill the unoccupied states of PCP222, as seen by the intense peaks near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This infers an orbital interaction between Ti and C atom of PCP222 mediated by charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The fact is also obvious because Ti has the relatively lower ionization potential than the C atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' ESP and Hirshfeld charges To get a picture of electronic charge distribution over the PCP222 during Ti functionalization, we plotted the electrostatic potential (ESP) map over the total electron density, as shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The variation of electron density in the ESP map is shown by using different colour codes, which follows the pattern of accumulation and reduction of electron density as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' red (maximum electron density) >orange > yellow > green > blue (minimum electron density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In the ESP plot (Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='S1), the red region over the benzene ring of PCP222 implies the aggregation of electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' After 22 c PCP222-Ti 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Ti 18- Total 16- 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 12 - DOS 10- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 2 0 18 16 14 12 10 8 6 2 0 4 Energy (eV)the functionalization of the Ti atom, the region changed to dark blue, indicating the deficiency of electron density around the metal making it susceptible to bind with the guest molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Moreover, region around the carbon atoms of PCP222 turns from red to green supporting the charge transfer as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The estimated Hirshfeld charge on C and Ti atoms is computed to be -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='121 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='511 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u, respectively, which makes the Ti atom nearly ionic, opening the possibility for H2 adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 Diffusion energy barrier calculation Figure 3: Ti diffusion energy barrier over the PCP222 According to earlier reports, the aggregation of transition metal atoms on the substrate may lower the ability of the host material for hydrogen adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' So, before hydrogen adsorption on the surface of PCP222, it is necessary to study the possibility of metal clustering on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' If the Ti atom is displaced from its stable adsorption position on PCP222 due to an increase in temperature, there is a strong possibility of metal clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Since the Ti binding energy on PCP222 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 eV/Ti) is lower than the cohesive energy of an isolated single Ti atom (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='85 eV), we evaluated whether or not there is an energy barrier for Ti atom diffusion on PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The diffusion energy barrier is calculated by displacing Ti to a finite neighbourhood (δr) over the adsorption site of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The difference in energy calculated between the initial and that of the close neighbourhood is then plotted with the diffusion coordinates as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The figure illustrates the diffusion energy barrier to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='97 eV, which is sufficient to prevent the diffusion of the Ti atom over PCP222 and therefore avoid Ti-Ti clustering which is also supported AE= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='97 eV 6 5 4 ev 4 2 1 0 2 3 5 0 1 4 Diffusion coordinatesby the works of Dewangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [26] and Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Therefore, the present Ti functionalized PCP222 can be considered a suitable candidate for hydrogen adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 Adsorption of H2 molecules on PCP222-Ti Figure 4: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-1H2, (b) PCP222-Ti-2H2, (c) PCP222-Ti-3H2, (d) PCP222-Ti-4H2, (e) PCP222-Ti-5H2, (f) PCP222-Ti-6H2 Figure 5: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-2H, (b) PCP222-Ti-2H-1H2, (c) PCP222-Ti-2H-2H2, (d) PCP222-Ti-2H-3H2, (e) PCP222-Ti-2H-4H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' (a) (b) (c) d) e f)a b (c)Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen Hydrogen (H-H) in Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Average adsorption energy of H2 on PCP222-Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Name of complex Bridge C-C Rc -Ti Ti-H H-H Eads (eV) PCP222-Ti 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='566 PCP222-Ti-2H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='800 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='750 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='796 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='797 PCP222-Ti-2H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='765 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='953 PCP222-Ti-3H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='798 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='784 PCP222-Ti-4H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='818 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='672 PCP222-Ti-5H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='842 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='554 PCP222-Ti-6H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='842 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='467 PCP222-Ti-2H-1H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='822 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='480 PCP222-Ti-2H-2H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='837 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='474 PCP222-Ti-2H-3H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='851 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='406 PCP222-Ti-2H-4H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='837 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='256 To investigate the hydrogen adsorption on the surface of Ti functionalized PCP222, we added the H2 molecules sequentially to PCP222-Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' First, we added a single H2 molecule at about 2 Å above the Ti atom functionalized on PCP222 and allowed the system to relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' It is observed that the H2 molecule dissociates into two fragments of H atoms and forms chemical bond with the Ti atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The Ti-H bond length is found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='75 Å which is close to the experimental result for titanium monohydride [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The H-H bond distance is noted to be about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 Å (Figure 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The binding energy between Ti and H is calculated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='79 eV which lies in the range of chemisorption mechanized by Kubas’s interaction [2, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Similar result was also reported by Ciraci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' for the adsorption of a single H2 molecule on Ti-decorated SWNT8 ( and SWBNT ) where the H2 molecules dissociate into individual H atoms with a binding energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='83 eV/H (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='93 eV/H) and H-H- distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='71Å (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='38 Å)[51, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, when two H2 molecules are simultaneously added to the sorption center, the calculated average adsorption energy is reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='95 eV/H2, with the average H-H bond length stretching from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='74 Å to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This result clearly indicates the adsorption process to be physisorptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This is because of reduced interaction strength between Ti atoms and H2 molecules caused due to screening effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' From the ESP analysis (7) it is obvious that simultaneous presence of two H2 molecules reduces the charge densities of Ti and H2 thereby inducing a weak charge polarization which causes the physisorption of hydrogen on the surface of Ti functionalized PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Another way of generating similar isomeric configuration is chemisorption induced physisorption of H2 molecules on Ti functionalized PCP222 in which one H2 molecule is adsorbed over n PCP222-Ti-2H (Figure 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Interestingly, this configuration is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 eV lower in energy than that of PCP222-Ti-2H2, and the H2 adsorbed with lower adsorption energy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='48 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Therefore, we proceed with both configurations for further hydrogen adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Sequential adsorption of H2 molecules on PCP222-Ti results in the maximum adsorption up to 6H2 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The adsorption of 3rd, 4th, 5th, and 6th H2 molecules to PCP222-Ti reduces the average H2 adsorption energy to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='554, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='467 eV/H2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On the other hand, successive addition of H2 molecules to PCP222-Ti-2H leads to maximum adsorption of four hydrogen molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' More addition of H2 molecules beyond maxima in both the cases causes them to fly away from the sorption center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' It is observed that the average adsorption energy decreases with an increase in the number of H2 molecules in the system which is due to the steric hindrance among the adsorbed H2 crowed and the increase in distances between the H2 and sorption centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The estimated data of adsorption energy and geometrical parameters of all the bare hydrogenated systems and presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 Partial density of states Figure 6: Partial density of state on Ti and H atoms of (a) PCP222-Ti-2H, (b) PCP222-Ti-2H-1H2, (c) PCP222-Ti-2H2, and (d) PCP222-Ti-6H2 The partial density of states (PDOS) of Ti and H atoms of the hydrogen adsorbed PCP222-Ti with the chemisorbed, and physisorbed hydrogen is plotted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The adsorption of 1H2 to the host resulting in chemisorption is contributed from the strong overlapping of H and Ti orbital near 9 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Upon adsorption of another H2 molecule over PCP222-Ti-2H, the peaks of σ-orbital (HOMO) of hydrogen and Ti orbital appears at around -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='7 eV below the Fermi level and σ* (LUMO) of hydrogen interacts with the orbital of Ti and chemisorbed H above the Fermi level (figure 6(b)) which can be explained by the Kubas mechanism in which a small charge transfer occurs from the σ(HOMO) orbital of H2 to the vacant 3d orbital of the Ti atom, followed by a back-donation of charges in the other direction from the partially filled 3d orbitals of Ti to σ* (LUMO) of H2 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' When two H2 molecules are introduced simultaneously to the PCP222-Ti, similar DOS peaks are observed, suggesting the H2 adsorption via the Kubas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 Ti (a) PCP222-Ti-2H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 H (chemisorbed) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 - V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 2 18 16 14 12 10 8 6 4 2 0 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 Ti (b) PCP222-Ti-2H-1H2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='6 H (chemisorbed) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 H (physisorbed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 人 PDOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 18 16 14 12 10 8 6 4 2 0 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 Ti (c) PCP222-Ti-2H, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 H (physisorbed) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 - 人 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 16 14 12 10 8 6 18 4 2 2 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 Ti (d) PCP222-Ti-6H2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 H (physisorbed) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 1 T2 18 16 14 12 10 8 6 4 2 0 4 Energy (eV)mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, here the σ orbital of H2 splits into several peaks in the range of -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 to - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 eV and moves closer to the Fermi level inferring lower in the interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On adsorption of 6H2 molecules to Ti functionalized PCP222, the σ orbitals split into numerous peaks in a broad range of -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 eV to -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 eV with enhanced intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This signifies that the adsorption strength gets weaker with an increase in the quantity of H2 molecules in the host systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 Electrostatics potential and Hirshfeld charges Figure 7: Electrostatics potential map of (a) PCP222-Ti, (b) PCP222-Ti-2H, (c) PCP222-Ti-2H2, (d) PCP222-Ti-3H2, (e) PCP222-Ti-4H2, (f) PCP222-Ti-5H2, (f) PCP222-Ti-6H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To obtain a qualitative depiction of electronic charge distribution over the bare and hydrogenated PCP222-Ti, we generated and plotted the electrostatic potential (ESP) map on the total electron density as shown in Figure 7 The charge distribution is used to determine the active adsorption region for the guest hydrogen molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The dark blue zone above the Ti atom on PCP222-Ti (Figure 7(a)) and the dark red region over the first adsorbed hydrogen atom indicates a strong interaction between them leading to chemisorption of hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Upon adsorption of two H2 molecules simultaneously, the region over Ti turns from dark blue to light blue, suggesting the fact that, positive charge get transferred from the Ti atom to the adsorbed H2 and C atom of PCCP222 thereby inducing charge polarization which causes physisorption of the second H2 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further addition of H2 molecules to PCP222-Ti, the region over Ti atom turns to bluish-green and then to green inferring further charge transfer (depletion of electron density near Ti ) and the yellow region over the adsorbed H2 represents a little accumulation of electron density at hydrogen molecules[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='000e-2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='000 e-2 (a) (b) (c) (d) (e) (f) (g) Sideview Topview Figure 8: Hirshfeld charges before and after hydrogen adsorption on PCP222-Ti Figure 8 shows the average Hirshfeld charges on the Ti atom, the adsorbed H2 molecules, and the C atoms of the benzene ring (Ti functionalized site) as a function of the number of H2 adsorbed on the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The average charges on the C atom of the benzene ring are initially computed to be - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='031 e which then raises to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='121 e with the functionalization of the Ti atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The charge on the Ti atom of PCP222-Ti is found to be +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='511 e, indicating the transfer of electronic charges from the Ti atom to the C atom of the benzene ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On chemisorption of the first hydrogen on PCP222- Ti, the electronic charges on the Ti and H atoms are +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='41 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='24 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u implying a strong attractive interaction between them as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Adding more H2 molecules gradually lessen the Hirshfeld charges over the Ti and H atoms implying polarization induced weak interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 Bader’s topological analysis The topological analysis at the bond critical point (BCP) is used to investigate the nature of interactions between the Ti-functionalized PCP222 and the adsorbed H2 molecules employing Bader’s quantum theory of atoms in molecules (QTAIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The topological descriptors associated with the electronic distribution, such as electron density (\uf072), Laplacian (\uf0d12\uf072), and total energy density (ℌ) (calculated as the sum of local kinetic G(\uf072) potential energy density V(\uf072) ), at BCPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 Ring C before Ti decoration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='7 Ring C after Ti decoration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='6 Ti atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' H atom Hirshfeld Charges (eu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 0 2 5 6 Number of H, molecules, nare presented in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' reported that the positive value of the Laplacian of electron density (\uf0d12\uf072>0) at BCP indicates a decrease in \uf072 at the bonding region, suggesting an interaction of closed-shell (non-covalent) type [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For PCP222-Ti-6H2, the value of \uf072 and \uf0d12\uf072 at BCP of Ti and adsorbed H2 are found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='057 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='208 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u, respectively which infers a closed-shell interaction between Ti and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Moreover, the negative value of ℌBCP and − G(\uf072) V(\uf072) > 1 at BCP of Ti and H2 confirm the closed-shell interaction among sorption center and H2 as proposed by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' (Table S1) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For C–C and C-Ti bond, the average \uf072 value shows very nominal changes after the hydrogen adsorption which suggests the post-adsorption chemical stability of the host material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Additionally, the average \uf072 on BCP of the H-H bond in PCP222-Ti-6H2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='231 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u which is almost the same as on isolated bare H2 molecule (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='263 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This implies that the adsorbed hydrogens are in quasi-molecular form during the adsorption which also reflected in H- H bond elongation by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='06-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='14 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 Thermodynamically usable H2 capacity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 Storage capacity Figure 9: Optimized geometry of hydrogen saturated 3Ti functionalized PCP222 To examine the maximum H2 gravimetric storage capacity of the system, we have functionalized the Ti atom on each benzene ring of PCP222 resulting in the structure of PCP222-3Ti as shown in Figure 9 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further, we added H2 molecules to each Ti atom functionalized on PCP222 sequentially as discussed in previous section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The calculated average H2 adsorption energy and the change in geometrical parameters are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The adsorption of H2 on PCP222-3Ti is observed to behave similar to that of on single Ti atom on PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On saturation of the H2 uptake capacity of PCP222-3Ti, each sorption center is found holding a maximum of 6H2 molecules with a gravimetric storage capacity of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Since the first H2 molecule on each Ti atom dissociate into two H atom and bonded strongly with Ti atoms, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='31 wt% of hydrogen adsorbed via the chemisorption process is difficult to desorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' However, the concurrent addition of two or more H2 molecules to each Ti atom over PCP222, results in physisorption kind of adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further, to confirm the stability of maximum hydrogenated systems, the energy gap (Eg) (gap between HOMO-LUMO) and global reactivity parameters such as η, χ, and ω were estimated using the Koopmans theorem[58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Notwithstanding, the studied system follow the “maximum hardness and minimum electrophilicity principle,” ensuring their chemical stability (Figure S4)[59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Table 2: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen-hydrogen (H-H) in Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Average adsorption energy and successive desorption energy of PCP222-3Ti- nH2 (n=3,6,9,12,15,18) Name of complex Bridge C-C Rc-Ti Ti-H H-H Eads (eV) Edes (eV) PCP222_3Ti 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='543 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='590 PCP222_3Ti-3H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='537 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='799 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='747 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='824 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='824 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='824 PCP222_3Ti-6H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='537 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='152 PCP222_3Ti-9H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='537 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='790 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='832 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='464 PCP222_3Ti-12H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='536 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='824 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='360 PCP222_3Ti-15H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='535 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='825 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='050 PCP222_3Ti-18H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='536 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='838 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='043 Figure 10: Hydrogen occupation number for PCP222-3Ti at various T and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For a practically usable hydrogen medium, a substantial amount of H2 molecules should be adsorbed by the host material at attainable adsorption conditions and the adsorbed H2 molecules should be desorbed effectively at a suitable temperature (T) and pressure (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Thus, it is essential to estimate the number of hydrogen molecules usable at a wide variety of T and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' We have estimated the usable hydrogen gravimetric density of the studied system by calculating the number of H2 molecules stored in PCP222-3Ti at different T and P using the empirical value of H2 gas chemical potential (μ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The H2 gravimetric density is estimated from the occupation number (N) by the following equation and plotted with various T and P in Figure 10[60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 𝑁 = ∑ 𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] 𝑁𝑚𝑎𝑥 𝑛=0 ∑ 𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] 𝑛𝑚𝑎𝑥 𝑛=0 (4) Here Nmax is the maximum number of H2 molecules adsorbed on each Ti atom on PCP222, n and gn represents the number of H2 molecules adsorbed and configurational degeneracy for a n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' kB is the Boltzmann constant and -Eads (>0) indicates the average adsorption energy of H2 molecules over PCP222-3Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' μ is the empirical value of chemical potential of H2 gas at specific T and P, obtained by using the following expression [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln ( 𝑃 𝑃0) (5) Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='380 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='772 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='164 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='556 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='948 5- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='340 wt% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='732 4 工 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='124 3- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='516 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='908 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='300 50 100 150 30 e(bar) 200 250 Pressure 300 400From the Figure 10 it is clear that, the PCP222-3Ti can store 18H2 molecules at temperatures up to 80 K and 10-60 bar pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Up-to these thermodynamic conditions, the maximum H2 storage capacity of the studied system is estimated as 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 wt%, which is consistent the experimentally reported value for Pd functionalized carbon nanotubes [62] and is fairly above the target set by US-DOE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 wt% by 2025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On raising the temperature above 80 K, the H2 molecules start to desorb from the PCP222-3Ti and retain >5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 wt% of H2 till the temperature of 120 K under 30-60 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Further, rise in temperature, the system maintains an H2 gravimetric density of 5 wt% (close to the target of US-DOE) throughout a temperature range of 120-300 K and a pressure range of 3- 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This thermodynamic condition may be treated as an ideal storage condition for H2 on PCP222- 3Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' At the temperature of 400 K and pressure of 1-10 bar, the system retains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='31 wt% of hydrogen, that are adsorbed via the chemisorption process and may be desorbed at very high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Thus, a total gravimetric density of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='06 wt% (difference in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='D at 80 K and 400 K) H2 molecules are usable under ambient conditions, which is fairly higher than the US-DOE target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This result justifies that the Ti functionalization over PCP222 can be used as a potential reversible hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 Molecular dynamics simulations Figure 11: (a) Potential energy trajectories of hydrogenated PCP222-3Ti and (b) Time evolution trajectory of average bond length between the Ti atom and C atoms of PCP222 at 300K and 500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='14 300K (Hartree) 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='16 500K 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='18 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='20 Potentialenergy 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='22 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='24 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='26 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='28 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='30 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='32 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='34 0 100 200 300 400 500 600 700 800 900 1000 Time (fs) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='7 C-Tidistance@300K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='6 C-Tidistance@500K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='8 100 200 300 400 500 600 700 800 900 1000 Time (fs)We have performed molecular dynamic (MD) simulations using the atom-centered density matrix propagation (ADMP) to check the desorption of hydrogen from the PCP222-3Ti-nH2and the structural integrity of the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' During the simulations, the temperature was maintained by the velocity scaling method, and the temperature was checked and corrected at every time step of 10 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Figure 11(a) and S5, show the time variation potential energy trajectories and system snapshots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The MD simulations at 300K and 1 ps reveal that 2H2 molecules from each Ti atom fly away, and each Ti continues to hold three physisorbed H2 molecules and two chemisorbed hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' When the temperature is elevated to 500 K, almost all the H2 molecules get desorbed and each sorption center hold one physisorbed H2 and two chemisorbed H atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Since the first physisorbed H2 is bound strongly with the host material, it may desorb at a higher temperature and time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This indicates that the system PCP222-3Ti is not complete reversible at normal temperatures and may show 100% desorption at a higher temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' For a practical hydrogen storage material, it is necessary that the host material must keep the structural integrity above the average desorption temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' To examine the structural integrity of the host material (PCP222-3Ti), we carried out the MD simulations with the host material at 300 K and significantly above the room temperature (500 K) using ADMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' With a time step of 1 fs, the ADMP-MD simulations are carried out for 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Figure 11(b) depicts the time variation trajectory of the average distance between the Ti atom and the carbon atoms of PCP222 benzene rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' We observe that the PCP222-3Ti maintains its structural stability at 500 K, and the distances between the C-C and C-H bonds essentially remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The time evolution trajectories of the average distance between the Ti and C atom of PCP222 were noticed to oscillate about the mean value (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='32 Å) with little variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' This illustrates that the host material’s structural stability is maintained significantly above room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' In light of this, we believe that PCP222-3Ti can be a viable option for hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' 4 Conclusion In this study, we investigated the thermodynamical stability and hydrogen storage properties of Ti-functionalized [2,2,2]paracyclophane, using the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The Ti atoms are strongly bonded to the PCP222 via Dewar mechanism, and no clustering of Ti atoms over PCP222 was noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The first H2 molecule is chemisorbed with binding energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='797 eV, while the remaining H2 molecules are physisorbed with an average H2 adsorption energy in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='467 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='953 eV/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' On saturation with the H2, the Ti atom on PCP222 could adsorb up to 6H2 molecules, while the Ti-2H on PCP222 could adsorb up to 4H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The average H-H bond distance elongated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='06-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='14 Å during the adsorption process which implied that the adsorbed H2 molecules were in quasi-molecular form and the fact is supported by the Hirshfeld charge distribution analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' When three Ti atoms were functionalized on PCP222, the H2 gravimetric capacity of the system was up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='37 wt%, which was fairly above the US-DOE requirements for practical hydrogen applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' During saturation of H2 adsorption, the host material displayed no significant change in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' The thermodynamic usable hydrogen capacity was found to be up to 5 wt% throughout a temperature range of 120-300 K and a pressure range of 3-60 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' At the temperature of 400 K and pressure of 1-10 bar, the system retains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='31 wt% of hydrogen which could be desorbed at very high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' A total gravimetric density of up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='06 wt% H2 molecules are usable under ambient conditions which is fairly higher than the US-DOE target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' MD simulations at 500 K revealed the structural integrity and reversibility of the host and also showed that chemisorbed hydrogens are retained at this temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Since, there is no experimental works reported on Ti-functionalized PCP222 for hydrogen storage, we hope our computational work will contribute significantly to the research of hydrogen storage in macrocyclic compounds and provide supporting reference for the future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' References [1] Sachin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Shet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Shanmuga Priya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Sudhakar, Muhammad 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content='096402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' [62] Mehrabi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=', Parvin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=', Reyhani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=', Mortazavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Hydrogen storage in multi-walled carbon nanotubes decorated with palladium nanoparticles using laser ablation/chemical reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} +page_content=' Materials Research Express, 2017, 4(9), 095030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQflgfM/content/2301.03990v1.pdf'} diff --git a/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf b/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ef16137d3f88379f08c067a756bb6c13546247e5 --- /dev/null +++ b/9dAzT4oBgHgl3EQfgvzi/content/2301.01475v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:320b918c8554efa43ccea233a6295c757ecbf630db285825c5e2d4a1880de31b +size 1201031 diff --git a/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/2301.01713v1.pdf.txt b/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/2301.01713v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a90c92d887701005dc683ad04dcbda7e37a2d0e --- /dev/null +++ b/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/2301.01713v1.pdf.txt @@ -0,0 +1,609 @@ +Robust Surface Reconstruction from +Orthogonal Slices + +Radek Sviták1, Václav Skala2 +Department of Computer Science and Engineering, +University of West Bohemia in Pilsen, Univerzitní 8, 306 14 Plzeň, Czech Republic +E-mail: rsvitak@kiv.zcu.cz + +Abstract +The surface reconstruction problem from sets of planar parallel slices representing cross sections +through 3D objects is presented. The final result of surface reconstruction is always based on the correct +estimation of the structure of the original object. This paper is a case study of the problem of the structure +determination. We present a new approach, which is based on considering mutually orthogonal sets of slices. +A new method for surface reconstruction from orthogonal slices is described and the benefit of orthogonal +slices is discussed too. The properties and sample results are presented as well. + + + +This work is was supported by the Ministry of Education of the Czech Republic – projects: +1FRVŠ 1348/2004/G1 +2MSM 235200005 +1. Introduction + +The crucial task of the surface reconstruction +from slices is a correct estimation of the +original object structure, i.e. the solution of the +contour correspondence problem. Most of the +existing methods simply consider the overlap +of contours in a pair of consecutive parallel +slices as the only correspondence criterion. +Therefore, they produce unacceptable structure +estimation when the angle between the axis of +the object and the normal of the slices +increases. +Higher density of slices can help to +solve this problem, but it is not always +possible because of the resolution limit of the +scanning device, etc. It is obvious that other +slices in non-parallel planes offer an additional +information. In this paper we will concentrate +on the benefit of orthogonal slices for the +reconstruction process. In comparison to the +existing methods, our currently achieved +results show, that for a set of objects the +resultant surface is significantly more accurate +with respect to the similarity to the original +surface. +The concept of the new proposed +method +is +presented +and +results +of +comparisons with the existing methods are +discussed as well. + + +A +C +B + +D +Figure 1: Problematic cases when solving the contour +correspondence problem. Expected problems using the +overlapping +criterion: +A, B, C, D; +generalized +cylinders: B, D; +MST: C, D; +Reeb graph +based +methods: D. +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + +2. Brief survey of existing methods + + +Several methods for surface reconstruction +from slices have been developed since about +1970. In this section we will classify them +according to their approach to solving the +contour correspondence problem. For more +extensive study of the existing methods from +the other viewpoints, see [2, 4, 6, 7]. +The simplest methods estimate the +contour correspondence locally between each +consecutive pair of contours. Typically, +contours +that +overlap +each +other +are +considered as correspondent. This works if the +density of slice is high, i.e. the distance +between slices is low, and the axis of the input +object is nearly perpendicular to the slices +planes. +A +more +advanced +method +uses +generalized elliptical cylinder to solve the +correspondence problem [1, 11, 12]. Contours +are first classified as elliptical or complex by +determining how well the vertices of their +perimeter can be fit by an ellipse. If the fit is +too poor, a contour is classified as complex, +and can not be incorporated into an elliptical +cylinder. Then the ellipses are grouped to the +cylinders. When as many contours as possible +have been organized into cylinders, then the +algorithm uses the geometric relationship +between cylinders to group them into objects. +This method is most useful for elongated +smooth objects with roughly elliptical cross +section. +Apparently +the +best +existing +approaches that have been published are two +graph-based methods. The first of them +presented by Skinner [10] computes a +minimum spanning tree based on contour +shape and position. In the first step a graph is +constructed by representing each contour as a +node and connecting each node to all nodes +representing contours in adjacent sections. The +best fitting ellipse is computed for each +contour. The cost of an edge of the graph +relies on the mutual position and size of two +ellipses: + +( ) +2 +2 +2 +2 +) +b +(b +) +a +(a +) +y +(y +) +x +(x +i,j +c +j +i +j +i +j +i +j +i +− ++ +− ++ +− ++ +− += +, + +where (xi, yi, zi), (xj, yj, zj) represent the +centers of the ellipses of contours i, j, +respectively, and ai, bi, aj, bj are their major +and minor axis lengths. +The minimum spanning tree computed +for the graph represents the solution to the +correspondence problem. The method works +well for naturally tree-structured objects, the +main limitation is its inability to solve the +correspondence problem correctly for general +graph topologies, e.g. genus > 0. + + +A + + + + + +B +Figure 2: A) Data set of slices of the cochlea. Using the +Reeb graph it is possible to detect and represent the +right contour correspondence. The advantage consists in +the possibility of considering the correspondence +among contours of one slice (B). Taken from [8]. + +The second graph based method +presented by Shinagawa [8, 9] uses surface +coding based on Morse Theory to construct a +Reeb graph [14] representing the contour +connectivity. Each contour represents a node +in the graph, edges of the graph represent the +contour correspondence relation. Edges are +added to the graph in the manner to avoid +making connections that would result in a +surface that is not a 2-manifold. For each pair +of contours that can be legally connected, a +weight function is evaluated, and its value is +used to establish a priority for connecting that +pair of contours. The algorithm proceeds by +making the highest priority connections in +regions where the number of contours in each +section does not change, and then adds +connections in order of decreasing priority +with respect to the a priori knowledge of the +number of connected components and the +topological genus. +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + +2 +0 +3It is necessary to note that all the +existing solutions just estimate the contours +correspondence, i.e. the structure of the +original object, should be emphasized. In +Figure 1 there are some typical example data +sets to illustrate capabilities of the approaches +mentioned in this section. + +3. Orthogonal slices + +One set of parallel planar slices is one of the +well-known boundary representations of a 3D +object. Usually the planes of such slices set are +perpendicular to the z-axis, and thus called z- +slices. +If we slice an object by more then one +set of parallel slices and moreover when these +sets +are +mutually +orthogonal, +we +get +orthogonal sets of slices. Consider now that +we have z-slices, x-slices and y-slices of an +object, see Figure 3. Note, that the contours +are supposed to be polygonal, oriented the +way that when looking from the positive +direction of the given slices set axis, the +contours have the interior on its left side and +the exterior on the right side, see Figure 4. + +3.1 Contour correspondence + +The main advantage of orthogonal slices +consists in the approach how the contour +correspondence can be determined. It is +important to emphasize that two orthogonal +contours which intersect each other comes +aparently from one and the same surface +component of the input object. It means that +the intersection of contours is very important +since it provides accurate information about +the correct structure, see Figure 5. +It is obvious that if the slices in the +orthogonal sets sample the object sufficiently, +then the intersections of contours from the +orthogonal slices identify the correspondence +relation accurately, i.e. the correct structure of +the original object. + +4. The algorithm + +The planes of slices divide space into a set of +spatial cells of a spatial grid M. In Figure 3 +can be seen three mutually orthogonal planes +of grid M. We distinguish two kinds of cells of +M, the surface-crossing and the surface- +passing cells. There are parts of contours on +some sides of a surface-crossing cell, which +means that the resultant surface intersects the +cell, see Figure 6. + + +Figure 3: An example of three orthogonal slices sets. + + +Figure 4: Correct contour orientation. + + +Figure 5: The mutual crossings of orthogonal contours +define the correspondence relation. + + +Figure 6: A surface-crossing cell. Parts of contours on +the sides of the cell together with node points form +spatial polygons. Node points are denoted as white +circles. Each edge of G is adjacent with two cells of M. +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + +The intersection of two orthogonal +slices consisting of curvilinear contours is a set +of points and we call them node points, see +Figure 7. Now we focus on surface-crossing +cell. An important observation is that parts of +input contours and the node points form +spatial polygons. Each such polygon is +enclosed in a surface-crossing cell, its patch is +part of the resultant surface, see Figure 6. + +4.1 The correspondence problem + +At this moment we suppose that the +correspondence of contours is identified +sufficiently by the intersections of orthogonal +contours as it has been discussed in +section 3.1. +Consider the intersection of two +contours as the relation of correspondence. +Note that the number of components of a +graph +constructed +of +such +a +relation +corresponds to the number of disjoint +components of the resultant surface. + +4.2 Node points computation + +A node point is geometrically the intersection +of two contours. Topologically it is the +representation of a contour correspondence +relation. It holds that each node point must lie +on the edge of the grid M. Since the contours +are supposed as polygonal curves, we cannot +compute the intersections of two orthogonal +sections directly. We obtain them in two +phases. + + +A +B +Figure 7: A) An input contour, the lattice represents +positions of orthogonal slices planes. B) The contour +formed by its node points (black spots). + +In the first step intersections of each +contour and the grid M are computed. These +intersections are added among the current +contour vertices on the appropriate position. +They are registered on the corresponding edge +of the grid M simultaneously. Our algorithm +works on the same principle as the Cohen- +Sutherland’s line clipping algorithm [3]. +An intersection of a slice plane and all +other orthogonal slice planes forms a lattice + 1 + 10 + 100 +1 000 +10 000 +100 000 +2 +3 +4 +5 +6 +7 +8 +9 +10 +frequency of +occurence + +Figure 8: Polygon size (number of edges) histogram. +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + +with cells, see Figure 7. Each node point arises +as the intersection of a contour and a side of a +cell. Since the contour is supposed to be +polygonal, a node point is simply computed as +an intersection of two segments. Singular +cases when a contour crosses a cell at its +corner are handled separately [13]. +In the second step the node point +construction is completed. The correspondent +vertex, which is a member of the orthogonal +contour and also a member of the same edge +of M, must be found. As it was said before it is +done very fast searching the auxiliary +registrations of contour intersections on the +appropriate edge of M. Each two nearest +intersections coming from orthogonal contours +registered on an edge of M are qualified as +correspondent vertices building together a +node point. + +4.3. Constructing the surface + +Now suppose graph G, whose set of vertices +consists of a set of the node points and whose +edges represent the parts of contours between +two node vertices. Note that the geometrical +shape of the edges still corresponds to the +appropriate parts of contours. Now the task is +to find such cycles of graph G, which have the +property that their geometrical representation +lies within one cell of M. Those cycles +represents spatial polygons that lie on the +surface. +We suppose each edge e of G is +adjacent with cells B1 and B2, see Figure 6. +Each cell from {B1, B2} includes one cycle c +of our interest, which is adjacent with e (that +results from the consideration of 2-manifold +objects). The circle c represents the spatial +polygon being searched. Thus for each e two +cycles +e +B +c +1 , +e +B +c +2 must be searched and then +polygons +1cp , +2 +cp correspondent to those +cycles are constructed. +As soon as all polygons are obtained, +we can start to patch them. We can use any +arbitrary patching technique. Note that the +number of sides of such polygon can be high, +but in cases of our data sets it is in range 2 – +10, see the graph in Figure 8. +The proposed method starts with +finding a suitable point in the center of each +polygon. Then using the center point each +polygon is divided into set of quadrilaterals, +which are easier to patch, see Figure 9. + + +Figure 9: Partition of a generic polygon in the set of +quadrilaterals. +Requires +the +central +point +C +determination. + + +A + +B + +C + +D +Figure 10: Results of the surface reconstruction. A) An +input data set (courtesy of Martin Čermák), B) VTK +surface reconstruction from slices class, C) A common +volume based method, D) Proposed method for surface +reconstruction from orthogonal slices. +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + +5. Results + +All the problematic data sets mentioned in +section 1 and many more have been processed +using: +- +surface reconstructing from slices class +from VTK, +- +a common volume based method; see +[5] for more details, +- +our proposed method for surface +reconstruction from orthogonal slices. +The results of the reconstruction of one data +set are illustrated in Figure 10, the complete +documentation and experimental results can be +found at http://herakles.zcu.cz/research/slices. + +6. Conclusion and further research + +Our +current +research +proves +that +the +advantages of orthogonal slices in the process +of surface reconstruction are significant. There +is a set of objects for which the orthogonal +slices are almost the only way to reconstruct +them correctly. +The proposed method supposes that the +object is sampled well enough, so that the +number of components of the correspondence +graph G equals to the number of disjoint +components of the original surface. + +The main point of our further research +is the solution of problems caused by under- +sampling, i.e. to deal with data sets that do not +sample +the +input +object +sufficiently. +Furthermore we would like to study the +influence of contour inaccuracy on the node +point computation. + + +References + +[1] Bresler, Y., Fessler, J.A., Macovski, A.: A +Bayesian approach to reconstruction from +incomplete projections of a multiple object 3D +domain. IEEE Trans. Pat. Anal. Mach. Intell., +11(8):840-858, August 1989. +[2] Cong, G., Parvin, B.: Robust and efficient +surface reconstruction from contours. The +Visual Computer, (17):199-208, 2001 +[3] Foley, J. D., van Dam, A., Feiner, S. K. +and Hughes, J. F., Computer Graphics: +Principles and Practice, Addison-Wesley, +1990. +[4] Jones, M., Chen, M.: A new approach to +the construction of surfaces from contour data. +Computer Graphics Forum (13): 75-84, 1994 +[5] Klein, R., Schilling, A.: Fast Distance +Interpolation for Reconstruction of Surfaces +from +Contours. +In +proceedings +of +Eurographics '99, Short Papers and Demos, +September 1999. +[6] Meyers, D.: Multiresolution tiling. In +Proceedings, Graphics Interface '94, pages 25- +32, Banff, Alberta, May 1994. +[7] Meyers, D.: Reconstruction of Surfaces +From Planar Contours. PhD thesis, University +of Washington, 1994. +[8] Shinagawa, Y., Kunii, T.L.: Constructing a +Reeb graph automatically from cross sections. +IEEE Comuter Graphics and Applications, +11(6): 44-51, November 1991. +[9] Shinagawa, Y., Kunii, T.L., Kergosien, +Y.L.: Surface coding based on Morse theory. +IEEE Comuter Graphics and Applications, +11(5): 66-78, September 1991. +[10] Skinner, +S.M.: +The +correspondence +problem: Reconstruction of objects from +contours in parallel sections. Master’s thesis, +Department +of +Computer +Science +and +Engineering, University of Washington, 1991. +[11] Soroka, B.I.: Understanding Objects From +Slices: +Extracting +Generalised +Cylinder +Descriptions From Serial Sections. PhD thesis, +University of Kansas Dept of Computer +Science, March 1979. TR-79-1. +[12] Soroka, B.I.: Generalized cones from +serial sections. Computer Graphics and Image +Processing, (15): 54-166, 1981. +[13] Svitak, +R., +Skala, +V.: +Surface +Reconstruction +from +Orthogonal +Slices, +ICCVG 2002, Zakopane, Poland, 2002 +[14] Wood, Z. J.: Computational Topology +Algorithms +For +Discrete +2-Manifolds. +California Institute of Techology, PhD Thesis, +May 2003 + +Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004 + diff --git a/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/load_file.txt b/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a63eae0d51b30337992d57527b8930a0cbb3f2b --- /dev/null +++ b/AdAzT4oBgHgl3EQfvv7L/content/tmp_files/load_file.txt @@ -0,0 +1,246 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf,len=245 +page_content='Robust Surface Reconstruction from Orthogonal Slices Radek Sviták1, Václav Skala2 Department of Computer Science and Engineering, University of West Bohemia in Pilsen, Univerzitní 8, 306 14 Plzeň, Czech Republic E-mail: rsvitak@kiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='cz Abstract The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The final result of surface reconstruction is always based on the correct estimation of the structure of the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' This paper is a case study of the problem of the structure determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' We present a new approach, which is based on considering mutually orthogonal sets of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A new method for surface reconstruction from orthogonal slices is described and the benefit of orthogonal slices is discussed too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The properties and sample results are presented as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' This work is was supported by the Ministry of Education of the Czech Republic – projects: 1FRVŠ 1348/2004/G1 2MSM 235200005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Introduction The crucial task of the surface reconstruction from slices is a correct estimation of the original object structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' the solution of the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Most of the existing methods simply consider the overlap of contours in a pair of consecutive parallel slices as the only correspondence criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Therefore, they produce unacceptable structure estimation when the angle between the axis of the object and the normal of the slices increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Higher density of slices can help to solve this problem, but it is not always possible because of the resolution limit of the scanning device, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' It is obvious that other slices in non-parallel planes offer an additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In this paper we will concentrate on the benefit of orthogonal slices for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In comparison to the existing methods, our currently achieved results show, that for a set of objects the resultant surface is significantly more accurate with respect to the similarity to the original surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The concept of the new proposed method is presented and results of comparisons with the existing methods are discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A C B D Figure 1: Problematic cases when solving the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Expected problems using the overlapping criterion: A, B, C, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' generalized cylinders: B, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' MST: C, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Reeb graph based methods: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='221-233, ISSN 1230-0535, 2004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Brief survey of existing methods Several methods for surface reconstruction from slices have been developed since about 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In this section we will classify them according to their approach to solving the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' For more extensive study of the existing methods from the other viewpoints, see [2, 4, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The simplest methods estimate the contour correspondence locally between each consecutive pair of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Typically, contours that overlap each other are considered as correspondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' This works if the density of slice is high, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' the distance between slices is low, and the axis of the input object is nearly perpendicular to the slices planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A more advanced method uses generalized elliptical cylinder to solve the correspondence problem [1, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Contours are first classified as elliptical or complex by determining how well the vertices of their perimeter can be fit by an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' If the fit is too poor, a contour is classified as complex, and can not be incorporated into an elliptical cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Then the ellipses are grouped to the cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' When as many contours as possible have been organized into cylinders, then the algorithm uses the geometric relationship between cylinders to group them into objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' This method is most useful for elongated smooth objects with roughly elliptical cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Apparently the best existing approaches that have been published are two graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The first of them presented by Skinner [10] computes a minimum spanning tree based on contour shape and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In the first step a graph is constructed by representing each contour as a node and connecting each node to all nodes representing contours in adjacent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The best fitting ellipse is computed for each contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The cost of an edge of the graph relies on the mutual position and size of two ellipses: ( ) 2 2 2 2 ) b (b ) a (a ) y (y ) x (x i,j c j i j i j i j i − + − + − + − = , where (xi, yi, zi), (xj, yj, zj) represent the centers of the ellipses of contours i, j, respectively, and ai, bi, aj, bj are their major and minor axis lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The minimum spanning tree computed for the graph represents the solution to the correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The method works well for naturally tree-structured objects, the main limitation is its inability to solve the correspondence problem correctly for general graph topologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' genus > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A B Figure 2: A) Data set of slices of the cochlea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Using the Reeb graph it is possible to detect and represent the right contour correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The advantage consists in the possibility of considering the correspondence among contours of one slice (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Taken from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The second graph based method presented by Shinagawa [8, 9] uses surface coding based on Morse Theory to construct a Reeb graph [14] representing the contour connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each contour represents a node in the graph, edges of the graph represent the contour correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Edges are added to the graph in the manner to avoid making connections that would result in a surface that is not a 2-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' For each pair of contours that can be legally connected, a weight function is evaluated, and its value is used to establish a priority for connecting that pair of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The algorithm proceeds by making the highest priority connections in regions where the number of contours in each section does not change, and then adds connections in order of decreasing priority with respect to the a priori knowledge of the number of connected components and the topological genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='221-233, ISSN 1230-0535, 2004 2 0 3It is necessary to note that all the existing solutions just estimate the contours correspondence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' the structure of the original object, should be emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In Figure 1 there are some typical example data sets to illustrate capabilities of the approaches mentioned in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Orthogonal slices One set of parallel planar slices is one of the well-known boundary representations of a 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Usually the planes of such slices set are perpendicular to the z-axis, and thus called z- slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' If we slice an object by more then one set of parallel slices and moreover when these sets are mutually orthogonal, we get orthogonal sets of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Consider now that we have z-slices, x-slices and y-slices of an object, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Note, that the contours are supposed to be polygonal, oriented the way that when looking from the positive direction of the given slices set axis, the contours have the interior on its left side and the exterior on the right side, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='1 Contour correspondence The main advantage of orthogonal slices consists in the approach how the contour correspondence can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' It is important to emphasize that two orthogonal contours which intersect each other comes aparently from one and the same surface component of the input object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' It means that the intersection of contours is very important since it provides accurate information about the correct structure, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' It is obvious that if the slices in the orthogonal sets sample the object sufficiently, then the intersections of contours from the orthogonal slices identify the correspondence relation accurately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' the correct structure of the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The algorithm The planes of slices divide space into a set of spatial cells of a spatial grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In Figure 3 can be seen three mutually orthogonal planes of grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' We distinguish two kinds of cells of M, the surface-crossing and the surface- passing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' There are parts of contours on some sides of a surface-crossing cell, which means that the resultant surface intersects the cell, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Figure 3: An example of three orthogonal slices sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Figure 4: Correct contour orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Figure 5: The mutual crossings of orthogonal contours define the correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Figure 6: A surface-crossing cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Parts of contours on the sides of the cell together with node points form spatial polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Node points are denoted as white circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each edge of G is adjacent with two cells of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='221-233, ISSN 1230-0535, 2004 The intersection of two orthogonal slices consisting of curvilinear contours is a set of points and we call them node points, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Now we focus on surface-crossing cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' An important observation is that parts of input contours and the node points form spatial polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each such polygon is enclosed in a surface-crossing cell, its patch is part of the resultant surface, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='1 The correspondence problem At this moment we suppose that the correspondence of contours is identified sufficiently by the intersections of orthogonal contours as it has been discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Consider the intersection of two contours as the relation of correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Note that the number of components of a graph constructed of such a relation corresponds to the number of disjoint components of the resultant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='2 Node points computation A node point is geometrically the intersection of two contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Topologically it is the representation of a contour correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' It holds that each node point must lie on the edge of the grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Since the contours are supposed as polygonal curves, we cannot compute the intersections of two orthogonal sections directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' We obtain them in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A B Figure 7: A) An input contour, the lattice represents positions of orthogonal slices planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' B) The contour formed by its node points (black spots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In the first step intersections of each contour and the grid M are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' These intersections are added among the current contour vertices on the appropriate position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' They are registered on the corresponding edge of the grid M simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Our algorithm works on the same principle as the Cohen- Sutherland’s line clipping algorithm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' An intersection of a slice plane and all other orthogonal slice planes forms a lattice 1 10 100 1 000 10 000 100 000 2 3 4 5 6 7 8 9 10 frequency of occurence Figure 8: Polygon size (number of edges) histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='221-233, ISSN 1230-0535, 2004 with cells, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each node point arises as the intersection of a contour and a side of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Since the contour is supposed to be polygonal, a node point is simply computed as an intersection of two segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Singular cases when a contour crosses a cell at its corner are handled separately [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' In the second step the node point construction is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The correspondent vertex, which is a member of the orthogonal contour and also a member of the same edge of M, must be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' As it was said before it is done very fast searching the auxiliary registrations of contour intersections on the appropriate edge of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each two nearest intersections coming from orthogonal contours registered on an edge of M are qualified as correspondent vertices building together a node point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Constructing the surface Now suppose graph G, whose set of vertices consists of a set of the node points and whose edges represent the parts of contours between two node vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Note that the geometrical shape of the edges still corresponds to the appropriate parts of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Now the task is to find such cycles of graph G, which have the property that their geometrical representation lies within one cell of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Those cycles represents spatial polygons that lie on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' We suppose each edge e of G is adjacent with cells B1 and B2, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Each cell from {B1, B2} includes one cycle c of our interest, which is adjacent with e (that results from the consideration of 2-manifold objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The circle c represents the spatial polygon being searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Thus for each e two cycles e B c 1 , e B c 2 must be searched and then polygons 1cp , 2 cp correspondent to those cycles are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' As soon as all polygons are obtained, we can start to patch them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' We can use any arbitrary patching technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Note that the number of sides of such polygon can be high, but in cases of our data sets it is in range 2 – 10, see the graph in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The proposed method starts with finding a suitable point in the center of each polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Then using the center point each polygon is divided into set of quadrilaterals, which are easier to patch, see Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Figure 9: Partition of a generic polygon in the set of quadrilaterals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Requires the central point C determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A B C D Figure 10: Results of the surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' A) An input data set (courtesy of Martin Čermák), B) VTK surface reconstruction from slices class, C) A common volume based method, D) Proposed method for surface reconstruction from orthogonal slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='221-233, ISSN 1230-0535, 2004 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Results All the problematic data sets mentioned in section 1 and many more have been processed using: - surface reconstructing from slices class from VTK, - a common volume based method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' see [5] for more details, - our proposed method for surface reconstruction from orthogonal slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The results of the reconstruction of one data set are illustrated in Figure 10, the complete documentation and experimental results can be found at http://herakles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='cz/research/slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Conclusion and further research Our current research proves that the advantages of orthogonal slices in the process of surface reconstruction are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' There is a set of objects for which the orthogonal slices are almost the only way to reconstruct them correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The proposed method supposes that the object is sampled well enough, so that the number of components of the correspondence graph G equals to the number of disjoint components of the original surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' The main point of our further research is the solution of problems caused by under- sampling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' to deal with data sets that do not sample the input object sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' Furthermore we would like to study the influence of contour inaccuracy on the node point computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=' References [1] Bresler, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=', Fessler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=', Macovski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'} +page_content=': 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100644 index 0000000000000000000000000000000000000000..a12086d85d6d8f318e455a35db8c41163c89e0a5 --- /dev/null +++ b/B9E2T4oBgHgl3EQfRwdj/content/tmp_files/2301.03784v1.pdf.txt @@ -0,0 +1,2250 @@ +Inside the Black Box: Detecting and Mitigating Algorithmic Bias across +Racialized Groups in College Student-Success Prediction + +Denisa Gándara +The University of Texas at Austin +1912 Speedway, Stop D5000; Austin, Texas 78712 +denisa.gandara@austin.utexas.edu +Hadis Anahideh* +University of Illinois Chicago +1200 West Harrison St., Chicago, Illinois 60607 +hadis@uic.edu +Matthew P. Ison +Northern Illinois University +1425 W. Lincoln Hwy., DeKalb, Illinois 60115 +mison@niu.edu +Anuja Tayal +University of Illinois Chicago +1200 West Harrison St., Chicago, Illinois 60607 +atayal4@uic.edu + +Acknowledgments: The research reported here was supported, in whole or in part, +by the Institute of Education Sciences, U.S. Department of Education, through +grant R305D220055 to the University of Illinois Chicago and by grant, +P2CHD042849 awarded to the Population Research Center at The University of +Texas at Austin by the Eunice Kennedy Shriver National Institute of Child Health +and Human Development. The content is solely the responsibility of the authors. +Abstract: Colleges and universities are increasingly turning to algorithms that +predict college-student success to inform various decisions, including those +related to admissions, budgeting, and student-success interventions. Because +predictive algorithms rely on historical data, they capture societal injustices, +including racism. A model that includes racial categories may predict that racially +minoritized students will have less favorable outcomes. In this study, we explore +bias in education data by modeling bachelor’s degree attainment using various +machine-learning modeling approaches. We also evaluate the utility of leading +bias-mitigating techniques in addressing unfairness. Using nationally +representative data from the Education Longitudinal Study of 2002, we +demonstrate how models incorporating commonly used features to predict +college-student success produce racially biased results. + +*Corresponding author + + + +1 + +Since the emergence of “big data” in the 1990s, efforts to use advanced +statistical techniques to predict outcomes of interest have proliferated across +various social domains, education notwithstanding (Baker et al., 2019; +Government Accountability Office [GAO], 2022). The suite of techniques used to +forecast outcomes and inform decision-making within organizations is broadly +known as “predictive analytics.” Although largely unseen, predictive analytics +fuel myriad decisions within educational institutions, from college admissions +(Hutt et al., 2019) and student retention interventions (Baker et al., 2019), to fiscal +health and resource allocation (Wayt, 2019; Yanosky & Arroway, 2015). +A key component within the vast array of predictive statistical techniques +is the predictive model, a computational tool that maps the input set of attributes +of individuals (e.g., high school GPA and demographic features) to their +outcomes (e.g., college credits accumulated) in order to identify underlying +associations and patterns in the data. The predictive model is especially useful +with large datasets, where it is impossible or inefficient to identify associations +and patterns manually. +In recent years, observers have raised concerns that predictive models in +education may perpetuate social disparities, especially when they ignore how +extant societal injustices can bias historical data (GAO, 2022). For instance, a +model that includes socially relevant attributes, such as race, gender, and income, +will often predict that students from socially disadvantaged categories (e.g., + + + +2 +women in STEM) will have less favorable outcomes. Such a model will be +extrapolating from prior relationships between socially relevant attributes (e.g., +race) and educational outcomes (e.g., graduation) that are partly shaped by +societal injustices, such as racism, sexism, and classism (e.g., López et al., 2018). +In this study, we appraise predictive models within the higher education +context. We begin by modeling bachelor’s degree attainment to explore biases in +educational data. We then assess the utility of bias-mitigating techniques in +addressing unfairness. This analysis focuses on disparities in college-student +success predictions across racialized groups, since educational attainment rates +across racial/ethnic groups remain markedly unequal (U.S. Department of +Education, 2021). Given these inequities in educational attainment levels, +predictive models that are agnostic to racial bias may penalize groups that have +been subject to racialized social disadvantages. +We situate our statistical analyses within relevant historical and social +contexts (Zuberi, 2001), recognizing that racially minoritized groups are +disadvantaged in the educational context through various interlocking social +systems of oppression (Reskin, 2012). Although an exhaustive review is beyond +the scope of this paper, we refer readers to examples of systems, structures, and +practices that penalize racially minoritized groups. In the education domain, +oppressive barriers to educational success include educational tracking (Oakes, +1985), deepening school segregation (Orfield et al., 2012), teacher racial bias + + + +3 +(Gershenson & Papageorge, 2018), racial disparities in school funding that track +with levels of segregation (Weathers & Sosina, 2019), and disparate punishment +of Black and Latinx students (Davison et al., 2019). Racially minoritized students’ +educational success is also conditioned by racialized barriers outside education, +including constraints on wealth accumulation and income, which limit students’ +ability to pay for higher education (Mitchell et al., 2019). +It is important to understand this background since the state of the world, +which is rooted in various societal injustices, affects the data distribution. These +historical injustices condition educational opportunities and experiences for +racially minoritized students. Subsequently, when predictive models make +predictions on students who are racially minoritized, they may be predicted to fail, +reinforcing historical biases. Amidst this backdrop, this study addresses the +following questions: +1. To what extent are college student-success predictions biased across +racial/ethnic groups? +2. How effective are computational strategies in mitigating racial/ethnic +bias? +Predictive models warrant greater attention in education not only because +they are ubiquitous, but also because they have the potential to reinforce and +legitimize societal inequities. Decisions grounded in biased predictions can yield +significant societal consequences. For instance, college admission may unfairly be + + + +4 +denied to racially minoritized students if the model shows they have lower +predicted likelihoods of success (Hutt et al., 2019). With course +recommendations, predictions could lead to educational tracking, encouraging +students from racially minoritized groups to pursue courses or majors that are +perceived as less challenging (Ekowo & Palmer, 2017). Such consequences may +go undetected since automated sorting mechanisms remain both obfuscated (due +to their invisibility to educational stakeholders) and legitimized through +perceptions that statistical models are objective (Hirschman & Bosk, 2020). +Literature on Fairness in College-Student Success Prediction + +In recent years, educational researchers and data scientists have begun to +develop insights into fairness and bias within various stages of the machine +learning (ML) process. Among the most important discernments from these +studies are the importance of representation of socially relevant groups in training +datasets (Riazy et al., 2020),1 and novel statistical techniques intended to measure +and enhance predictive fairness between groups (Gardner et al., 2019; Hutt et al., +2019). A small number of studies have examined algorithmic fairness in college- +student success (Anderson et al., 2019; Hu & Rangwala, 2020; Hutt et al., 2019; +Lee & Kizilcec, 2020; Yu et al., 2020). Most of these studies have detected bias in +existing data, particularly with models using institutional (college or university) +administrative data (see Hutt et al., 2019 for an exception). For instance, +Anderson and colleagues (2019), who used administrative data from a single + + + +5 +institution, found that their predictive models advantaged White students (e.g., +higher rates of predicted success for students who failed and lower rates of +predicted failure for students who succeeded) and disadvantaged Hispanic/Latinx +and male students. Yu and colleagues (2020) examined how the data source (e.g., +learning management system [LMS], institutional data, or survey data) affected +predictions on college outcomes, concluding that institutional data were more +likely to be biased against disadvantaged groups. +Gardner and colleagues (2019) also used institutional data to examine the +fairness of models used to predict course success in higher education. Their +analysis, which focused on gender, showed that model fairness varied according +to the algorithm used, the variables included in models, the specific course +examined, and the gender imbalance ratio in a given course. Importantly, they did +not identify a meaningful tradeoff between fairness and accuracy. These results +contradict arguments and other evidence signaling that as predictive algorithms +implement bias-mitigation efforts, the predictive accuracy of the algorithm +declines (e.g., Lee & Kizilcec, 2020). +Expanding upon this prior work, the present study offers a more holistic +picture of bias in college-student success predictions. Most research on this topic +is situated in the Learning Analytics literature, predicting outcomes within +courses (Gardner et al., 2019; Hu & Rangwala, 2020; Lee & Kizilcec, 2020; +Riazy et al., 2020; Yu et al., 2020). In this study, we adopt a broader + + + +6 +conceptualization of college-student success by modeling an educational +attainment outcome. Predictions of attainment-related outcomes are more likely to +be used to inform practices related to admissions and campus-wide student- +success interventions (Ekowo & Palmer, 2017; Wayt, 2019). We extend prior +work by using nationally representative data instead of course data (e.g., Hu & +Rangwala, 2020), single-institution data (e.g., Anderson et al. 2019), or non- +representative national data (e.g., Hutt et al., 2019). Moreover, beyond exploring +bias in the data, we test various approaches for mitigating bias, both in data +preparation (preprocessing) and in models (in-processing). Finally, we bolster our +empirical contribution by exploring various notions of fairness, presenting +conceptual models that can be used for further exploration of bias in educational +data. +Data Sources +Data come from the Education Longitudinal Study of 2002 (ELS), a +nationally representative, longitudinal study of students who were 10th graders in +2002. Given our focus on bachelor’s degree attainment, the dataset is filtered +based on the institution type to only include students who attended four-year +postsecondary institutions. The outcome variable captures the students’ highest +level of education as of the third follow-up interview (eight years after expected +high-school graduation). To construct a binary classification problem, we label + + + +7 +students with a bachelor’s degree and higher as the favorable outcome (label=1), +and all others as the unfavorable outcome (label=0). +Predictive variables include features commonly used for student-success +prediction, including student demographic characteristics, socioeconomic traits, +grades, college preparation, and school experience. Since category labels are not +ordinal, we create binary variables for each level of the categorical variables +following National Center for Education Statistics (NCES) documentation +(NCES, n.d.). The complete list of variables appears in Supplementary Materials +(Appendix A). While our dataset does not include all possible variables that could +be incorporated in a model predicting college-student success, our dataset has the +advantage of being large (n = 15,244) and nationally representative, and including +the most commonly used features (p = 29) based on our review of literature on +college-student success prediction. +Since we have a high number of missing values, we ran the models +separately with multiple imputation (Rubin, 1996) and without imputation +(listwise deleted rows with missing data).2 To avoid the confounding impact of +imputation on both unfairness and model performance, we stratified on the +response variable (bachelor’s degree attainment) and racial groups for the +training-testing splits, retaining the distribution of the historical data in both +partitions. For simplicity, we present results without imputation in our main +results. Results with imputation appear in supplementary materials (Appendix B). + + + +8 +Those results indicate that imputation is inconsequential for all models except for +support vector machine (SVM), where it reduces the variance of unfairness, +resulting in a more robust model. A deeper investigation of how imputation +affects the unfairness of the prediction outcome appears elsewhere (Anahideh et +al., 2021). +First, we randomly split the dataset into training and testing subsets with +an 80:20 ratio (80% training, 20% testing). The ML models are trained on the +training data and evaluated on the testing data to demonstrate their +generalizability. To evaluate the fairness of the prediction outcome using various +fairness notions (described below), we stratified the training and testing datasets +by the outcome variable class labels (1, 0) and racial/ethnic categories, ensuring +that we have enough observations from each group. The results are averaged over +30 different splits of the data. Table 1 presents the distribution of the outcome +variable by racial/ethnic category after dropping observations with missing +values. +<> +Analysis Methods +Evaluating Unfairness +We employed the most widely used ML models in higher education, +including Decision Tree (Hamoud et al., 2018), Random Forest (Pelaez, 2018), +Logistic Regression (Thompson et al., 2018), and SVM (Agaoglu, 2016). Each + + + +9 +ML model has predefined parameters known as hyperparameters that must be +provided before the training phase (e.g., depth of the tree in Decision Trees). +Since the optimal values of such hyperparameters are data-dependent, we +performed a five-fold cross-validation (CV) for each model to determine the best +set of hyperparameters. In this process the dataset was divided into five partitions, +four of which were utilized for training and one for validation. Cross-validation +repeats this process and selects a different partition for validation each time. A +grid of feasible hyperparameters was assessed based on the CV schema described +above to choose the optimum set. Under 30 distinct random splits of training and +testing datasets, we obtained the best set of hyperparameters before we performed +model training. To evaluate model performance, we report the average and +variance of the accuracy as well as unfairness towards different racial/ethnic +groups using various notions of unfairness. +Fairness Notions. We consider four different conceptions of fairness +commonly used in algorithmic fairness: statistical parity, equal opportunity, +predictive equality, and equalized odds (Barocas et al., 2017). In practice, users +can select the measure of fairness that is preferred based on context, knowledge +of social disparities, use case, and regulations. We briefly describe each fairness +notion in turn; the probabilistic definitions of these notions appear in the +supplemental materials (Appendix C). + + + +10 +Statistical Parity is achieved by having equal favorable outcomes (degree +attainment) received by the unprivileged group (e.g., Black) and the privileged +group (e.g., White). Said differently, under the notion of statistical parity, we +consider a model fair if being a member of a racially minoritized group is not +correlated with the probability of bachelor’s degree attainment. +The next three fairness measures build on the statistical notions of +true/false positives/negatives (for a visual, see Confusion Matrix in Appendix D). +Specifically, +• A true positive result would correctly predict success for a student who +succeeds (in our case, attains a bachelor’s degree). +• A true negative result would correctly predict failure for a student who +does not succeed. +• A false positive result (Type I error) would incorrectly predict success for +a student who does not succeed. +• A false negative result (Type II error) would incorrectly predict failure for +a student who does succeed. +Building on these statistical notions, Equal Opportunity represents equal +false negative rates between groups. This fairness notion requires that each group +receive the negative outcome at equal rates, conditional on their success. In other +words, under this notion, the model should (incorrectly) predict failure for those +who succeeded (attained at least a bachelor’s degree) at the same rate for + + + +11 +students across racial/ethnic groups. This notion assumes knowledge of the true +outcome values (whether a student attained at least a bachelor’s degree) and aims +to satisfy parity across socially relevant groups, subject to the true values. +A third fairness notion is Predictive Equality, which represents equal +false positive rates. To satisfy this criterion, positive predictions (that a given +student will attain a bachelor’s degree) for students who do not actually attain a +bachelor’s degree should be the same across racial/ethnic groups. +Finally, Equalized Odds represents the average difference in false +positive and true positive rates between groups. To achieve fairness under this +notion, both the false positive rate (wrongly predicting success) and the true +positive rate (correctly predicting success) should be the same across +racial/ethnic groups. We use these notions to evaluate fairness in college-student +success predictions. +Mitigating Bias +In addition to evaluating unfairness, we implement statistical techniques +to mitigate bias. Literature on bias-mitigation techniques for ML models is +burgeoning. Such techniques can be categorized into three groups: preprocessing, +in-processing, and post-processing approaches (Pessach et al., 2022). +Preprocessing techniques entail fairness evaluation in the data-preparation step, +which should, in turn, mitigate bias for downstream tasks. We apply two + + + +12 +preprocessing techniques: Reweighting (Kamiran & Calders, 2012) and +Disparate Impact Remover (DIR) (Feldman et al., 2015). +Reweighting assigns different weights to the training samples in each +combination of racial/ethnic group and outcome-variable class label (e.g., Black +X outcome label=1). It does so before training a model to adjust the bias across +groups. Because individual observations from the unprivileged +groups with positive outcomes are underrepresented in the training data (see +Table 1), classifiers are susceptible to bias. In this preprocessing approach, the +data points representing successful outcomes for unprivileged groups are +identified and upweighted, so they have a larger influence on model training. +In contrast to Reweighting, DIR changes the distributions of other +features in the model (not race/ethnicity) to force distributions to overlap at the +group level. This process removes the ability to distinguish between group +membership from a feature that otherwise offers a good indication of which +group a data point may belong to. +In-processing techniques generally involve modifying the ML algorithms +to account for fairness during the training process, such that the parameter +estimation of the classifier forces the prediction outcome to be fair toward all +(racial/ethnic) groups. The enforcement is accomplished in the optimization +subproblem by adding a fairness metric as a constraint similar to the +Exponentiated Gradient Descent approach (Agarwal, 2018). + + + +13 +We also employ a second in-processing technique, Meta Fair Classifier +(Celis et al., 2018), which takes a large class of fairness metrics as inputs and +returns an optimal classifier that is fair with respect to constraints on the given +set of metrics. This approach works for various fairness criteria and provides +theoretical guarantees by developing a general form of constrained optimization +problems, which encompasses many existing fair classification problems. This +stands in contrast to earlier work on fair classification, which focused on +constructing classifiers that are constrained with respect to a single fairness +metric (e.g., Zafar et al., 2017). +Post-processing techniques for mitigating bias adjust the prediction +outcome after training a regular ML model, changing the values across different +groups. We exclude these techniques since post-processing mechanisms are +implemented at a later phase in the learning process, often producing inferior +results (Woodworth et al., 2017). These approaches are also more controversial +than in-process and preprocess strategies in the domains of “affirmative action” +and are thus less likely to be used in education settings (Hirschman & Bosk, +2020). +In the results section, we refer to Reweighting as ReW, Disparate Impact +Remover as DIR, Exponentiated Gradient Reduction as ExGR, and Metaclassifier +as MetaC. For comparison, we also consider the baseline classification scenario, +where no mitigation strategy is used. + + + +14 +Comparisons +We use two comparison approaches to appraise model unfairness and test +mitigation techniques, namely, at 1) the subgroup level (i.e., each racial/ethnic +group versus the rest), and 2) the aggregate level (i.e., privileged versus +unprivileged). First, we compared each racial/ethnic group against all others and +consider 1 for a certain group (e.g., Black) and 0 for every other group (e.g., +White, Asian, Hispanic, and Two or More Races) to calculate gaps as discussed +previously. +To evaluate the limitations of aggregation, which is common in this type +of work, we also aggregate White and Asian groups in the privileged category +and Black, Hispanic, and Two or More Races (2+) groups in the unprivileged +category. These comparisons represent an extension over prior work as they +allow us to investigate the impact of existing mitigation techniques at both the +subgroup and aggregate levels. Most existing techniques only work with binary +sensitive attributes (e.g., “White” and “Non-White”), requiring the researcher to +specify the privileged group and forcing other subgroups to be aggregated as the +unprivileged group (Pessach et al., 2022). +Although some existing unfairness mitigation techniques have the +potential to incorporate non-binary sensitive attributes, such extension has not +been implemented in the literature. Binarizing sensitive attributes (1: privileged, +0: unprivileged) for the mitigation processes may not reduce fairness gaps for + + + +15 +each group. This is important in educational settings where research shows that +students from different racial/ethnic groups have distinct experiences and +outcomes (e.g., López et al., 2018). Hence, it is critical to evaluate unfairness +after applying mitigation techniques at the subgroup levels, as there might be +significant differences between unprivileged subgroups. +Results +We find no significant difference between the performance (accuracy) of +different ML classifiers, although there are some differences in levels of +unfairness across fairness notions and models. To facilitate comparison, Figure 1 +presents results for all ML models. We discuss the main findings for our +assessments of unfairness and the effectiveness of bias-mitigation techniques in +turn. +<
> +Evaluating Unfairness +Subgroup Level: Each Group Versus the Rest. Figure 1 shows a +comparison of unfairness levels using all four fairness notions and ML models +for the baseline (without bias mitigation). The testing accuracy across these +models is 78%, on average. These results indicate that Black and Hispanic groups +are treated unfairly across models. Generally, the SVM model yields less unfair +results, across fairness notions, compared to the other ML models. Under the +fairness notions of Statistical Parity (Figure 1a), Predictive Equality (Figure 1c), + + + +16 +and Equalized Odds (Figure 1d), the boxes for Black and Hispanic students are at +a lower level across all ML models, indicating that these students receive +favorable outcomes (i.e., bachelor’s attainment or higher) at a lower rate than +students in other categories. For the notion of Equal Opportunity (Figure 1b), +higher levels in the box plots, which we observe for Black and Hispanic groups, +represent more unfairness. +For a concrete example of unfairness with respect to Statistical Parity, in +one of the test splits, students in the Asian and White categories have a 91% +probability of attainment, while those in the Black and Hispanic categories have +63% and 68% probabilities, respectively. Without correcting for bias, predictive +models will be more likely to predict that students categorized as Black and +Hispanic are less likely to attain a bachelor’s degree or higher when compared to +more privileged peers. +Findings for Predictive Equality further illustrate bias in the predictions. +Among the students who did not complete their degree (y=0), the probability of +attainment is estimated as 78% for White and 83% for Asian, while it is +estimated as 33% for Hispanic, and 0% for Black. 3 As illustrated in Figure 1b, +the models are also more likely to falsely predict failure for Black and Hispanic +students than for White and Asian students. Illustratively, for a single split, +among the students who completed their degree (y=1), the probability of failure + + + +17 +is estimated as 4.6% for White and 5.5% for Asian, while it is estimated as 20% +for Hispanic and 8% for Black. +Moreover, the plots show that the variation of values for the White and +Asian groups is minimal, especially for the White group, whereas the variation of +unfairness gaps for the other groups is significantly larger. Variation for the +category of two or more races is especially large, suggesting this is not a +meaningful category and should be used with caution in student-success +prediction. Differences in variation across racial/ethnic groups indicate that +models for minoritized groups are more sensitive to the train/test splits. Due to +the population bias across different racial/ethnic groups in the ELS dataset (i.e., +statistical underrepresentation of Black and Hispanic students), the train/test +splits can significantly change the distribution and presence of underrepresented +individuals in each partition, significantly impacting the unfairness of the model +for each split scenario. In practice, this will result in less stable and fair model +performance for predicting the success of an unobserved individual from a +statistically underrepresented group. +<
> +Aggregate Level: Privileged vs. Unprivileged. Figure 2 presents the box +plots for all four unfairness notions at the aggregated level of privileged (Asian +and White) versus unprivileged (Black, Hispanic, and two or more racial/ethnic +categories) for all prediction models. The first evident pattern from all four plots + + + +18 +is the mean difference between the two groups. Similar to results at the subgroup +level, we observe higher false negative rates for the unprivileged group. In other +words, the models are more likely to predict failure for Black and Hispanic +students who succeed compared to White and Asian students. +Comparing findings at the subgroup and aggregate levels, we observe that +aggregate results mask substantial differences we can glean from the subgroup +analysis. For instance, in Figure 1a, the DT and RF models show similar levels of +unfairness for Black and Hispanic groups, but LR and SVM are more unfair for +the Black group than the Hispanic group. At the aggregate level of analysis, this +variation cannot be observed (all models are unfair toward the unprivileged +group). We now turn to results for bias-mitigation techniques. +Mitigating Bias +Given space constraints and for ease of interpretability, we present +mitigation results using one predictive model, RF, which is a non-linear classifier +commonly used in the education literature. These results appear in Figure 3 +(findings from other ML models are in Appendix E). Our first observation is that +the preprocessing and in-processing mitigation methods only minimally decrease +accuracy (by 1% to 2%). One technique, MetaC, significantly improves accuracy +(by 10-to-17-points over the baseline model without bias mitigation). +The bias-mitigation techniques we used required us to specify the +privileged and unprivileged groups and to treat the sensitive attribute as binary. + + + +19 +The results demonstrate that the mitigation techniques are generally not effective +at reducing bias at the aggregate level. At the subgroup level, we do not find a +mitigation technique that improves fairness across all racial/ethnic subgroups; +when a technique reduces unfairness for one subgroup, it harms another. We first +present findings for the preprocessing mitigation techniques, ReW and DIR. +The results (in Figure 3) indicate that the ReW technique does not +effectively reduce bias for unprivileged groups when compared to the baseline +model. If the goal of the education data analyst is to reduce unfairness in student- +success predictions, it is not enough to increase the influence of datapoints that +represent successful students from unprivileged groups (e.g., Black students who +succeed) in the training process. This finding suggests that the +underrepresentation of successful students from unprivileged groups in the +training data is not a key source of bias in student-success predictions. +The second preprocessing mitigation technique we employed, DIR, +decreases unfairness for the Black group but leads to more unfair predictions for +the Hispanic group. This approach modifies the distributions of other features in +the model (e.g., students’ native language and family composition) to reduce +their correlation with racial/ethnic categorizations. A feature can provide a strong +hint as to which group a data point might belong to. DIR aims to eliminate this +capacity to distinguish between group membership. In addition to reducing +unfairness for the Black group, DIR diminishes the advantage of the Asian group + + + +20 +relative to that of other groups. However, the advantage for the White group is +actually exacerbated in two of the fairness notions (Statistical Parity and Equal +Opportunity). Contrary to expectations, applying DIR increases the Equal +Opportunity gap between the White group and all other groups, indicating a +decrease in the number of successful White students who are falsely predicted to +be unsuccessful. +Note that the DIR approach corrects the dataset measuring and +considering the Statistical Parity notion at the aggregate level. Hence, it is +expected to observe equal proportions of positive prediction from each group at +the aggregated level of privileged versus unprivileged. Our results show that DIR +cannot effectively achieve statistical parity for each subgroup using ELS data. +Even at the aggregate level (Figure 4a), DIR slightly removes the advantage for +the privileged group but does not improve fairness for the unprivileged group. +These findings also highlight differences between two groups that are often +considered privileged (Asian and White) and two groups that are often +considered unprivileged (Black and Hispanic), underscoring the importance of +disaggregation. +Turning to the in-processing techniques, ExGR did not significantly alter +the privilege of the White group or diminish the unfairness of the Black or +Hispanic groups for any of the four notions. Instead, both in-processing +techniques (ExGR and MetaC) result in greater variation, which indicates that the + + + +21 +repaired model is less robust to data splits. The MetaC technique effectively +reduces all four types of biases for the Hispanic group but is more unfair for +Black students with respect to Statistical Parity and Equalized Odds, again +highlighting the need to disaggregate education data across racialized groups. +The results confirm that even at the aggregated level, unfairness is not +mitigated significantly and the only technique that is slightly effective is MetaC. +Even then, MetaC only works for Hispanic students and is ineffective at reducing +bias for Black students. These preprocessing and in-processing techniques do not +significantly reduce demographic bias, demonstrating the need for better bias- +mitigation techniques. Future work should examine bias-mitigation when both +preprocessing and in-processing techniques are applied simultaneously. +Discussion + +The ubiquity of predictive analytics in higher education demands greater +attention to the “black box” of student-success-prediction models. This work +shows how such models produce unfair outcomes across various notions of +fairness. Further, we illustrate the limitations of existing techniques to reduce +bias. Using a nationally representative dataset with student-level data, we +demonstrate that across notions of fairness and various common ML models, +Black and Hispanic groups are treated unfairly. Not only are they more likely to +predict success for White and Asian groups (Statistical Parity) but they are also +significantly more likely to predict failure for Black and Hispanic students who + + + +22 +succeeded. This work illustrates how, without correcting for bias, Black and +Hispanic students may be offered fewer opportunities (e.g., admission) as a result +of student-success prediction models. We also show how bias-mitigation +techniques—both those that correct the dataset before modeling and those that +apply fairness constraints in the modeling process—generally fail to improve +fairness across subgroups. +One such technique, Reweighting, increases the influence of observations +representing racially minoritized students who are successful (e.g., Black students +who graduate). This technique is ineffective at reducing bias, indicating that the +main source of bias is not statistical underrepresentation but underlying, +unobservable sources of systemic and historical discrimination. + +In evaluating student-success prediction models, it is important to +understand the use case. While bias-agnostic models may reproduce social +inequities in college-admissions use cases, they may lead to greater support for +students when used to inform student-success interventions. Even then, +practitioners must take care not to produce deficit narratives of minoritized +students, treating them as though they have a lower likelihood of success. + +Despite widespread perceptions that statistical analysis is independent of +human judgment and error, this work demonstrates myriad decisions researchers +must make that have significant consequences for fairness, including which ML +model to use and which bias-mitigation techniques to employ. For example, if a + + + +23 +predictive algorithm closes the gap between a comparison group while benefiting +the majoritized group (rising tide metaphor), should such an algorithm be +considered fair (Kizilcec & Lee, in press)? +As higher education institutions strive to better serve students by +becoming more data-informed (Gagliardi & Turk, 2017), it is imperative that +predictive models are designed with attention to their potential social +consequences. It is critical to be aware of historical discrimination embedded in +the data and consider fairness measures to reduce bias in the outcomes of the +models. This paper demonstrates that more work is needed to develop fairness +measures to reduce bias across racialized groups. Future research should also +examine the influence of training/testing splits in the data. Another important +avenue for future work is understanding how feature selection (which variables +to include in the model) affects predictions and fairness across racialized groups. +Such work could expand on existing and conflicting recommendations +concerning the inclusion of race/ethnicity variables in student-success prediction +models (Hu & Rangwala, 2020). Finally, while we demonstrate the importance of +disaggregating beyond privileged/unprivileged, the ELS categories are severely +limited. Future work should disaggregate further to lead us toward more racially +just student-success practices in higher education. + + + + + +24 +Notes +1. In ML, a training dataset includes the data you use to train the model or +algorithm to predict the outcome of interest. +2. In all versions, we avoid imputing socially relevant (sensitive) attributes and +outcome variables, hence observations with missing values for these variables are +always dropped before imputation. +3. These estimated probabilities are based on RF modeling on a single train/test +split. + + + + + + + +25 +References +Agaoglu, M. (2016). Predicting instructor performance using data mining +techniques in higher education. IEEE Access, 4, 2379-2387. + Agarwal, A., Beygelzimer, A., Dudik, M., Langford, J., and Wallach, H. (Eds.). +(2018). A reductions approach to fair classification. International +conference on machine learning, 2018. +Anahideh, H., Nezami, N., & Gandara, D. (2021). 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U of Minnesota +Press. + + + +32 +Table 1: Distribution of bachelor’s degree or higher variable by racial/ethnic +category +Race +Bachelor’s or Higher +% of data +Asian, Hawaiian/Pacific Islander +1 +0.73984 + +0 +0.26016 +Black or African American +1 +0.62766 + +0 +0.37234 +Hispanic +1 +0.67470 + +0 +0.32530 +More than one race +1 +0.69697 + +0 +0.30303 +White +1 +0.76005 + +0 +0.23995 + + + + + + + +33 +Figure 1: Baseline with all ML models for all racial/ethnic groups + + + + + + +Figure 1d Equalized Odds of baseline for different ML models +Figure 1c Predictive Equality of baseline for different ML models +Figure 1b Equal Opportunity of baseline for different ML models +Figure 1a Statistical Parity of baseline for different ML models + +0.6 +LR +SVM +0.4 +DT +RF +0.2 +QQ +0.0 +百白 +-0.2 +-0.4 +-0.6 +Asian +Black +Hispanic +2+ +White0.6 +LR +SVM +0.4 +DT +RF +0.2 +0.0 +广 +-0.2 +0.4 +0.6 +Asian +Black +Hispanic +2+ +White0.6 +0.4 +0.2 +0.0 +-0.2 +LR +-0.4 +SVM +DT +0.6 +RF +Asian +Black +Hispanic +2+ +White1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +-0.50 +LR +SVM +0.75 +DT +RF +-1.00 +Asian +Black +Hispanic +2+ +White + +34 +Figure 2: Baseline with all ML models for privileged vs. unprivileged groups + + + +Figure 2d Equalized Odds of baseline for different ML models +Figure 2c Predictive Equality of baseline for different ML models +Figure 2b Equal Opportunity of baseline for different ML models +Figure 2a Statistical Parity of baseline for different ML models + +0.6 +LR +SVM +0.4 +DT +RF +0.2 +0°0 +0.2 +0.4 +0.6 +privileged +unprivilegec0.6 +LR +SVM +0.4 +DT +RF +0.2 +0.0 +0.2 +0.4 +0.6 +privileged +unprivileged1.00 +LR +0.75 +SVM +DT +0.50 +RF +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +privileged +unprivileqed0.6 +LR +SVM +0.4 +DT +RF +0.2 +0.0 +0.2 +0.4 +0.6 +privileged +unprivileged + +35 +Figure 3: Mitigation for all racial/ethnic groups + +Figure 3d Equalized Odds of RF using bias-mitigation techniques +Figure 3c Predictive Equality of RF using bias mitigation techniques +Figure 3b Equal Opportunity of RF using bias mitigation techniques +Figure 3a Statistical Parity of RF using bias mitigation techniques + +0.6 +Baseline +ReW +0.4 - +DIR +ExGR +0.2 +MetaC +oo +8 +白白 +0.0 +8 +0.2 +0.4 +0.6 +Asian +Black +Hispanic +2+ +White0.6 +Baseline +ReW +0.4 +DIR +ExGR +0.2 +MetaC +88 +0.0 +0.2 +0.4 +0.6 +Asian +Black +Hispanic +2+ +White1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Baseline +-0.50 +ReW +DIR +0.75 +ExGR +MetaC +-1.00 +Asian +Black +Hispanic +2+ +White0.6 +0.4 +0.2 +白白 +0.0 +-0.2 +Baseline +ReW +0.4 +DIR +ExGR +-0.6 +MetaC +Asian +Black +Hispanic +2+ +White + +36 +Figure 4: Mitigation for privileged vs. unprivileged groups +Figure 4d Equalized Odds of RF using bias mitigation techniques +Figure 4c Predictive Equality of RF using bias mitigation techniques +Figure 4a Statistical Parity of RF using bias mitigation techniques +Figure 4b Equal Opportunity of RF using bias mitigation techniques + +0.6 +Baseline +ReW +0.4 +DIR +ExGR +0.2 +MetaC +0.0 +0.2 +0.4 +0.6 +privileged +unprivileged0.6 +Baseline +ReW +0.4 +DIR +ExGR +0.2 +MetaC +白臣 +0.0 +0.2 +0.4 +0.6 +privileged +unprivileged0.6 +Baseline +ReW +0.4 +DIR +ExGR +0.2 +MetaC +0.0 +0.2 +0.4 +0.6 +privileged +unprivileged1.00 +Baseline +0.75 +ReW +DIR +0.50 +ExGR +0.25 +MetaC +0.00 +0.25 +0.50 +0.75 +1.00 +privileged +unprivileged +1 +Supplementary Materials +Appendix A: Variable List +Student's native language- +composite +Student’s perception of teacher-student +relationships in the school +Family Composition +BY highest level of participation in +interscholastic athletics +Generational status (immigration) +BY-F1 high school attendance pattern: by school +control +Parents Education +Number of school-sponsored activities +participated in 01-02 +Current (2006) marital-parental +status +Transcript: GPA in first year of known attendance +Sex-composite +Transcript: First year known enrollment: credits +earned +Student's race/ethnicity-composite +F1 hours worked per week during 03-04 school +year +Total family income from all +sources -composite +Offered scholarship/grant for first year at first PS +institution +Sector of first postsecondary +institution +College entrance exam scores relative to average +scores at 1st PS institution +School Urbanicity +Units in mathematics (SST) - categorical +% of full-time teachers are +Hispanic +F1 math standardized score + + +2 +% of full-time teachers are Black +Transfer student +% of full-time teachers are White +GPA for all courses taken in the 9th - 12th grades +- categorical +% of full-time teachers are +Hawaiian +Standardized test composite score-math/reading +% of full-time teachers are Indian +Highest level of education earned as of F3 + + +3 +Appendix B: Results with Imputation + + + +Statistical Parity of RF using bias mitigation techniques +Equal Opportunity of RF using bias mitigation techniques +Equalized Odds of RF using bias mitigation techniques +Predictive Equality of RF using bias mitigation techniques 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='edu Hadis Anahideh* University of Illinois Chicago 1200 West Harrison St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Chicago, Illinois 60607 hadis@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='edu Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Ison Northern Illinois University 1425 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Lincoln Hwy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', DeKalb, Illinois 60115 mison@niu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='edu Anuja Tayal University of Illinois Chicago 1200 West Harrison St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Chicago, Illinois 60607 atayal4@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='edu Acknowledgments: The research reported here was supported, in whole or in part, by the Institute of Education Sciences, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Department of Education, through grant R305D220055 to the University of Illinois Chicago and by grant, P2CHD042849 awarded to the Population Research Center at The University of Texas at Austin by the Eunice Kennedy Shriver National Institute of Child Health and Human Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The content is solely the responsibility of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Abstract: Colleges and universities are increasingly turning to algorithms that predict college-student success to inform various decisions, including those related to admissions, budgeting, and student-success interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Because predictive algorithms rely on historical data, they capture societal injustices, including racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A model that includes racial categories may predict that racially minoritized students will have less favorable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In this study, we explore bias in education data by modeling bachelor’s degree attainment using various machine-learning modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We also evaluate the utility of leading bias-mitigating techniques in addressing unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Using nationally representative data from the Education Longitudinal Study of 2002, we demonstrate how models incorporating commonly used features to predict college-student success produce racially biased results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Corresponding author 1 Since the emergence of “big data” in the 1990s, efforts to use advanced statistical techniques to predict outcomes of interest have proliferated across various social domains, education notwithstanding (Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Government Accountability Office [GAO], 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The suite of techniques used to forecast outcomes and inform decision-making within organizations is broadly known as “predictive analytics.” Although largely unseen, predictive analytics fuel myriad decisions within educational institutions, from college admissions (Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019) and student retention interventions (Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019), to fiscal health and resource allocation (Wayt, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Yanosky & Arroway, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A key component within the vast array of predictive statistical techniques is the predictive model, a computational tool that maps the input set of attributes of individuals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', high school GPA and demographic features) to their outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', college credits accumulated) in order to identify underlying associations and patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The predictive model is especially useful with large datasets, where it is impossible or inefficient to identify associations and patterns manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In recent years, observers have raised concerns that predictive models in education may perpetuate social disparities, especially when they ignore how extant societal injustices can bias historical data (GAO, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For instance, a model that includes socially relevant attributes, such as race, gender, and income, will often predict that students from socially disadvantaged categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2 women in STEM) will have less favorable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Such a model will be extrapolating from prior relationships between socially relevant attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', race) and educational outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', graduation) that are partly shaped by societal injustices, such as racism, sexism, and classism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', López et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In this study, we appraise predictive models within the higher education context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We begin by modeling bachelor’s degree attainment to explore biases in educational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We then assess the utility of bias-mitigating techniques in addressing unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This analysis focuses on disparities in college-student success predictions across racialized groups, since educational attainment rates across racial/ethnic groups remain markedly unequal (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Department of Education, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Given these inequities in educational attainment levels, predictive models that are agnostic to racial bias may penalize groups that have been subject to racialized social disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We situate our statistical analyses within relevant historical and social contexts (Zuberi, 2001), recognizing that racially minoritized groups are disadvantaged in the educational context through various interlocking social systems of oppression (Reskin, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Although an exhaustive review is beyond the scope of this paper, we refer readers to examples of systems, structures, and practices that penalize racially minoritized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In the education domain, oppressive barriers to educational success include educational tracking (Oakes, 1985), deepening school segregation (Orfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2012), teacher racial bias 3 (Gershenson & Papageorge, 2018), racial disparities in school funding that track with levels of segregation (Weathers & Sosina, 2019), and disparate punishment of Black and Latinx students (Davison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Racially minoritized students’ educational success is also conditioned by racialized barriers outside education, including constraints on wealth accumulation and income, which limit students’ ability to pay for higher education (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' It is important to understand this background since the state of the world, which is rooted in various societal injustices, affects the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These historical injustices condition educational opportunities and experiences for racially minoritized students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Subsequently, when predictive models make predictions on students who are racially minoritized, they may be predicted to fail, reinforcing historical biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Amidst this backdrop, this study addresses the following questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To what extent are college student-success predictions biased across racial/ethnic groups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' How effective are computational strategies in mitigating racial/ethnic bias?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Predictive models warrant greater attention in education not only because they are ubiquitous, but also because they have the potential to reinforce and legitimize societal inequities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Decisions grounded in biased predictions can yield significant societal consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For instance, college admission may unfairly be 4 denied to racially minoritized students if the model shows they have lower predicted likelihoods of success (Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' With course recommendations, predictions could lead to educational tracking, encouraging students from racially minoritized groups to pursue courses or majors that are perceived as less challenging (Ekowo & Palmer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Such consequences may go undetected since automated sorting mechanisms remain both obfuscated (due to their invisibility to educational stakeholders) and legitimized through perceptions that statistical models are objective (Hirschman & Bosk, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Literature on Fairness in College-Student Success Prediction In recent years, educational researchers and data scientists have begun to develop insights into fairness and bias within various stages of the machine learning (ML) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Among the most important discernments from these studies are the importance of representation of socially relevant groups in training datasets (Riazy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2020),1 and novel statistical techniques intended to measure and enhance predictive fairness between groups (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A small number of studies have examined algorithmic fairness in college- student success (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hu & Rangwala, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Lee & Kizilcec, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Most of these studies have detected bias in existing data, particularly with models using institutional (college or university) administrative data (see Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019 for an exception).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For instance, Anderson and colleagues (2019), who used administrative data from a single 5 institution, found that their predictive models advantaged White students (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', higher rates of predicted success for students who failed and lower rates of predicted failure for students who succeeded) and disadvantaged Hispanic/Latinx and male students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Yu and colleagues (2020) examined how the data source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', learning management system [LMS], institutional data, or survey data) affected predictions on college outcomes, concluding that institutional data were more likely to be biased against disadvantaged groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Gardner and colleagues (2019) also used institutional data to examine the fairness of models used to predict course success in higher education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Their analysis, which focused on gender, showed that model fairness varied according to the algorithm used, the variables included in models, the specific course examined, and the gender imbalance ratio in a given course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Importantly, they did not identify a meaningful tradeoff between fairness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These results contradict arguments and other evidence signaling that as predictive algorithms implement bias-mitigation efforts, the predictive accuracy of the algorithm declines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Lee & Kizilcec, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Expanding upon this prior work, the present study offers a more holistic picture of bias in college-student success predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Most research on this topic is situated in the Learning Analytics literature, predicting outcomes within courses (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hu & Rangwala, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Lee & Kizilcec, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Riazy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In this study, we adopt a broader 6 conceptualization of college-student success by modeling an educational attainment outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Predictions of attainment-related outcomes are more likely to be used to inform practices related to admissions and campus-wide student- success interventions (Ekowo & Palmer, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Wayt, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We extend prior work by using nationally representative data instead of course data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Hu & Rangwala, 2020), single-institution data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 2019), or non- representative national data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Hutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Moreover, beyond exploring bias in the data, we test various approaches for mitigating bias, both in data preparation (preprocessing) and in models (in-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Finally, we bolster our empirical contribution by exploring various notions of fairness, presenting conceptual models that can be used for further exploration of bias in educational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Data Sources Data come from the Education Longitudinal Study of 2002 (ELS), a nationally representative, longitudinal study of students who were 10th graders in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Given our focus on bachelor’s degree attainment, the dataset is filtered based on the institution type to only include students who attended four-year postsecondary institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The outcome variable captures the students’ highest level of education as of the third follow-up interview (eight years after expected high-school graduation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To construct a binary classification problem, we label 7 students with a bachelor’s degree and higher as the favorable outcome (label=1), and all others as the unfavorable outcome (label=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Predictive variables include features commonly used for student-success prediction, including student demographic characteristics, socioeconomic traits, grades, college preparation, and school experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Since category labels are not ordinal, we create binary variables for each level of the categorical variables following National Center for Education Statistics (NCES) documentation (NCES, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The complete list of variables appears in Supplementary Materials (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' While our dataset does not include all possible variables that could be incorporated in a model predicting college-student success, our dataset has the advantage of being large (n = 15,244) and nationally representative, and including the most commonly used features (p = 29) based on our review of literature on college-student success prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Since we have a high number of missing values, we ran the models separately with multiple imputation (Rubin, 1996) and without imputation (listwise deleted rows with missing data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='2 To avoid the confounding impact of imputation on both unfairness and model performance, we stratified on the response variable (bachelor’s degree attainment) and racial groups for the training-testing splits, retaining the distribution of the historical data in both partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For simplicity, we present results without imputation in our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Results with imputation appear in supplementary materials (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 8 Those results indicate that imputation is inconsequential for all models except for support vector machine (SVM), where it reduces the variance of unfairness, resulting in a more robust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A deeper investigation of how imputation affects the unfairness of the prediction outcome appears elsewhere (Anahideh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' First, we randomly split the dataset into training and testing subsets with an 80:20 ratio (80% training, 20% testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The ML models are trained on the training data and evaluated on the testing data to demonstrate their generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To evaluate the fairness of the prediction outcome using various fairness notions (described below), we stratified the training and testing datasets by the outcome variable class labels (1, 0) and racial/ethnic categories, ensuring that we have enough observations from each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The results are averaged over 30 different splits of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Table 1 presents the distribution of the outcome variable by racial/ethnic category after dropping observations with missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' <
> Analysis Methods Evaluating Unfairness We employed the most widely used ML models in higher education, including Decision Tree (Hamoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2018), Random Forest (Pelaez, 2018), Logistic Regression (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2018), and SVM (Agaoglu, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Each 9 ML model has predefined parameters known as hyperparameters that must be provided before the training phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', depth of the tree in Decision Trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Since the optimal values of such hyperparameters are data-dependent, we performed a five-fold cross-validation (CV) for each model to determine the best set of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In this process the dataset was divided into five partitions, four of which were utilized for training and one for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Cross-validation repeats this process and selects a different partition for validation each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A grid of feasible hyperparameters was assessed based on the CV schema described above to choose the optimum set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Under 30 distinct random splits of training and testing datasets, we obtained the best set of hyperparameters before we performed model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To evaluate model performance, we report the average and variance of the accuracy as well as unfairness towards different racial/ethnic groups using various notions of unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Fairness Notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We consider four different conceptions of fairness commonly used in algorithmic fairness: statistical parity, equal opportunity, predictive equality, and equalized odds (Barocas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In practice, users can select the measure of fairness that is preferred based on context, knowledge of social disparities, use case, and regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We briefly describe each fairness notion in turn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' the probabilistic definitions of these notions appear in the supplemental materials (Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 10 Statistical Parity is achieved by having equal favorable outcomes (degree attainment) received by the unprivileged group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Black) and the privileged group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', White).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Said differently, under the notion of statistical parity, we consider a model fair if being a member of a racially minoritized group is not correlated with the probability of bachelor’s degree attainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The next three fairness measures build on the statistical notions of true/false positives/negatives (for a visual, see Confusion Matrix in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Specifically, • A true positive result would correctly predict success for a student who succeeds (in our case, attains a bachelor’s degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' • A true negative result would correctly predict failure for a student who does not succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' • A false positive result (Type I error) would incorrectly predict success for a student who does not succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' • A false negative result (Type II error) would incorrectly predict failure for a student who does succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Building on these statistical notions, Equal Opportunity represents equal false negative rates between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This fairness notion requires that each group receive the negative outcome at equal rates, conditional on their success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In other words, under this notion, the model should (incorrectly) predict failure for those who succeeded (attained at least a bachelor’s degree) at the same rate for 11 students across racial/ethnic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This notion assumes knowledge of the true outcome values (whether a student attained at least a bachelor’s degree) and aims to satisfy parity across socially relevant groups, subject to the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A third fairness notion is Predictive Equality, which represents equal false positive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To satisfy this criterion, positive predictions (that a given student will attain a bachelor’s degree) for students who do not actually attain a bachelor’s degree should be the same across racial/ethnic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Finally, Equalized Odds represents the average difference in false positive and true positive rates between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To achieve fairness under this notion, both the false positive rate (wrongly predicting success) and the true positive rate (correctly predicting success) should be the same across racial/ethnic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We use these notions to evaluate fairness in college-student success predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Mitigating Bias In addition to evaluating unfairness, we implement statistical techniques to mitigate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Literature on bias-mitigation techniques for ML models is burgeoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Such techniques can be categorized into three groups: preprocessing, in-processing, and post-processing approaches (Pessach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Preprocessing techniques entail fairness evaluation in the data-preparation step, which should, in turn, mitigate bias for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We apply two 12 preprocessing techniques: Reweighting (Kamiran & Calders, 2012) and Disparate Impact Remover (DIR) (Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Reweighting assigns different weights to the training samples in each combination of racial/ethnic group and outcome-variable class label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Black X outcome label=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' It does so before training a model to adjust the bias across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Because individual observations from the unprivileged groups with positive outcomes are underrepresented in the training data (see Table 1), classifiers are susceptible to bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In this preprocessing approach, the data points representing successful outcomes for unprivileged groups are identified and upweighted, so they have a larger influence on model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In contrast to Reweighting, DIR changes the distributions of other features in the model (not race/ethnicity) to force distributions to overlap at the group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This process removes the ability to distinguish between group membership from a feature that otherwise offers a good indication of which group a data point may belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In-processing techniques generally involve modifying the ML algorithms to account for fairness during the training process, such that the parameter estimation of the classifier forces the prediction outcome to be fair toward all (racial/ethnic) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The enforcement is accomplished in the optimization subproblem by adding a fairness metric as a constraint similar to the Exponentiated Gradient Descent approach (Agarwal, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 13 We also employ a second in-processing technique, Meta Fair Classifier (Celis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2018), which takes a large class of fairness metrics as inputs and returns an optimal classifier that is fair with respect to constraints on the given set of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This approach works for various fairness criteria and provides theoretical guarantees by developing a general form of constrained optimization problems, which encompasses many existing fair classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This stands in contrast to earlier work on fair classification, which focused on constructing classifiers that are constrained with respect to a single fairness metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Zafar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Post-processing techniques for mitigating bias adjust the prediction outcome after training a regular ML model, changing the values across different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We exclude these techniques since post-processing mechanisms are implemented at a later phase in the learning process, often producing inferior results (Woodworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These approaches are also more controversial than in-process and preprocess strategies in the domains of “affirmative action” and are thus less likely to be used in education settings (Hirschman & Bosk, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In the results section, we refer to Reweighting as ReW, Disparate Impact Remover as DIR, Exponentiated Gradient Reduction as ExGR, and Metaclassifier as MetaC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For comparison, we also consider the baseline classification scenario, where no mitigation strategy is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 14 Comparisons We use two comparison approaches to appraise model unfairness and test mitigation techniques, namely, at 1) the subgroup level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', each racial/ethnic group versus the rest), and 2) the aggregate level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', privileged versus unprivileged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' First, we compared each racial/ethnic group against all others and consider 1 for a certain group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Black) and 0 for every other group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', White, Asian, Hispanic, and Two or More Races) to calculate gaps as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To evaluate the limitations of aggregation, which is common in this type of work, we also aggregate White and Asian groups in the privileged category and Black, Hispanic, and Two or More Races (2+) groups in the unprivileged category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These comparisons represent an extension over prior work as they allow us to investigate the impact of existing mitigation techniques at both the subgroup and aggregate levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Most existing techniques only work with binary sensitive attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', “White” and “Non-White”), requiring the researcher to specify the privileged group and forcing other subgroups to be aggregated as the unprivileged group (Pessach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Although some existing unfairness mitigation techniques have the potential to incorporate non-binary sensitive attributes, such extension has not been implemented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Binarizing sensitive attributes (1: privileged, 0: unprivileged) for the mitigation processes may not reduce fairness gaps for 15 each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This is important in educational settings where research shows that students from different racial/ethnic groups have distinct experiences and outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', López et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hence, it is critical to evaluate unfairness after applying mitigation techniques at the subgroup levels, as there might be significant differences between unprivileged subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Results We find no significant difference between the performance (accuracy) of different ML classifiers, although there are some differences in levels of unfairness across fairness notions and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' To facilitate comparison, Figure 1 presents results for all ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We discuss the main findings for our assessments of unfairness and the effectiveness of bias-mitigation techniques in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' <
> Evaluating Unfairness Subgroup Level: Each Group Versus the Rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Figure 1 shows a comparison of unfairness levels using all four fairness notions and ML models for the baseline (without bias mitigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The testing accuracy across these models is 78%, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These results indicate that Black and Hispanic groups are treated unfairly across models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Generally, the SVM model yields less unfair results, across fairness notions, compared to the other ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Under the fairness notions of Statistical Parity (Figure 1a), Predictive Equality (Figure 1c), 16 and Equalized Odds (Figure 1d), the boxes for Black and Hispanic students are at a lower level across all ML models, indicating that these students receive favorable outcomes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', bachelor’s attainment or higher) at a lower rate than students in other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For the notion of Equal Opportunity (Figure 1b), higher levels in the box plots, which we observe for Black and Hispanic groups, represent more unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For a concrete example of unfairness with respect to Statistical Parity, in one of the test splits, students in the Asian and White categories have a 91% probability of attainment, while those in the Black and Hispanic categories have 63% and 68% probabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Without correcting for bias, predictive models will be more likely to predict that students categorized as Black and Hispanic are less likely to attain a bachelor’s degree or higher when compared to more privileged peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Findings for Predictive Equality further illustrate bias in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Among the students who did not complete their degree (y=0), the probability of attainment is estimated as 78% for White and 83% for Asian, while it is estimated as 33% for Hispanic, and 0% for Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 3 As illustrated in Figure 1b, the models are also more likely to falsely predict failure for Black and Hispanic students than for White and Asian students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Illustratively, for a single split, among the students who completed their degree (y=1), the probability of failure 17 is estimated as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='6% for White and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='5% for Asian, while it is estimated as 20% for Hispanic and 8% for Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Moreover, the plots show that the variation of values for the White and Asian groups is minimal, especially for the White group, whereas the variation of unfairness gaps for the other groups is significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Variation for the category of two or more races is especially large, suggesting this is not a meaningful category and should be used with caution in student-success prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Differences in variation across racial/ethnic groups indicate that models for minoritized groups are more sensitive to the train/test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Due to the population bias across different racial/ethnic groups in the ELS dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', statistical underrepresentation of Black and Hispanic students), the train/test splits can significantly change the distribution and presence of underrepresented individuals in each partition, significantly impacting the unfairness of the model for each split scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In practice, this will result in less stable and fair model performance for predicting the success of an unobserved individual from a statistically underrepresented group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' <
> Aggregate Level: Privileged vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Unprivileged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Figure 2 presents the box plots for all four unfairness notions at the aggregated level of privileged (Asian and White) versus unprivileged (Black, Hispanic, and two or more racial/ethnic categories) for all prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The first evident pattern from all four plots 18 is the mean difference between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Similar to results at the subgroup level, we observe higher false negative rates for the unprivileged group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In other words, the models are more likely to predict failure for Black and Hispanic students who succeed compared to White and Asian students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Comparing findings at the subgroup and aggregate levels, we observe that aggregate results mask substantial differences we can glean from the subgroup analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For instance, in Figure 1a, the DT and RF models show similar levels of unfairness for Black and Hispanic groups, but LR and SVM are more unfair for the Black group than the Hispanic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' At the aggregate level of analysis, this variation cannot be observed (all models are unfair toward the unprivileged group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We now turn to results for bias-mitigation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Mitigating Bias Given space constraints and for ease of interpretability, we present mitigation results using one predictive model, RF, which is a non-linear classifier commonly used in the education literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These results appear in Figure 3 (findings from other ML models are in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Our first observation is that the preprocessing and in-processing mitigation methods only minimally decrease accuracy (by 1% to 2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' One technique, MetaC, significantly improves accuracy (by 10-to-17-points over the baseline model without bias mitigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The bias-mitigation techniques we used required us to specify the privileged and unprivileged groups and to treat the sensitive attribute as binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 19 The results demonstrate that the mitigation techniques are generally not effective at reducing bias at the aggregate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' At the subgroup level, we do not find a mitigation technique that improves fairness across all racial/ethnic subgroups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' when a technique reduces unfairness for one subgroup, it harms another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We first present findings for the preprocessing mitigation techniques, ReW and DIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The results (in Figure 3) indicate that the ReW technique does not effectively reduce bias for unprivileged groups when compared to the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' If the goal of the education data analyst is to reduce unfairness in student- success predictions, it is not enough to increase the influence of datapoints that represent successful students from unprivileged groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Black students who succeed) in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This finding suggests that the underrepresentation of successful students from unprivileged groups in the training data is not a key source of bias in student-success predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The second preprocessing mitigation technique we employed, DIR, decreases unfairness for the Black group but leads to more unfair predictions for the Hispanic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This approach modifies the distributions of other features in the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', students’ native language and family composition) to reduce their correlation with racial/ethnic categorizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' A feature can provide a strong hint as to which group a data point might belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' DIR aims to eliminate this capacity to distinguish between group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In addition to reducing unfairness for the Black group, DIR diminishes the advantage of the Asian group 20 relative to that of other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' However, the advantage for the White group is actually exacerbated in two of the fairness notions (Statistical Parity and Equal Opportunity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Contrary to expectations, applying DIR increases the Equal Opportunity gap between the White group and all other groups, indicating a decrease in the number of successful White students who are falsely predicted to be unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Note that the DIR approach corrects the dataset measuring and considering the Statistical Parity notion at the aggregate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Hence, it is expected to observe equal proportions of positive prediction from each group at the aggregated level of privileged versus unprivileged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Our results show that DIR cannot effectively achieve statistical parity for each subgroup using ELS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Even at the aggregate level (Figure 4a), DIR slightly removes the advantage for the privileged group but does not improve fairness for the unprivileged group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These findings also highlight differences between two groups that are often considered privileged (Asian and White) and two groups that are often considered unprivileged (Black and Hispanic), underscoring the importance of disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Turning to the in-processing techniques, ExGR did not significantly alter the privilege of the White group or diminish the unfairness of the Black or Hispanic groups for any of the four notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Instead, both in-processing techniques (ExGR and MetaC) result in greater variation, which indicates that the 21 repaired model is less robust to data splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The MetaC technique effectively reduces all four types of biases for the Hispanic group but is more unfair for Black students with respect to Statistical Parity and Equalized Odds, again highlighting the need to disaggregate education data across racialized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' The results confirm that even at the aggregated level, unfairness is not mitigated significantly and the only technique that is slightly effective is MetaC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Even then, MetaC only works for Hispanic students and is ineffective at reducing bias for Black students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These preprocessing and in-processing techniques do not significantly reduce demographic bias, demonstrating the need for better bias- mitigation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Future work should examine bias-mitigation when both preprocessing and in-processing techniques are applied simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Discussion The ubiquity of predictive analytics in higher education demands greater attention to the “black box” of student-success-prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This work shows how such models produce unfair outcomes across various notions of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Further, we illustrate the limitations of existing techniques to reduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Using a nationally representative dataset with student-level data, we demonstrate that across notions of fairness and various common ML models, Black and Hispanic groups are treated unfairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Not only are they more likely to predict success for White and Asian groups (Statistical Parity) but they are also significantly more likely to predict failure for Black and Hispanic students who 22 succeeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This work illustrates how, without correcting for bias, Black and Hispanic students may be offered fewer opportunities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', admission) as a result of student-success prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' We also show how bias-mitigation techniques—both those that correct the dataset before modeling and those that apply fairness constraints in the modeling process—generally fail to improve fairness across subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' One such technique, Reweighting, increases the influence of observations representing racially minoritized students who are successful (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=', Black students who graduate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This technique is ineffective at reducing bias, indicating that the main source of bias is not statistical underrepresentation but underlying, unobservable sources of systemic and historical discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In evaluating student-success prediction models, it is important to understand the use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' While bias-agnostic models may reproduce social inequities in college-admissions use cases, they may lead to greater support for students when used to inform student-success interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Even then, practitioners must take care not to produce deficit narratives of minoritized students, treating them as though they have a lower likelihood of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Despite widespread perceptions that statistical analysis is independent of human judgment and error, this work demonstrates myriad decisions researchers must make that have significant consequences for fairness, including which ML model to use and which bias-mitigation techniques to employ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' For example, if a 23 predictive algorithm closes the gap between a comparison group while benefiting the majoritized group (rising tide metaphor), should such an algorithm be considered fair (Kizilcec & Lee, in press)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' As higher education institutions strive to better serve students by becoming more data-informed (Gagliardi & Turk, 2017), it is imperative that predictive models are designed with attention to their potential social consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' It is critical to be aware of historical discrimination embedded in the data and consider fairness measures to reduce bias in the outcomes of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' This paper demonstrates that more work is needed to develop fairness measures to reduce bias across racialized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Future research should also examine the influence of training/testing splits in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Another important avenue for future work is understanding how feature selection (which variables to include in the model) affects predictions and fairness across racialized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Such work could expand on existing and conflicting recommendations concerning the inclusion of race/ethnicity variables in student-success prediction models (Hu & Rangwala, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Finally, while we demonstrate the importance of disaggregating beyond privileged/unprivileged, the ELS categories are severely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' Future work should disaggregate further to lead us toward more racially just student-success practices in higher education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 24 Notes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In ML, a training dataset includes the data you use to train the model or algorithm to predict the outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' In all versions, we avoid imputing socially relevant (sensitive) attributes and outcome variables, hence observations with missing values for these variables are always dropped before imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' These estimated probabilities are based on RF modeling on a single train/test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 25 References Agaoglu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' (2016).' metadata={'source': 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mitigation techniques Equal Opportunity of SVM using bias mitigation techniques Equalized Odds of SVM using bias mitigation techniques Predictive Equality of SVM using bias mitigation techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='6 Baseline ReW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='4 DIR 0 ExGR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='8 MetaC 08 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='2 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='6 privileged unprivileged 7 Appendix C: Probabilistic Definitions of Fairness Notions Fairness Notion Formulation Statistical Parity (SP) |𝑃 (𝑌̂ = 1|𝑆 = 1) − 𝑃 (𝑌̂ = 1|𝑆 = 0)| Equal Opportunity (EoP) |𝑃 (𝑌̂ = 0|𝑌 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='𝑆 = 1) − 𝑃 (𝑌̂ = 0|𝑌 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 𝑆 = 0)| Predictive Equality (PE) |𝑃 (𝑌̂ = 1|𝑌 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 𝑆 = 1) − 𝑃 (𝑌̂ = 1|𝑌 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 𝑆 = 0)| Equalized Odds (EO) |𝑃 (𝑌̂ = 1|𝑌 = 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 𝑆 = 1) − 𝑃 (𝑌̂ = 1|𝑌 = 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 𝑆 = 0)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' ∀𝑦 ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content=' 1} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='Appendix D: Confusion Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='Predicted Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='True Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='𝑌̂ = 1 𝑌̂ = 0 𝑌 = 1 True Positive False Negative 𝑌 = 0 False Positive True Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQfRwdj/content/2301.03784v1.pdf'} +page_content='9 Appendix E: Plots with Mitigation for Alternative ML Models (No 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a/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/2301.00074v1.pdf.txt b/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/2301.00074v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..13044b3e648b2786a228de696db6c735329b0245 --- /dev/null +++ b/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/2301.00074v1.pdf.txt @@ -0,0 +1,2249 @@ +Matrix Multiplication: +Verifying Strong Uniquely Solvable Puzzles⋆ +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Department of Computer Science +Union College +Schenectady, New York, USA +{andersm2, jiz, xua}@union.edu +Abstract. Cohn and Umans proposed a framework for developing fast +matrix multiplication algorithms based on the embedding computation +in certain groups algebras [12]. In subsequent work with Kleinberg and +Szegedy, they connected this to the search for combinatorial objects +called strong uniquely solvable puzzles (strong USPs) [11]. We begin +a systematic computer-aided search for these objects. We develop and +implement constraint-based algorithms build on reductions to SAT and +IP to verify that puzzles are strong USPs, and to search for large strong +USPs. We produce tight bounds on the maximum size of a strong USP for +width k ≤ 5, construct puzzles of small width that are larger than previ- +ous work, and improve the upper bounds on strong USP size for k ≤ 12. +Although our work only deals with puzzles of small-constant width, the +strong USPs we find imply matrix multiplication algorithms that run +in O(nω) time with exponent ω ≤ 2.66. While our algorithms do not +beat the fastest algorithms, our work provides evidence and, perhaps, a +path to finding families of strong USPs that imply matrix multiplication +algorithms that are more efficient than those currently known. +Keywords: matrix multiplication · strong uniquely solvable puzzle · +arithmetic complexity · integer programming · satisfiability · satisfiability +benchmark · upper bounds · reduction · application +1 +Introduction +An optimal algorithm for matrix multiplication remains elusive despite substan- +tial effort. We focus on the square variant of the matrix multiplication problem, +i.e., given two n-by-n matrices A and B over a field F, the goal is to com- +pute the matrix product C = A × B. The outstanding open question is: How +many field operations are required to compute C? The long thought-optimal +na¨ıve algorithm based on the definition of matrix product is O(n3) time. The +groundbreaking work of Strassen showed that it can be done in time O(n2.808) +[30] using a divide-and-conquer approach. A long sequence of work concluding +with Coppersmith and Winograd’s algorithm (CW) reduced the running time +⋆ An extended abstract of this paper appeared in the Proceedings of SAT 2020 [5]. +arXiv:2301.00074v1 [cs.CC] 30 Dec 2022 + +2 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Fig. 1: The leftmost diagram is a width-4 size-5 puzzle P. The middle three diagrams are +the three sets of subrows of P. The rightmost diagram is the puzzle P ′ resulting from +reordering the subrows of P as indicated by the arrows and then recombining them. +Since P can be rearranged as P ′ ̸= P without overlap, P is not uniquely solvable. +to O(n2.376) [26,28,31,13]. Recent computer-aided refinements of CW by others +reduced the exponent to ω ≤ 2.3728639 [16,32,22]. +Approach Cohn and Umans [12] introduced a framework for developing faster +algorithms for matrix multiplication by reducing this to a search for groups +with subsets that satisfy an algebraic property called the triple-product property, +which allows matrix multiplication to be embedded in the group algebra. Their +approach takes inspiration from the O(n log n) algorithm for multiplying degree- +n univariate polynomials by embedding into the group algebra of the fast Fourier +transform, c.f., e.g., [14, Chapter 30]. Subsequent work [11] elaborated on this +idea and developed the notion of combinatorial objects called strong uniquely +solvable puzzles (strong USPs). These objects imply a group algebra embedding +for matrix multiplication, and hence give a matrix multiplication algorithm as +well. +A width-k puzzle P is a subset of {1, 2, 3}k, and the cardinality of P is the +puzzle’s size. Each element of P is called a row of P, and each row consists +of three subrows that are elements of {1, ∗}k, {2, ∗}k, {3, ∗}k respectively. In- +formally, a puzzle P is a uniquely solvable puzzle (USP) if there is no way to +permute the subrows of P to form a distinct puzzle P ′ without cells with num- +bers overlapping. Figure 1 demonstrates a puzzle that is not a USP. A uniquely +solvable puzzle is strong if a tighter condition for non-overlapping holds (see +Definition 3). For a fixed width k, the larger the size of a strong USP, the faster +matrix multiplication algorithm it gives [11]. In fact, Cohn et al. show that there +exist an infinite family of strong USPs that achieves ω < 2.48. +We follow Cohn et al.’s program by developing: (i) verification algorithms +and heuristics to determine whether a puzzle is a strong USP, (ii) search algo- +rithms to find large strong USPs, (iii) practical implementations1 of these +1 Source code available here: https://bitbucket.org/paraphase/matmult + +3232 +2 +2 +3 +3 +3232 +１132 +11 +2 +3 +1123 +1213 +1 +1 +2 +3 +1312 +3113 +11 +3 +3 +3113 +1 +321 +1 +1 +2 +3 +１231Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +3 +algorithms, and (iv) new upper bounds on the size of strong USPs. The most +successful of our verification algorithms work by reducing the problem through +3D matching to the satisfiability (SAT) and integer programming (IP) prob- +lems that are then solved with existing tools. The algorithms we develop are not +efficient—they run in worst-case exponential time in the natural parameters. +However, the goal is to find a sufficiently large strong USP that would provide +a faster matrix multiplication algorithm, and the resulting algorithm’s running +time is independent of the running time of our algorithms. The inefficiency of +our algorithms limit the search space that we can feasibly examine. +Results Our theoretical results and implementation produces new bounds on +the size of the largest strong USP for small-width puzzles. For small-constant +width, k ≤ 12, we beat the largest sizes of [11, Proposition 3.8]. Our lower +bounds on maximum size are witnessed by strong USPs we found via search. +For k ≤ 5 we give tight upper bounds determined by exhaustively searching all +puzzles after modding out common symmetries. For k ≤ 12, we improve the +upper bounds on the size of strong USPs. Although our current results do not +beat [11] for unbounded k, they give evidence that there may exist families of +strong USPs that give matrix multiplication algorithms that are more efficient +than those currently known. The best strong USP we can produce imply matrix +multiplication algorithms with ω ≤ 2.66. +We also create a benchmark data set of SAT/UNSAT instances based on our +reductions from strong-USP verification and examine the performance of solvers +from the 2021 SAT Competition [6]. +Related Work For background on algorithms matrix multiplication problem, +c.f, e.g., [9]. There are also a number of negative results known. Na¨ıvely, the +dimensions of the output matrix C implies that the problem requires at least +Ω(n2) time. Slightly better lower bounds are known in general and also for +specialized models of computation, c.f., e.g., [29,20]. There are also lower bounds +known for a variety of algorithmic approaches to matrix multiplication. Ambainis +et al. showed that the laser method cannot alone achieve an algorithm with +ω ≤ 2.3078 [4]. A recent breakthrough on arithmetic progressions in cap sets [15] +combined with a conditional result on the Erd¨os-Szemeredi sunflower conjecture +[3] imply that Cohn et al.’s strong USP approach cannot achieve ω = 2 + ϵ for +some ϵ > 0 [10]. Subsequent work has generalized this barrier [1,2] to a larger +class of algorithmic techniques. Despite this, we are unaware of a concrete lower +bound on ϵ implied by these negative results. There remains a substantial gap in +our understanding between what has been achieved by the positive refinements +of LeGall, Williams, and Stothers, and the impossibility of showing ω = 2 using +the strong USP approach. +Recently Fawzi et al. showed how reinforcement learning techniques can be +used to develop new matrix multiplication algorithms [17]. Their work produces +matrix multiplication algorithms with ω < 2.77, which is faster than Strassen’s + +4 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +original algorithm (ω < 2.81), but far from the refinements of Coppersmith- +Winograd (ω < 2.372) or the results achieved in this work. +Organization Section 2 begins with the formal definition of a strong USP and +the Cohn-Umans framework. Sections 3 & 4, respectively, discuss our algorithms +and heuristics for verifying that and searching for a puzzle that is a strong USP. +Section 5 describes several upper bounds on the size of strong USPs. Sections 6 +& 7 discuss our implementation and experimental results. +2 +Preliminaries +For an integer k, we use [k] to denote the set {1, 2, . . . , k}. For a set Q, SymQ de- +notes the symmetric group on the elements of Q, i.e., the group of permutations +acting on Q. Cohn et al. introduced the idea of a puzzle [11]. +Definition 1 (Puzzle). For s, k ∈ N, an (s, k)-puzzle is a subset P ⊆ [3]k with +|P| = s. We call s the size of P, and k the width of P. +We say that an (s, k)-puzzle has s rows and k columns. The columns of a puzzle +are inherently ordered and indexed by [k]. The rows of a puzzle have no inherent +ordering, however, it is often convenient to assume that they are ordered and +indexed by the set of natural numbers [s]. +Cohn et al. establish a particular combinatorial property of puzzles that +allows one to derive group algebras that matrix multiplication can be efficiently +embedded into. Such puzzles are called strong uniquely solvable puzzles. However, +to give some intuition we first explain a simpler version of the property called +uniquely solvable puzzles. +Definition 2 (Uniquely Solvable Puzzle (USP)). An (s, k)-puzzle P is +uniquely solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or +(ii) there exists r ∈ P and c ∈ [k] such that at least two of the following hold: +(π1(r))c = 1, (π2(r))c = 2, (π3(r))c = 3. +Informally, a puzzle is not uniquely solvable if each row of the puzzle can be +broken into ones, twos, and threes pieces and then the rows can be reassembled +in a different way so that each new row is a combination of a ones, a twos, and a +threes piece where there is exactly one element of [3] for each column. Observe +that uniquely solvable puzzles can have at most 2k rows because each ones piece, +twos piece, and threes piece must be unique, as otherwise the duplicate pieces +can be swapped making the puzzle not uniquely solvable. +The definition of strong uniquely solvable puzzle is below, it is nearly the same +except that it requires that there be a collision on a column between exactly two +pieces, not two or more pieces like in the original definition. +Definition 3 (Strong USP (SUSP)). An (s, k)-puzzle P is strong uniquely +solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or (ii) there exists +r ∈ P and c ∈ [k] such that exactly two of the following hold: (π1(r))c = 1, +(π2(r))c = 2, (π3(r))c = 3. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +5 +Finally, Cohn et al. defined a strengthening of SUSP which requires that every +triple of rows witness the necessary overlap. +Definition 4 (Local SUSP). A local strong uniquely solvable puzzle is an +(s, k)-puzzle where for each triple of rows u, v, w ∈ P with u, v, w not all equal, +there exists c ∈ [k] such that (uc, vc, wc) is an element of +L = {(1, 2, 1), (1, 2, 2), (1, 1, 3), (1, 3, 3), (2, 2, 3), (3, 2, 3)}. +Every SUSP P corresponds to a much larger local SUSP P ′, which, informally, +is the result of concatenating and duplicating the rows of P to explicitly demon- +strate the ∀π1, π2, π3 part of Definition 3. +Proposition 1 ([11, Proposition 6.3]). Let P be a (s, k)-SUSP, then there +is a local (s!, s · k)-SUSP P ′. +Note that in all of the definitions, local, strong, uniquely solvability is invariant +to the ordering of the rows of the puzzle, because P is a set—we use this fact +implicitly. +Cohn et al. show the following connection between the existence of strong +USPs and upper bounds on the exponent of matrix multiplication ω. +Lemma 1 ([11, Corollary 3.6]). Let ϵ > 0, if there is a strong uniquely solv- +able (s, k)-puzzle, there is an algorithm for multiplying n-by-n matrices in time +O(nω+ϵ) where +ω ≤ min +m∈N≥3 +� +3 log m +log(m − 1) − +3 log s! +s · k log(m − 1) +� +. +This result motivates the search for large strong USPs that would result in faster +algorithms for matrix multiplication. In the same article, the authors also demon- +strate the existence of an infinite family of strong uniquely solvable puzzles, for +width k divisible by three, that achieves a non-trivial bound on ω. +Lemma 2 ([11, Proposition 3.8]). There is an infinite family of strong uniquely +solvable puzzles that achieves ω < 2.48. +Finally, they conjecture that strong uniquely solvable puzzles provide a route to +achieving quadratic-time matrix multiplication. Unfortunately, as mentioned in +the introduction, this conjecture was shown to be false. +Lemma 3 ([10]). Strong uniquely solvable puzzles cannot show ω < 2 + ϵ, for +some ϵ > 0. +That said, there remains hope that the uniquely solvable puzzle approach could +beat the refinements of Coppersmith-Winograd even if it cannot reach ω = 2. + +6 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Algorithm 1 : Brute Force Verification +Input: An (s, k)-puzzle P. +Output: YES, if P is a strong USP and NO otherwise. +1: function VerifyBruteForce(P) +2: +for π2 ∈ SymP do +3: +for π3 ∈ SymP do +4: +if π2 ̸= 1 ∨ π3 ̸= 1 then +5: +found = false. +6: +for r ∈ P do +7: +for i ∈ [k] do +8: +if δri,1 + δ(π2(r))i,2 + δ(π3(r))i,3 = 2 then found = true. +9: +if not found then return NO. +10: +return YES. +3 +Verifying Strong USPs +The core focus of this article is the problem of verifying strong USPs, i.e., given +an (s, k)-puzzle P, output YES if P is a strong USP, and NO otherwise. In this +section we discuss the design of algorithms to solve this computational problem +as a function of the natural parameters s and k. +All of the exact algorithms we develop in this section have worst-case expo- +nential running time. However, asymptotic worst-case running time is not the +metric we are truly interested in. Rather we are interested in the practical per- +formance of our algorithms and their capability for locating new large strong +USPs. The algorithm that we ultimately develop is a hybrid of a number of +simpler algorithms and heuristics. +We begin by discussing a na¨ıve brute force algorithm based on the defini- +tion of strong USP (Subsection 3.1), see how it motivations a reduction to the +3D matching problem (Subsection 3.2), and then how we might formulate a re- +duction to the satisfiability and integer programming problems (Subsections 3.4 +& 3.5). We then describe several verification heuristics based on properties of +strong USP (Subsection 3.6) and combine them with the verification algorithms +to produce a hybrid algorithm Verify (Subsection 3.7). As we discuss in Sub- +section 7.2, our hybrid algorithm is quickly able to check whether a given puzzle +is a strong USP and aid in the search for strong USP. +3.1 +Brute Force +The obvious algorithm for verification comes directly from the definition of a +strong USP. Informally, we consider all ways of permuting the twos and threes +pieces relative to the ones pieces and check whether the non-overlapping con- +dition of Definition 3 is met. A formal description of the algorithm is found in +Algorithm 1. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +7 +The ones in Line 4 of Algorithm 1 denote the identity in SymP , and δa,b is +the Kronecker delta function which is one if a = b and zero otherwise. Observe +that Algorithm 1 does not refer to the π1 of Definition 3. This is because the +strong USP property is invariant to permutations of the rows and so π1 can be +thought of as an arbitrary phase. Hence, we fix π1 = 1 to simplify the algorithm. +Seeing that |SymP | = s!, we conclude that the algorithm runs in time O((s!)2 · +s · k · poly(s)) where the last factor accounts for the operations on permutations +of s elements. The dominant term in the running time is the contribution from +iterating over all pairs of permutations. Finally, notice that if P is a strong USP, +then the algorithm runs in time Θ((s!)2·s·k·poly(s)), and that if P is not a strong +USP the algorithm terminates early. The algorithm’s poor performance made it +unusable in our implementation, however, its simplicity and direct connection to +the definition made its implementation a valuable sanity check against later more +elaborate algorithms (and it served as effective onboarding to the undergraduate +students collaborating on this project). +Although Algorithm 1 performs poorly, examining the structure of a seem- +ingly trivial optimization leads to substantially more effective algorithms. Con- +sider the following function on triples of rows a, b, c ∈ P: f(a, b, c) = ∨i∈[k](δai,0+ +δbi,1+δci,2 = 2). We can replace the innermost loop in Lines 7 & 8 of Algorithm 1 +with the statement found = found ∨ f(r, π1(r), π2(r)). Observe that f neither +depends on P, r, nor the permutations, and that Algorithm 1 no longer depends +directly on k. To slightly speed up Algorithm 1 we can precompute and cache f +before the algorithm starts and then look up values as the algorithm runs. We +precompute f specialized to the rows in the puzzle P, and call it fP . +3.2 +Strong USP Verification to 3D Matching +It turns out to be more useful to work with fP than with P. It is convenient +to think of fP as a function fP : P × P × P → {0, 1} that is the complement +of the characteristic function of the relations of a tripartite hypergraph HP = +⟨P ⊔ P ⊔ P, ¯ +fP ⟩ where the vertex set is the disjoint union of three copies of P +and fP indicates the edges that are not present in HP . +Let H = ⟨P ⊔ P ⊔ P, E ⊆ P 3⟩ be a tripartite 3-hypergraph. We say H has +a 3D matching (3DM) iff there exists a subset M ⊆ E with |M| = |P| and for +all distinct edges e1, e2 ∈ M, e1 and e2 are vertex disjoint, i.e., e1 ∩ e2 = ∅. +Determining whether a hypergraph has a 3D matching is a well-known NP- +complete problem (c.f., e.g., [18]). We say that a 3D matching is non-trivial if +it is not the set {(r, r, r) | r ∈ P}. Figure 2 demonstrates a 3-hypergraph with a +non-trivial 3D matching. +The existence of non-trivial 3D matchings in HP is directly tied to whether +P is a strong USP. +Lemma 4. A puzzle P is a strong USP iff HP has no non-trivial 3D matching. +Proof. We first argue the reverse. Suppose that Hp has a non-trivial 3D matching +M. We show that P is not a strong USP by using M to construct π1, π2, π3 ∈ + +8 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Fig. 2: An example hypergraph G with edges E = {(r1, r1, r2), (r1, r3, r3), (r2, r2, r1), +(r2, r3, r1), (r3, r2, r3)}. The highlighted edges are a non-trivial 3D matching M = +{(r1, r1, r2), (r2, r3, r1), (r3, r2, r3)} of G. +SymP that witness this. Let π1 be the identity permutation. For each r ∈ P, +define π2(r) = q where (r, q, ∗) ∈ M. Note that q is well defined and unique +because M is 3D matching and so has vertex disjoint edges. Similarly define +π3(r) = q where (r, ∗, q) ∈ M. Observe that by construction +M = {(π1(r), π2(r), π3(r)) | r ∈ P}. +Since M is a matching of HP , M ⊆ ¯ +fP . Because M is a non-trivial matching at +least one edge in (a, b, c) ∈ M has either a ̸= b, a ̸= c, or b ̸= c. This implies, +respectively, that as constructed π1 ̸= π2, π1 ̸= π3, or π2 ̸= π3. In each case we +have determined that π1, π2, and π3 are not all identical. Thus we determined +permutations such that for all r ∈ P, f(π1(r), π2(r), π3(r)) = 0. This violates +Condition (ii) of Definition 3, hence P is not a strong USP. +The forward direction is symmetric. Suppose that P is not a strong USP. We +show that HP has a 3D matching. For P not to be a strong USP there must exist +π1, π2, π3 ∈ SymP not all identical such that Condition (ii) of Definition 3 fails. +Define e(r) = (π1(r), π2(r), π3(r)) and M = {e(r) | r ∈ P}. Since Condition (ii) +fails, we have that fP (e(r)) = false for all r ∈ P. This means that for all r ∈ P, +e(r) ∈ ¯ +fP and hence M ⊆ ¯ +fP . Since π1 is a permutation, |M| = |P|. Observe +that M is non-trivial because not all of the permutations are identical and there +must be some r ∈ P with e(r) having non-identical coordinates. Thus M is a +non-trivial 3D matching. +⊓⊔ +As a consequence of Definition 3, strong-USP verification is in coNP. Note +that although 3D matching is an NP-complete problem, Lemma 4 does not im- +mediately imply that verification of strong USPs is coNP-complete because HP +is not an arbitrary hypergraph. It remains open whether strong-USP verification +is coNP-complete. Lemma 4 implies that to verify P is a strong USP it suffices to +determine whether HP has a non-trivial 3D matching. In the subsequent subsec- +tions we examine algorithms for the later problem. We can, in retrospect, view +Algorithm 1 as an algorithm for solving 3D matching. +We note that the parameters s and k are not fully independent. First, s ≤ 3k +because the maximum number of rows in a puzzle of width k is |[3]k| = 3k. Sec- +ond, we eliminate the dependence on k entirely by transforming an (s, k)-puzzle + +GMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +9 +Algorithm 2 : Bidirectional Dynamic Programming Verification +Input: An (s, k)-puzzle P. +Output: YES, if P is a strong USP and NO otherwise. +1: function VerifyDynamicProgramming(P) +2: +Let T = ∅. +3: +Construct 3D matching instance HP . +4: +function SearchHalf(ℓ, Q, ℓQ, R, ℓR, δ, t) +5: +if ℓ = t then +6: +if δ = 1 then +▷ Forward Base Case +7: +Insert (Q, R) into T. +8: +return false. +9: +else +▷ Reverse Base Case +10: +if (P − Q, P − R) ∈ T then +11: +return true. +12: +else +13: +return false. +14: +res = false. +▷ Recursive Case +15: +for ℓ′ +Q = ℓQ + 1 to s do +16: +for ℓ′ +R = ℓR + 1 to s do +17: +if (pℓ, pℓ′ +Q, pℓ′ +R) ∈ HP ∧ ¬res then +18: +res = SearchHalf(ℓ + δ, Q ∪ {pℓ′ +Q}, ℓ′ +Q, R ∪ {pℓ′ +R}, ℓ′ +R, δ, t). +19: +return res. +20: +SearchHalf(1, ∅, 0, ∅, 0, 1, ⌊s/2⌋ + 1). +21: +return SearchHalf(s, ∅, 0, ∅, 0, −1, ⌊s/2⌋). +into a 3D matching instance on the vertex set [s]3. However, this transformation +is not without cost, because the size of HP is a function of the cube of s rather +than linear in the size of the puzzle s · k. +3.3 +Dynamic Programming +The realization that the verification of strong USPs is a specialization of 3D +matching leads to a dynamic programming algorithm for verification that runs +in linear-exponential time O(22spoly(s)+poly(s, k)). The reduction allows us to +replace the permutations from SymP with subsets of P and effectively reduce +the cost of the outer loops of Algorithm 1 from s! = Θ(2s log s) to 2s. +Algorithm 2 describes a recursive bidirectional dynamic programming al- +gorithm for strong-USP verification that uses the 3D matching instance. The +algorithm consists of two phases. Let t = ⌊s/2⌋. The first phase determines all +possible sets Q, R ⊆ P with |Q| = |R| = t such that there is 3D matching M1 +of HP when restricted to the vertices {p1, p2, . . . , pt} ⊔ Q ⊔ R. The sets Q, R +satisfying the requirement are stored in a table T during the first phase on Line +7. The second phase determines all possible sets Q, R ⊆ P with |Q| = |R| = s−t + +10 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +such that there is a 3D matching M2 of HP when restricted to the vertices +{pt+1, pt+2, . . . , ps} ⊔ Q ⊔ R. For each pair (Q, R) the algorithm considers in the +second phase, it checks whether (P − Q, P − R) was inserted into T during the +first phase. If the pair is present, it means that there is a 3D matching of HP +which is M = M1 ∪ M2. This works because, by Line 10, M1 and M2 are partial +3D matchings on {p1, . . . , pt} ⊔ (P − R) ⊔ (P − Q) and {pt+1, . . . ps} ⊔ R ⊔ Q, +respectively, which implies that M1 and M2 are vertex disjoint. The first phase +always returns false, which is ignored, and the second phase returns whether +a complete matching could be found, and, hence, by Lemma 4, whether P is a +strong USP. +The running time of this algorithm is dominated by the number of pairs of +sets (Q, R) it examines. Observe that rows of P are considered in order in Lines +15 & 16. Further, the algorithm tracks the index of the last elements added to Q +and R in ℓQ and ℓR, respectively. The algorithm only adds new elements to Q or +R that have higher indexes than ones previously added. Altogether this implies +that each pair of sets (Q, R) is only considered at most once during a phase. Since +Q, R ⊆ P, there are at most �t +i=0 +�s +i +� +· +�s +i +� +≤ (�t +i=0 +�s +i +� +)2 ≤ (2s)2 = 4s pairs +(Q, R). This means that SearchHalf is called at most 4s times during each +phase. Hence the running time of the algorithm is O(4s·s2·poly(s)+T3DM(s, k)) +where s2 factor comes from the inner loops, poly(s) time to manipulate the sets +and track the contents of T as a hash table, and T3DM(s, k) accounts for the +time to construct HP . The memory requirements of Algorithm 2 are similarly +high—the first phase uses O(4s · s) bits to store T. +Note that Algorithm 2 does not early terminate on P that are strong USP, +because it must search through all pairs before determining that none can be +found. The algorithm could be modified to allow early termination when P is +not a strong USP by causing the second phase of search to immediately return +in Line 18 once the first 3D matching witness has been located. However, this +still requires the first phase to run to completion. A remedy for this would be to +run both phases in parallel and have them check against each other. We chose +not to because it would substantially complicate the implementation and would +be unlikely to ultimately improve the performance of our combined algorithms. +For comparison, more advanced techniques like those of Bj¨orklund et al. can +achieve a better asymptotic time of O(2spoly(s)) [8]. We chose not to implement +their algorithm, because we judged that it would not substantially increase the +domain for which verification was possible. +3.4 +3D Matching to Satisfiability +By Lemma 4, one can determine whether a puzzle P is a strong USP by con- +structing the graph HP and deciding whether it has a non-trivial 3D matching. +Here we reduce our 3D matching problem to the satisfiability (SAT) problem on +conjunctive normal form (CNF) formulas and then use a state-of-the-art SAT +solver to resolve the reduced problem. To perform the reduction, we convert +the graph HP into a CNF formula ΨP , a depth-2 formula that is the AND of + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +11 +ORs of Boolean literals. We construct ΨP so that ΨP is satisfiable iff HP has a +non-trivial 3D matching. +Let HP = ⟨V = P ⊔ P ⊔ P, E ⊆ P 3⟩ be the 3D matching instance associated +with the puzzle P. Our goal is to determine whether there is a non-trivial 3D +matching M ⊆ E. A na¨ıve reduction would be to have variables Mu,v,w indicating +inclusion of each edge (u, v, w) ∈ P 3 in the matching. This results in a formula +ΨP with s3 variables and size Θ(s5) because including an edge e ∈ P 3 excludes +the Θ(s2) edges e′ with e∩e′ ̸= ∅. To decrease the size of ΨP we instead use sets of +variables to indicate which vertices in the second and third part of V are matched +with each vertex in the first part. In particular we have Boolean variables M 1 +u,v +and M 2 +u,w for all u, v, w ∈ P, and these variable map to assignments in the na¨ıve +scheme in the following way: M 1 +u,v ∧ M 2 +u,w ⇔ Mu,v,w. +We now write our CNF formula for 3D matching. First, we have clauses that +prevents non-edges from being in the matching: +Ψ non-edge +P += +� +(u,v,w)∈E +(¬M 1 +u,v ∨ ¬M 2 +u,w). +(1) +Second, we add clauses require that every vertex in HP is matched with some +edge: +Ψ ≥1 +P += +� � +u∈P +(∨v∈P M 1 +u,v) ∧ (∨w∈P M 2 +u,w) +� +∧ +� � +v∈P +(∨u∈P M 1 +u,v) +� +∧ +� � +w∈P +(∨u∈P M 2 +u,w) +� +. +(2) +Third, we require that each vertex be matched with at most one edge and so +have clauses that exclude matching edges that overlap on one or two coordinates. +Ψ ≤1 +P += +� +i∈{1,2} +� +(u,v),(u′,v′)∈P 2 +(u = u′ ∨ v = v′) ∧ (u, v ̸= u′, v′) ⇒ ¬M i +u,v ∨ ¬M i +u′,v′. +(3) +Fourth, we exclude the trivial 3D matching by requiring that at least one of the +diagonal edges not be used: Ψ non-trivial +P += � +u∈P ¬M 1 +u,u∨¬M 2 +u,u. Finally, we AND +these into the overall CNF formula: ΨP = Ψ non-edge +P +∧ Ψ ≤1 +P +∧ Ψ ≥1 +P +∧ Ψ non-trivial +P +. +The size of the CNF formula ΨP is Θ(s3), has 2s2 variables, and is a factor of s2 +smaller than the na¨ıve approach. Thus we reduce 3D matching to satisfiability +by converting the instance HP into the CNF formula ΨP . +3.5 +3D Matching to Integer Programming +In parallel to the previous subsection, we use the connection between verifica- +tion of strong USPs and 3D matching to reduce the former to integer program- +ming, another well-known NP-complete problem (c.f., e.g., [21]) and then apply + +12 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +a state-of-the-art solver to resolve it. Again, let HP = ⟨V, E⟩ be the 3D match- +ing instance associated with P. We construct an integer program QP over {0, 1} +that is infeasible iff P is a strong USP. Here the reduction is simpler than the +previous one because linear constraints naturally capture matching. +We use Mu,v,w to denote a variable with values in {0, 1} to indicate whether +the edge (u, v, w) ∈ P 3 is present in the matching. To ensure that M is a subset +of E we add the following edge constraints to QP : ∀u, v, w ∈ P, ∀(u, v, w) ̸∈ +E, Mu,v,w = 0. We also require that each vertex in each of the three parts +of the graph is incident to exactly one edge in M. This is captured by the +following vertex constraints in QP : ∀w ∈ P, � +u,v∈P Mu,v,w = � +u,v∈P Mu,w,v = +� +u,v∈P Mw,u,v = 1. Lastly, since we need that the 3D matching be non-trivial +we add the constraint: � +u∈P Mu,u,u < |P|. +To check whether P is a strong USP we determine whether QP is not feasible, +i.e., that no assignment to the variables M satisfy all constraints. We note that +reduction from 3D matching to IP is polynomial time and that there are s3 +variables in QP , and that the total size of the constraints is s3 · Θ(1) + 3s · +Θ(s2) + 1 · Θ(s3) = Θ(s3), similar to size of ΨP in the SAT reduction. +3.6 +Heuristics +Although the exact algorithms presented in the previous sections make sub- +stantial improvements over the brute force approach, the resulting performance +remains impractical. To resolve this, we also develop several fast verification +heuristics that may produce the non-definitive answer MAYBE in place of YES +or NO. Then, to verify a puzzle P we run this battery of fast heuristics and +return early if any of the heuristics produce a definitive YES or NO. When all of +the heuristics result in MAYBE, we then run one of the slower exact algorithms +that were previously discussed. The heuristics have different forms, but all rely +on the structural properties of strong uniquely solvable puzzles. +Downward Closure The simplest heuristics we consider is based on the fact +that strong USPs are downward closed. +Lemma 5. If P is a strong USP, then so is every subpuzzle P ′ ⊆ P. +Proof. Let P be a strong USP and P ′ ⊆ P. By Definition 3, for every (π1, π2, π3) ∈ +Sym3 +P not all identity, there exist r ∈ P and i ∈ [k] such that exactly two of the +following hold: (π1(r))i = 1, (π2(r))i = 2, (π3(r))i = 3. Consider restricting the +permutations to those that fix the elements of P\P ′. For these permutations it +must be the case that r ∈ P ′ because otherwise r ∈ P\P ′ and there is exactly +one j ∈ [3] for which (πj(r))i = j holds. Thus we can drop the elements of P\P ′ +and conclude that for every tuple of permutations in SymP ′ the conditions of +Definition 3 hold for P ′, and hence that P ′ is a strong USP. +⊓⊔ +This leads to a polynomial-time heuristic that can determine that a puzzle is +not a strong USP. Informally, the algorithm takes an (s, k)-puzzle P and s′ ≤ s, + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +13 +Algorithm 3 : Downward-Closure Heuristic +Input: An (s, k)-puzzle P, and size s′ ≤ s. +Output: NO, if P has a set of s′ rows that do not form a strong USP, and +MAYBE otherwise. +1: function HeuristicDownwardClosed(P, s′) +2: +for P ′ ⊆ P, |P ′| = s′ do +3: +if P ′ is not a strong USP then return NO. +4: +return MAYBE. +and verifies that all subsets P ′ ⊆ P with size |P ′| = s′ are strong USPs. If any +subset P ′ is not a strong USP, the heuristic returns NO, and otherwise it returns +MAYBE. For completeness, this algorithm is described in Algorithm 3. +This algorithm runs in time O( +� s +s′ +� +·T(s′, k)) where T(s′, k) is the runtime for +verifying an (s′, k)-puzzle. In practice we did not apply this heuristic for s′ larger +than 3. When s′ is some constant d, the running time becomes O(sd · T(d, k)) = +O(sdk) using the brute force algorithm (Algorithm 1) for verification of the +puzzle P ′. +Unique Pieces Every strong uniquely solvable puzzle is a uniquely solvable +puzzle. A necessary condition for a puzzle to be a USP is that for each element +in [3], the collection of subrows contains no duplicates. +Lemma 6 (Implicit in [11]). If P is a USP, then for all e ∈ [3], and distinct +rows r1, r2 ∈ P, there is a column c ∈ [k] were one of the rows r1 or r2 has an +e and the other one does not. +Proof. Suppose, for the sake of contradiction, that this is not the case, and dis- +tinct rows r1, r2 ∈ P have e in exactly the same columns for some e ∈ [3]. We +show that P is not a USP. Choose πe = (r1r2), i.e., the permutations that trans- +poses the subrows for e in rows r1 and r2. Choose the other two permutations +for the elements of [3]\{e} to be the identity. Since the permutations are not all +the identity, the second half of Definition 2 applies. However, the puzzle that +results from the permutations is identical to P and for all c ∈ [k] and each row +r ∈ P there exists exactly on i ∈ [3] where (πi(r))c = i. Hence the definition of +uniquely solvable is not satisfied and we have a contradiction. +⊓⊔ +Note that the reverse direction of Lemma 6 does not hold. The puzzle in Figure 1 +is an example of this: It is not uniquely solvable, but the subrows for each element +are distinct. +We can make Lemma 6 effective as via a linear-time heuristic capable of +ruling out puzzles that are not (strong) USPs. Although straightforward, for +completeness we formalize our approach in Algorithm 4. When the sets are +implemented as hash tables, the expected running time of this algorithm is O(s· +k) time, which is linear in the size of the puzzle P. An alternative worst-case +O(s · k) time implementation uses radix sort to sort the characteristic sequences + +14 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Algorithm 4 : Unique Pieces Heuristic +Input: An (s, k)-puzzle P. +Output: NO, if a witness is found for P not being a (strong) USP, and MAYBE +otherwise. +1: function HeuristicUniquePieces(P) +2: +Initialize empty sets S1, S2, S3. +3: +for r ∈ P do +4: +for e ∈ [3] do +5: +Let h = {c ∈ [k] | rc = e}. +6: +if h ∈ Se then return NO. +7: +Se = Se ∪ {h}. +8: +return MAYBE. +of the subrows as binary numbers and then scans adjacent rows to to detect +duplication. +The unique pieces heuristic is equivalent to the downward-closure heuristic +for subpuzzles of size two. +Lemma 7. Let P be an (s, k)-puzzle, then HeuristicUniquePieces(P) = +HeuristicDownwardClosed(P, 2). +Proof. We show both directions. +Suppose that P fails the unique pieces heuristic for, w.l.o.g., e = 1, then there +are distinct rows r1, r2 ∈ P where the cells that contain 1 are all in the same +columns. This means we can swap those 1’s subrows without causing overlap +or changing the puzzle. This implies that P ′ = {r1, r2} is not a (strong) USP. +Since |P ′| = 2 and P ′ ⊆ P, the downward closure heuristic for s′ = 2 will also +conclude that P is not a (strong) USP. +Suppose that P fails the downward-closure heuristic for s′ = 2. Then there +is a pair of distinct rows r1, r2 ∈ P for which P ′ = {r1, r2} is not a strong +USP. Suppose there is no columns were r1 and r2 differ, then the subrows of +r1, r2 are the same for all elements, and so P fails the unique pieces heuristic. +For the other case, suppose there is a least one column c ∈ [k] where r1 and r2 +differ. W.l.o.g., let that column be ((r1)c, (r2)c) = (1, 2). Because P ′ is not an +SUSP and this column is (1, 2), there can be other no columns that are in from +the set {(1, 3), (2, 3), (3, 2), (3, 1)} otherwise they would form an SUSP with the +column (1, 2). This means the only columns that P ′ contains are from the set +{(1, 2), (2, 1), (1, 1), (2, 2), (3, 3)}. Therefore, the columns which contain 2 must +match and the subrows for 2 in r1 and r2 are identical. Thus, P ′, and so P, fails +the unique pieces heuristic. +⊓⊔ +A corollary of this proof is that for size-two puzzles, every USP is also a strong +USP. +Corollary 1. Let P be a (2, k)-puzzle, if P is a uniquely solvable puzzle, then +P is a strong uniquely solvable puzzle. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +15 +Since the unique pieces heuristic is equivalent to the downward-closure heuristic +for s′ = 2 and the running time of unique pieces is linear in the puzzle size, +O(s·k), and the running time of downward closed is O(s2 ·k), we use the unique +pieces heuristic in place of downward closed for s′ = 2. +Greedy This heuristic attempts take advantage of Lemma 4 and greedily search +for a 3D matching for the instance HP . The heuristic proceeds iteratively, de- +termining the vertex of the first part of the 3D matching instance with the +least edges and randomly selecting an edge of that vertex to put into the 3D +matching. If the heuristic successfully constructs a 3D matching it returns NO +indicating that the input puzzle P is not a strong USP. If the heuristic reaches a +point were prior commitments have made the matching infeasible, the heuristic +starts again from scratch. This process is repeated some number of times be- +fore it gives up and returns MAYBE. In our implementation we use s2 attempts +because it is similar to the running time of the reductions and it empirically re- +duced the number of instances requiring full verification in the domain of puzzles +with k = 6, 7, 8 while not increasing the running time by too much. The greedy +heuristic is formalized in Algorithm 5. +The array cts is used to store the number of edges cts[u] that remain associ- +ated with vertex u along the first coordinate. Much of the algorithm is devoted +to maintaining this invariant. The sets U, V, W store the vertices along the three +coordinates, respectively, that have already been incorporated into the partial +3D matching. Like in Algorithm 2 we do not store the matching itself, only the +vertices involved. The break at Line 10 triggers when the partial 3D matching +is a dead end and cannot be extended into a full 3D matching. The condition +of Line 23 is true when a full 3D matching has been constructed and causes the +algorithm to return that P is not a strong USP. +The running time of this algorithm is O(s3t+T3DM(s, k)), where T3DM(s, k) +is the time required to construct 3D matching instances from (s, k)-puzzles. +This algorithm has the potential to be considerably slower than the downward- +closure heuristic, and in practice we set t = s2. However, the main loop can +terminate early at Line 10 when it fails to extend the 3D matching, this permits +the expected time to much less than the worst case. For a puzzle P that is a +strong USP, the heuristic takes the full Ω(s3t + T3DM(s, k)) time. +Compared to the downward-closure and unique pieces heuristics this heuristic +is much less efficient. As a result we only run it when when the other heuristics +have failed. See Subsection 7.2 for a comparison of effectiveness these heuristics +in our experiments. +3.7 +Hybrid Algorithm +Our final verification algorithm (Algorithm 6) is a hybrid of several exact al- +gorithms and heuristics. The size thresholds for which algorithm and heuristic +to apply were determined experimentally for small k and are focused on the +values where our strong USP search algorithms are tractable k ≤ 6 (or nearly + +16 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Algorithm 5 : Greedy Heuristic +Input: An (s, k)-puzzle P, and iteration bound t. +Output: NO, if a witness is found for P not being a strong USP, and MAYBE +otherwise. +1: function HeuristicGreedy(P) +2: +Construct 3D matching instance HP . +3: +for i = 1 to t do +4: +for u ∈ P do +5: +cts[u] = � +v,w∈P HP (u, v, w). +▷ Number of edges incident vertex u. +6: +Let U, V, W = ∅. +7: +Let m = 0. +▷ Number of edges in matching. +8: +while m < s do +9: +Select u ∈ {w ∈ ¯U | cts[w] = maxv∈ ¯U cts[v]} uniformly at random. +10: +if cts[u] = 0 then break. +11: +Let D = {(v, w) ∈ ¯V × ¯W | HP (u, v, w) = 1}. +12: +Select (v, w) ∈ D uniformly at random. +13: +for v′ ∈ P do +▷ Update edge counts. +14: +for w′ ∈ P do +15: +if (v′, w′) ∈ ¯V × ¯W and HP (u, v′, w′) = 1 then +16: +cts[u]--. +17: +if (v′, w′) ∈ ¯U × ¯W and HP (v′, v, w′) = 1 and v′ ̸= u then +18: +cts[v′]--. +19: +if (v′, w′) ∈ ¯U × ¯V and HP (v′, w′, w) = 1 and v′ ̸∈ {u, v} then +20: +cts[v′]--. +21: +U, V, W = U ∪ {u}, V ∪ {v}, W ∪ {w}. +▷ Add edge to matching. +22: +m = m + 1. +23: +if m ≥ s then return NO. +▷ 3D matching found so not SUSP, halt. +24: +return MAYBE. +tractable k ≤ 8). We decide to run both of the reductions to SAT and IP in +parallel because it is not clear which algorithm performs better in general. Since +verification halts when either algorithm completes, the wasted effort is within a +factor of two of what the better algorithm could have done alone. We also chose +to do this because we experimentally observed that there were many instances +that one of the algorithms struggled with that the other did not—this resulted +in a hybrid algorithm that out performed the individual exact algorithms on +average. We show in Subsection 7.2 that our hybrid algorithm and heuristics +perform well in practice at quickly verifying strong USPs for small width k. Fur- +ther, Subsection 7.3 contains a discussion of the relative performance of the SAT +and IP approaches on different instance types from our benchmark experiments. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +17 +Algorithm 6 : Hybrid Verification +Input: An (s, k)-puzzle P. +Output: YES, if P is a strong USP, and NO otherwise. +1: function Verify(P) +2: +if s ≤ 2 then return VerifyBruteForce(P). +3: +Return result if HeuristicUniquePieces(P) is not MAYBE. +4: +if s ≤ 7 then return VerifyDynamicProgramming(P). +5: +Return result if HeuristicDownwardClosed(P, 3) is not MAYBE. +6: +Return result if HeuristicGreedy(P) is not MAYBE. +7: +Run VerifySAT(P) and VerifyIP(P) in parallel and return first result. +4 +Searching for Strong USPs +With a practical verification algorithm in hand, we consider the problem of +searching for large strong USPs. Because the set of strong USPs is downward +closed, a natural search strategy is: Start with the empty set and repeatedly +consider adding rows while maintaining the strong-USP property. However, while +this strategy will lead to a maximal-size strong USP, it is not guaranteed to +produce a maximum-size strong USP. This is because the set of strong USPs +does not form a matroid, rather it is only an independence system (c.f., e.g., +[25]). +In particular, (i) the empty puzzle is a strong USP and (ii) the set of strong +USP are downward closed by Lemma 5. The final property required to be a +matroid, the augmentation property, requires that for every pair of strong USPs +P1, P2 with |P1| ≤ |P2| there is a row of r ∈ P2\P1 such that P1 ∪ {r} is also a +strong USP. For a simple counterexample consider the strong USPs P1 = {32} +and P2 = {12, 23}. Using Lemma 6, we see that neither P1 ∪ {12} = {12, 32} +nor P1 ∪{23} = {23, 32} are strong USPs, and hence the augmentation property +fails. One consequence is that na¨ıve greedy algorithms will likely be ineffective +for finding maximum-size strong USPs. Furthermore, we do not currently know +of an efficient algorithm that can take a strong USP P and determine a row r +such that P ∪ {r} is a strong USP. +Despite that, we have had some success in applying general-purpose tree- +search techniques with pruning based on the symmetries of strong USPs together +with our practical verification algorithm to construct maximum-size strong USPs +for small k. +4.1 +Puzzle Symmetry +Since puzzles are defined as sets of rows, the ordering of the rows of a puzzle +P does not affect the SUSP property. Similarly, but slightly less obviously, the +SUSP property is invariant to reordering the columns of the puzzle, because the +required existential condition ∃c ∈ [k] st. (...) from Definition 3 is independent + +18 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +of the ordering of the columns. Lastly, the alphabet [3] typically used to repre- +sent the elements of a puzzle is completely arbitrary, any set of three distinct +values would suffice. These values are not interpreted mathematically, aside from +their convenience in expressing the SUSP definition concisely. This logic can be +formalized into the following lemma. +Lemma 8. Let ρ ∈ Sym[k], δ ∈ Sym[3]. A (s, k)-puzzle P is a strong USP iff +{(δ(rρ(c)))c∈[k] | r ∈ P} is a strong USP. +Proof. Follows immediately from Definition 1 and Definition 3. +⊓⊔ +This lemma implies that the SUSP property is invariant with respect to these +kinds of puzzle transformations. We call two puzzles P, P ′ that are related in this +way isomorphic, and use the notation P ∼= P ′ to denote this. The relation ∼= is +an equivalence relation, because permutations are invertable, and so it partitions +the set of puzzles into equivalence classes. +This notion of isomorphism is naturally related to the same notion in graphs. +For each (s, k)-puzzle P we can define a colored, undirected graph GP . This +graph consists of vertices that are partitioned into four sets of different colors: +V = {rowr}r∈[s]⊔{colc}c∈[k]⊔{ei}i∈[3]⊔{vr,c}(r,c)∈[s]×[k]. There are s+k+3+s·k +vertices in GP . The first three parts are vertices representing the rows and +columns of P, and the elements of [3], respectively, and the fourth part are +vertices for each of the s·k cells in the P. The edge relation of GP is straightfor- +ward: Each vertex vr,c is connected to three vertices corresponding to the row, +columns and element that the cell indexed (r, c) contains in P. In particular, the +three edges attached to vr,c are (vr,c, rowr), (vr,c, colc), (vr,c, eltP (r,c)). In total, +GP has 3·s·k edges. Because the vertex sets for rows, columns, and elements are +each uniquely colored and each cell of P is connected to vertices representing its +row, column, and element, the automorphisms of GP are in 1-1 correspondence +to the automorphisms of P under permutations of rows, columns, and elements. +This implies that for two (s, k)-puzzles P, P ′, if GP ∼= GP ′ then there exists per- +mutations of the rows, columns, and elements of P which results in P ′. Further +by Lemma 8, if GP ∼= GP ′, then P ∼= P ′, and P is an SUSP iff P ′ is an SUSP. +4.2 +Symmetry-Pruned Tree Search +A natural way to search for strong USPs is based on breadth-first search and +uses the fact that strong USP are downward closed (Lemma 5): To find the +largest possible width-k strong USP, (i) start with all possible first rows – the 3k +(1, k)-puzzles, (ii) attempt to extend the resulting puzzles with all possible rows +keeping only the puzzles that are strong USPs and which are not isomorphic to +the strong USPs that have been seen before to form the new search frontier, and +(iii) repeat Step (ii) until the search frontier is empty. +To ensure the algorithm does not revisit isomorphic puzzles, we use canonical +graph representations [Gp] of the puzzle graphs GP . A canonical graph represen- +tation is a binary encoding of a graph with the property that for any two graphs +G1, G2, [G1] = [G2] iff G1 ∼= G2 (c.f., e.g., [24]). As the search algorithm runs we + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +19 +Algorithm 7 : Symmetry-Pruned Breadth-First Search +Input: An integer k ≥ 0. +Output: The number b, which is the size of the largest width-k strong USP. +1: function SP-BFS(k) +2: +Let Q be an empty queue. +3: +Let I be an empty set. +4: +Let b = 0. +5: +enqueue(Q, ∅). +6: +while Q is not empty do +7: +P = dequeue(Q). +8: +for r ∈ [3]k\P do +9: +Let P ′ = P ∪ {r}. +10: +if Verify(P ′) and [G′ +P ] ̸∈ I then +11: +enqueue(Q, P ′). +12: +I = I ∪ {[G′ +P ]}. +13: +b = |P ′|. +14: +return b. +record the set I of canonical graph representations [GP ] of each distinct puzzle +P that has been added to the search frontier. Each time a puzzle P ′ is consid- +ered for being added to the search frontier we first check whether its canonical +graph representation [GP ′] ∈ I, if it is, we do not add P ′ to the frontier. The use +of canonical representations of puzzles dramatically shrinks the search space by +searching from [P] rather than every P ′ ∼= P and by not allowing duplicates of +[P] to be enqueued. This algorithm SP-BFS is formalized in Algorithm 7. +We argue the correctness of this algorithm. +Lemma 9. For k ∈ N, SP-BFS(k) returns the maximum integer s for which +there exists an (s, k)-SUSP. +Proof. Ignoring the pruning that I performs for a moment, it is routine to argue +that SP-BFS behaves like a generic breadth-first search algorithm over the tree of +all strong USPs. This is because of the downward-closure property of strong USP +(Lemma 5), which makes any strong USP P reachable from the trivial strong +USP ∅ using a series of row inclusions. SP-BFS(k) results in an exhaustive +search of all strong USPs of width k and return the maximum size b of such +SUSPs. +We argue that when considering the pruning that I contributes to, SP- +BFS(k) enqueues exactly one element of each equivalence class of puzzles that +are SUSPs. Then, as a consequence of Lemma 8, the algorithm must explore ev- +ery equivalence class of width-k SUSPs. Hence, it explores an equivalence class +with SUSPs of maximum size and subsequently returns that size, which is the +expected output. +To complete the argument and show that the symmetry pruned search covers +the entire search space of equivalence classes, suppose, for the sake of contradic- + +20 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +tion, that there is some smallest s such that there is an (s, k)-puzzle P that does +not have its equivalence class [P] searched. We know that s > 1, because the +algorithm starts by considering all possible (1, k)-puzzles. Let P ′ be the (s−1, k)- +puzzle created from P by removing one of its rows r, P ′ has as least one row +because s > 1. By hypothesis, the equivalence class of [P ′] has been visited by +SP-BFS because P ′’s size is s − 1 < s. Consider [P] and remove the row that +corresponded to r to form [P]′. It must be the case that [P ′] ∼= [P]′. This isomor- +phism extends to [P] in that there must be a row r′ such that ([P ′]∪{r′}) ∼= [P], +where r′ is replaces the row remove from [P]. Therefore, since [P ′] is searched, +the algorithm must consider all possible rows to extend by, including r′. This is +means that the equivalence class of [P] is searched, a contradicting our assump- +tion. Therefore every equivalence class of SUSPs is searched by SP-BFS. +⊓⊔ +This approach reduces the size of the search space, improving both the run- +ning time of the search and the space required to keep track of the frontier puz- +zles. The worst case running time of SP-BFS is O(3k·#EQUIV (k)·(TVerify(sk+ +1, k)+TCanonize(sk, k)), where #EQUIV (k) is the number equivalence classes of +strong USP of width k, TVerify(sk + 1, k) is the time to verify the maximum size +(sk +1, k)-puzzles examined by the algorithm, and TCanonize(sk, k) is the time to +compute the canonical graph representation of each puzzle P considered by the +algorithm (assuming TVerify and TCanonize are monotone in their parameters). +See Subsection 7.1 for the experimental results of running SP-BFS and a +discussion of implementation issues. +5 +Upper Bounds +Although the main focus of this research line is to construct sufficiently large +strong USP that would imply faster matrix multiplication algorithms, our tech- +niques and approach can also be applied to search for tighter upper bounds on +the size of strong USP. We describe several SUSP-size upper bounds in this +section. +ω Bound. Prior work explicitly discusses bounds on the capacity of infinite +families of USP (c.f., [11, Lemma 3.2, Theorem 3.3]). Since every SUSP is a +USP, these bounds also apply to SUSP and can be restated to apply to individual +puzzles. The first bound, which we denote as the “ω bound”, results from (i) +Lemma 1, which is monotone non-increasing for fixed k, and (ii) the fact that ω ≥ +2. To compute this bound we evaluate the inequality of Lemma 1 on increasingly +large s until just before the consequence implies ω < 2 which is in contradiction +with ω ≥ 2. +Unique Pieces Bound. The second bound, which we denote as the “unique pieces +bound”, following directly from Lemma 6. Since that lemma requires that each +row of a (strong) USP have a unique ones, twos, and threes piece, the total +number of rows in a strong USP cannot be more than 2k. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +21 +USP Bound. The third bound, which we denote as the “USP bound”, results +from the proof of [11, Lemma 3.2]. Although not spelled out in that article, the +proof relies on the following subclaim that directly bounds s as a function of k. +Proposition 2. Let P be a (s, k)-USP, then +s ≤ +k +� +c1=0 +k−c1 +� +c2=0 +min +�� k +c1 +� +, +� k +c2 +� +, +� +k +k − (c1 + c2) +�� += O +� +k2 · +� 3 +22/3 +�k� +. +Note that the USP bound is asymptotically tighter than the unique pieces bound +as +3 +22/3 ≈ 1.8899 < 2. +Clique Bound. The fourth bound, which we denote as the “clique bound”, results +from the fact that SUSPs are downward closed (Lemma 5). In particular if P +is an SUSP, then for every P ′ ⊆ P with 2 rows must also be an SUSP. Fix +k ∈ N and consider a graph Gk whose vertices correspond to the possible rows +of a width-k puzzle, i.e., strings in [3]k, and where there is an edge between +r1, r2 ∈ [3]k if {r1, r2} is an SUSP. Observe that by downward closure, each +(s, k)-SUSP corresponds to a clique of size s in Gk. This approach naturally +generalizes from the Clique problem to h-HypergraphClique problem where the +graph Gh +k consists the same 3k vertices as Gk = G2 +k, but instead has the arity-h +edges {r1, r2, . . . , rh} which are (h, k)-SUSPs. +Proposition 3. Let P be an (s, k)-SUSP and 2 ≤ h ≤ s. Then for +Gh +k = ⟨V = [3]k, E = {P ′ ⊆ V | P ′ is a strong USP and |P ′| = h}⟩, +(Gh +k, s) ∈ h-HypergraphClique. +Therefore, the size of a maximum hypergraph clique in Gh +k is an upper bound of +size of width-k SUSP. We use “clique bound” to denote the specific instantiation +of this bound for h = 2. +Exhaustive Bound. For fifth bound, which we denote as the “exhaustive bound”, +we consider the results of Algorithm 7 when run in the domain of k where the +full search space can be feasibly explored. Because these bounds are based on +exhaustive search they are inherently tight. +Downward-Closure Bound. The final bound we consider follows from the downward- +closure property of SUSPs. +Proposition 4. Let P be an (s, k)-SUSP with k > 1, then there exists an +(⌈ s +3⌉, k − 1)-SUSP. +Proof. Fix any c ∈ [k] and consider the cth column of P, then, by averaging, +there must be an element of e ∈ [3] that appears at least ⌈ s +3⌉ times in that +column. Let P ′ ⊂ P be the subpuzzle of P whose rows have e in the cth column. +P ′ is a strong USP, because P is a strong USP and strong USPs are downward + +22 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +closed (Lemma 5). Form P ′′ by removing the cth column of P ′. P ′′ is a strong +USP, because P ′ is a strong USP and the strong-USP property is invariant to +addition or removal of constant columns. By construction, P ′′ is a (⌈ s +3⌉, k − 1)- +SUSP. +⊓⊔ +This bound is not as independently applicable like the others, but it can lift +upper bounds of s ≤ u at k to s ≤ 3u at k + 1. +See Subsection 7.1 for the results of evaluating the above bounds for small +width and a discussion of issues involved in concretely calculating them. +6 +Implementation +We implemented our verification algorithms, heuristics, and search algorithms, +along with various utilities and appropriate datastructures to represent under- +lying information such as puzzles in C++. The source code for our implementa- +tion is available under a MIT License at https://bitbucket.org/paraphase/ +matmult. +We use a number of external libraries with subroutines that are key to the +functioning of our algorithms. Our IP-based verifier and Clique bound calcula- +tor both use the commercial, closed-source mixed-integer programming solver +Gurobi to solve the integer programs produced by our reductions [19]. Our SAT- +based verifier uses, by default, the kissat-sc2021-sat solver from the 2021 +SAT Competition by A. Biere, M. Fleury, and M. Heisinger [6, page 10]. Note +that the conference version of this article used the MapleCOMSPS solver—see +Subsection 7.3 for a discussion of solver benchmarks, comparisons, and choice. +We implemented Algorithm 7 using our hybrid verifier, and the graph automor- +phism library Nauty [24] as a subroutine to perform the required graph canon- +ization on GP . The original versions of our SP-BFS implementation targeted a +high-performance computing cluster environment, because our brute force and +dynamic programming implementations were not efficient enough. Subsequent +improvements to our verification algorithms made this unnecessary. Despite this, +our SP-BFS implementation is still in MPI and uses a MapReduce framework +[27] to maintain a distributed search frontier. +Our code base also contains multiple implementations of depth-first-search- +inspired algorithms for locating strong USPs. These algorithms use our hybrid +verification implementation and puzzle symmetry pruning technique discussed +in Section 4. For brevity and to keep this article focused on strong-USP verifi- +cation, we elect not to discuss these algorithms and defer them to a subsequent +article. That said, some of the concrete puzzles we found and report in the next +section were generated by such algorithms. These puzzles once found were ex- +perimentally verified as strong USPs using the techniques discussed in detail in +Section 3. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +23 +k +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +[11] +s ≥ +1 +2 +3 +4 +4 +10 +10 +16 +36 +36 +36 +136 +ω ≤ 3.00 2.88 2.85 2.85 +2.80 +2.74 +2.70 +This work s ≥ +1 +2 +3 +5 +8 +14 +21 +30 +42 +64 +112 +196 +ω ≤ 3.00 2.88 2.85 2.81 2.78 2.74 2.73 2.72 2.72 2.71 2.68 2.66 +Table 1: Comparison with [11] of lower bounds on the maximum of size of width-k +strong USPs and the upper bounds on ω they imply. Bold font indicates tight results +for that k. +7 +Experimental Results +Our experimental results come in several flavors for small-constant width k: +(i) constructive lower bounds on the maximum size of width-k strong USPs +witnessed by found puzzles, (ii) upper bounds on the maximum size of width-k +strong USPs, (iii) the number of SUSPs and SUSP equivalence classes for width +k, (iv) experimental data comparing the run times of our verification algorithms +and distinguishing likelihood of our heuristics, and (v) a benchmark data set of +SAT/UNSAT instances that we use to compare the effectiveness of competitive +SAT solvers as subroutines for the SAT-based part of our verifier. +All of the results in this section were produced by running our algorithm +implementations on the same Ubuntu 20.04 PC with a 3.00 GHz Intel Core +i9-10980XE CPU and 128 GB of RAM. +7.1 +New Upper and Lower Bounds on the Size of Strong USPs +New Lower Bounds. Table 1 summarizes new lower bounds for maximum SUSP +size in comparison with [11]. The lower bounds of [11] are from the constructions +in their Propositions 3.1 and 3.8, which give families of strong USPs for even +k or k divisible by three. For k’s which are not divisible by two or three, we +extrapolate their construction by adding a new column, this preserves the SUSP +property. The upper bounds on ω in this table are computed by plugging s and +k into Lemma 1 and optimizing over m. For clarity we omit ω’s that would be +larger than previous columns. Our results in this table we produced by running +SP-BFS and other search algorithms which verify that the final result is a strong +USP. Our bounds are tight for all k ≤ 5, because of the exhaustive nature of +SP-BFS, and constructively improve the known lower bounds for 4 ≤ k ≤ 12. +Figure 3 contains representative examples of maximal-size strong USPs we +found for k ≤ 6. The strong uniquely solvable (14, 6)-puzzles we found represent +the greatest improvement in ω versus the construction of [11] for small k. Further, +our puzzle for k = 12 is the result of taking the Cartesian product of two copies +of a strong uniquely solvable (14, 6)-puzzles. Note that Proposition 3.8 of [11] + +24 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Fig. 3: Representative maximal-size strong USPs found for width k = 1, 2, . . . , 6. +gives an infinite family of strong USPs that achieves ω < 2.48 as k goes to +infinity, which is stronger than our results are directly able to achieve. +New Upper Bounds. Table 2 summarizes the results of evaluating the bounds +from Section 5 for puzzles of width k ≤ 12. The calculations were routine except +for the clique bound that required constructing Gk, converting it into a mixed +integer program, and solving that program using Gurobi [19]. This was feasible +on our test system up to k = 11. We also experimented with calculating the upper +bounds for the 3-HypergraphClique bound, but found it infeasible to compute +for k ≥ 5 and so have omitted the results. The final row of the table contains the +best upper bounds we achieved, including applying the downward-closure bound +to lift adjacent bounds at k = 6 and k = 12. These upper bounds are stronger +than those immediately implied by [11]. +Observe that exhaustive search produced the best and tightest bounds, and +that the clique bound is considerably stronger than the unique pieces, USP, and ω +bounds. The unique pieces bounds appears to be stronger than the USP bound, +but we know that that is an artifact of the small value of k. As k increase, +the USP bound will become tighter than the unique pieces bound. Based on +the processing time we spent on k = 6, we conjecture that s = 14 is tight +for k = 6 and that our lower bounds for k > 6 are not. Our results suggests +there is considerable room for improvement in the construction of strong USPs, +and that it is possible that there exist large puzzles for k = 7, 8, 9 that would +beat [11]’s constructions and perhaps come close to the Coppersmith-Winograd +refinements. That said, it seems that new insights into the SUSP search problem +are required to proceed for k > 6. +Counting Strong USP. Table 3 shows the number of strong USPs and equiva- +lence classes of SUSP exhaustively calculated using SP-BFS with and without +symmetric pruning. Observe that the number of strong USPs is many orders of +magnitude more than the number of equivalence classes of strong USPs, even for +(3, 3)-SUSPs. Exhaustive search became infeasible even with puzzle symmetry + +312 +331213 +2.2) +3.3Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +25 +k +Bound +1 2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +ω +3 7 15 31 62 120 230 438 831 1,575 2,890 5,637 +Unique +2 4 +8 16 32 +64 128 256 512 1,024 2,048 4,096 +USP +3 6 12 24 45 +87 168 312 597 1,140 2,112 4,023 +Clique +1 3 +5 +9 17 +30 +55 105 186 +348 +654 +Exhaustive 1 2 +3 +5 +8 +Best +1 2 +3 +5 +8 +24 +55 105 186 +348 +654 1,962 +Table 2: Upper bounds on the size of SUSPs for widths k ≤ 12. Bold font indicates the +bound is tight, and blanks indicate the calculation for this puzzle width was infeasible. +pruning for k ≥ 6 as the memory usage of Algorithm 7 for storing the search +frontier exceeds the 128GB available on our test system. +7.2 +Algorithm Performance +To measure the performance of our verification algorithms and heuristics we ran +them on 10,000 random puzzles at each point on a sweep through parameter +space for widths k = 5 . . . 12 and sizes s = 1 . . . 60. We chose to test performance +via random sampling because we do not have access to a large set of solved +instances. This domain coincides with the frontier of our search space, and we +tuned the parameters of the heuristics and algorithms in the hybrid algorithm to +perform well in this domain. We did not deeply investigate performance charac- +teristics outside of this domain. In Figures 4, 5, & 6 we plot results, for brevity, +that are representative of the parameter space only for k ∈ {6, 9}. +Running Time. Figure 4 shows the average running times of our verification +algorithms in seconds. The brute force and dynamic programming algorithms +perform poorly except for very small size, s ≤ 8, and their curves loosely match +the exponential-time bounds we expect. The plots for the two reduction-based +algorithms (SAT and IP) behave similarly to each other. They are slower than +brute force and dynamic programming for small values of s, and their behavior +for large s is quite a bit faster. We speculate that the former is due to the cost of +constructing the reduced instance and overhead of the third party tools. Further +observe that the SAT reduction handily beats the IP reduction on large size for +k = 6, but as k increases, the gap decreases. We also note that across the settings +of k the IP reduction has effectively the same running time and is independent +of k. This is likely because the size of the IP instance depends only on s. The +hybrid algorithm generally performs best or close to best at small values of s +and is clearly faster for large values of s. Notice that it matches the dynamic +programming algorithm closely for small values of s and then diverges when the + +26 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +k +s +1 +2 +3 +4 +5 +6 +1 1 3 +2 +9 +3 +27 +4 +81 +5 +243 +7 +729 +2 +2 24 +9 +408 +33 +4,848 +91 +50,160 +229 +486,024 +3 +9 1,800 +240 +182,304 +2,429 +8,361,000 +16,971 291,347,280 +4 +728 2,445,120 +59,149 992,377,400 +1,611,648 +? +5 +190 3,248,640 +707,029 +? +? +? +6 +2,337,715 +? +? +? +7 +1,359,649 +? +? +? +8 +89,196 +? +? +? +9 +? +? +Table 3: Number of equivalence classes (bold face, left) versus total number of encoded +SUSPs (normal face, right) by (s, k)-puzzle dimensions. Computed using Algorithm 7. +Empty cells indicate that the number of SUSPs and equivalence classes is zero. ?’s +indicate unknown values that were infeasible to compute. +reduction-based algorithms and heuristics are activated at larger s. Observe that +the hybrid algorithm is effectively constant time for large s, though the size for +which this happens increases as a function of k. We expect this is because the +density of strong USPs decreases rapidly with s, and that the randomly selected +puzzles are likely far from satisfying Definition 3 and, hence, they are quickly +rejected by the unique pieces heuristics. Further evidence of this is that running +time of the hybrid algorithm converges to the running time of the unique pieces +heuristic for large k. +Heuristic Effectiveness. Figure 5 shows the probability that each individual +heuristic distinguishes a random puzzle in our benchmark. Observe that the +distinguishing power of the downward closure heuristic for s′ = 2 and unique +pieces heuristics coincide, demonstrating experiment consistency with Lemma 7. +Further, and for the same reason, the downward closure heuristic for s′ = 3 has +at least as high a distinguishing likelihood as the unique pieces heuristic. In the +plots, these three heuristics achieve almost 100% probability of distinguishing +random puzzles by size s = 30. The greedy heuristic perform less well than the +others and get substantially worse as k increases. We do not plot the running +times of the heuristics here, but they behave as expected by the earlier analysis. +As we noted earlier, unique pieces is linear time in the size of the puzzle and +the fastest of the heuristics. Figure 4 shows how the running time of the hybrid +algorithm and unique pieces converges as essentially all random puzzles of large +size, which the benchmark examined, are verified as non-SUSPs by this heuristic. +Variation in Running Time. Finally, we look at the variation in the running +times of the hybrid algorithm in Figure 6. For small s, the running time dis- +tribution is far from a normal distribution–the average is far above the median + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +27 +Fig. 4: Log plots of the average running times for verifying 10,000 random (s, k)-puzzles +for each s ∈ [50], k ∈ {6, 9}. The plots describe the behavior of five verification algo- +rithms brute force (BF), dynamic programming (DP), reduction to satisfiability (SAT), +reduction to integer programming (IP), and our hybrid algorithm (Hybrid). The run- +ning time of the unique pieces heuristic is also included. +and middle 50% of running times. This effect becomes even more pronounced +as k increases. However, we find that as s increases, the median running time +converges with the median running time of the unique pieces heuristic, and then +for larger s, the average running time converges as well. This is a consequence +of the hybrid algorithm having to run the orders of magnitude slower reduction- +based algorithms when the fast heuristics fail to resolve the instance. Although +not plotted here, we found that the range of the distribution of running times +for the SAT-based verifier was larger than for the IP-based verifier, even though +the IP-based verifier was slower on average. +Overall, our hybrid verification algorithm performs reasonably well in prac- +tice on random instances, despite reductions through NP-complete problems. +7.3 +Choice of SAT Solver +In the conference version of this article we examined only one SAT solver for +use in our implementation, MapleCOMSPS, a conflict-driven solver that uses a +learning rate branching heuristic, and that was a top performer at the 2016 SAT +Competition [7,23,5]. In this article we create a set of benchmark satisfiability +instances, using the SUSP verification reduction on a variety of puzzles (recall + +Average Verification Time (sec) vs Puzzle Size +k=6 +k=9 +100 +1 +0 +4 +A +10-1 +口 +(sec) +00 +10-2 +A +Time +Hybrid +10-3 +BF +10-4 +DP +8 +SAT +10-5 +IP +Unique +10-6 +胰*★* +*★★ +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +Puzzle size +Puzzle size28 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Fig. 5: Plots of the likelihood that each of the heuristics produces a definitive +results on 10,000 random (s, k)-puzzles for each size s ∈ [50] and width k +∈ +{6, 9}. Here “row pairs” is HeuristicDownwardClosed(P, 2) and “row triples” is +HeuristicDownwardClosed(P, 3). The row pairs points are plotted, but are hard +to see, because the unique pieces points coincides with them. +Subsection 3.4), and examined the performance of 352 solvers submitted to the +main track of the 2021 SAT Competition [6]. +We select benchmark instances consisting of (s, k)-puzzle with sizes from the +set +{(2, 2), (3, 3), (5, 4), (8, 5), (14, 6), (21, 7), (30, 8), (42, 9)}. +We choose these sizes, because we want positive and negative instances and these +sizes represent the largest strong USPs of each width we have been able to locate +through search. For each size we created ten puzzles that are strong USPs and +ten puzzles that are not. To create the ten non-SUSPs we randomly generated a +puzzle of that size and verified it was not a strong USP. To create the ten strong +USPs we for each size we used the results of our search algorithms. Then we ran +all of the puzzles through our SAT reduction to create .dimacs files for each +instance. Note that the SUSPs correspond to UNSAT instances and non-SUSPs +correspond to SAT instances. In total there are 160 instances in this benchmark. +We then ran each of the 35 solvers on each the 160 instance files and check the +output of each run against the expected result. For each trial, we record the user +CPU time reported by the Linux time command, or a timeout if the program +runs more than 5000 seconds without halting (mimicking the rules of the real +SAT competition). For comparison, we also run the MapleCOMSPS solver (from +2 There were 39 SAT solvers submitted to the main track. We use the default build +configuration for each submission. We were unable to build three of them, and one +that builds repeatedly crashed on all benchmarks without producing a result. We +tested the remaining 35. + +Heuristic Definitive Result Likelihood vs Puzzle Size +k=6 +k=9 +100 +Definitive Result +80 +60 +40 +Greedy +Row Pairs +% +20 +Row Triples +Unique +0 +5 +10 +15 +20 +25 +30 +5 +10 +15 +20 +25 +30 +Puzzle size +Puzzle sizeMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +29 +Fig. 6: Log box plots of the distribution of the running times of the hybrid verification +algorithm on 10,000 random (s, k)-puzzles for each s ∈ [50], k ∈ {6, 9}. The blue +circles denote the average running times of the hybrid algorithm. The dark blue blocks +indicates the median times. The thick vertical lines indicate the middle 50% of times, +and the thin vertical lines indicate the full range of running times at each s. +earlier version of this article), our MIP-based verifier (recall Subsection 3.5) and +our final hybrid verification algorithm on the same set of benchmark puzzles. +To compare the results of each solver we calculate the maximum time to +complete each instance across all of the runs, which is 5000 seconds if a run +timed out, and then divide by that maximum time to normalize all of the running +times to the interval [0, 1]. We calculate a benchmark score for each solver by +summing their relative running times across all instances. Table 4 contains the +benchmark scores for each solver. +MapleCOMSPS, the solver we used in the conference version of this article, +performs similarly to the best scoring solvers from the 2021 competition. The +recorded timeouts across all solvers come almost exclusively from the UNSAT +instances derived from (30, 8)-SUSPs and (42, 9)-SUSPs. The Gurobi-based ver- +ifier performs substantially worse than the best performing satisfiability solvers +on SAT instances (non-SUSPs), but dramatically better on UNSAT instances +(SUSPs). +Figure 7 shows the performance of the Gurobi-based verifier against the five +solvers with the best SAT scores. In this plot the instance completion times +for each solver are sorted in increasing order, so that curves further to the left +are better. If this were not a log-plot, the area to the left of the curve would +be proportional to the benchmark scores from Table 4. Observe that for SAT +instances, the SAT solvers, including MapleCOMSPS, follow similar trajectories. +Gurobi performs an order of magnitude worse across all SAT instances. The +hybrid algorithm, although plotted, is not visible because of how effective the +heuristics are at identifying random SAT (non-SUSP) instances. For UNSAT + +Hybrid Running Time (sec) vs Puzzle Size +k=6 +k=9 +10 +10-3 +10-5 +10-6 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +Puzzle size +Puzzle size30 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +instances, the situation is different. Gurobi performs relatively more slowly for +small, easier instances, but substantially better than the SAT solvers for larger, +harder instances. The performance of the solvers on easier UNSAT instances is +more varied than the corresponding case for SAT instances, but this does not +translate into much of a difference in benchmark score because the magnitude +of the relative completion time is low. +For UNSAT instances, the benchmark score is dominated by the number of +timeouts, each of which effectively adds one to the score. Indeed, the plots for +the SAT solver cut off between instance numbers 60 to 70, because the remaining +instances cause timeouts. Finally, notice that hybrid algorithm out performs the +others for small UNSAT instances – these are instances of the sort where the +brute force and bi-directional search algorithms are applied. For larger instances +the hybrid algorithm tracks an order of magnitude worse than the Gurobi-based +verifier. This is because our algorithm is tuned to encounter many more SAT +instances (non-SUSPs) than UNSAT instances (SUSPs). Further, because the +one-sided heuristics rule out SAT instances quickly in practice, on UNSAT in- +stances the hybrid algorithm runs these heuristics first, but then has to fall back +on the Gurobi-based verifier causing some overhead. +Ultimately, the results of these benchmarking experiments suggest that there +is not a substantial difference between using the 2016 MapleCOMSPS and the +best solvers from the 2021 competition. Even so, we choose kissat-sc20221-sat +as the default solver in our implementation, because it performed the best on our +benchmark of SAT instances. Using our current approach, Gurobi is essential to +the feasible verification of SUSPs. +The benchmark instances and puzzles, and the entirety of the raw timing +data can be found in our repository3. +8 +Conclusions +We initiated the first study of the verification of strong USPs and developed +practical software for both verifying and searching for them. We give tight results +on the maximum size of width-k strong USPs for k ≤ 5 and improved upper and +lower bounds on maximum strong-USP size for k ≤ 12. We prove a number of +properties of strong USPs related the verification and search. We also produce +a new set of benchmark instances for SAT solvers. +Although our results do not produce a new upper bound on the running +time of matrix multiplication, they demonstrate there is promise in this ap- +proach. There are a number of open questions. Is strong-USP verification coNP- +complete? What is the maximum strong-USP capacity? Is there a way to bridge +the apparent gap between the values of ω implied by single SUSPs and the values +implied by infinite families of SUSPs? What are tight bounds on maximum-size +strong USPs for k ≥ 6 and do these bound lead to asymptotically faster algo- +rithms for matrix multiplication? +3 https://bitbucket.org/paraphase/matmult/src/main/data_set/ + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +31 +The main bottleneck in our work is the size of the search space—new insights +seem to be required to substantially reduce it. Are there subclasses of strong +USPs that can be more effectively searched? Are there search strategies that +would be more effective on this space? +Acknowledgments +The authors thank the anonymous reviewers for their detailed and thoughtful +suggestions for improving this work. +The second and third authors thank Union College for the Undergraduate +Summer Research Fellowships funding their work. The first author thanks the +many undergraduate students that have contributed in some form to this project +over the years, including: Jonathan Kimber, Akriti Dhasmana, Jingyu Yao, Kyle +Doney, Quoc An, Harper Lyon, Zachary Dubinsky, Talha Mushtaq, Jing Chin, +Diep Vu, Hung Duong, Vu Le, Siddhant Deka, Baibhav Barwal, Aavasna Ru- +pakheti. +References +1. Alman, +J., +Williams, +V.V.: +Further +limitations +of +the +known +approaches +for matrix multiplication. In: 9th Innovations in Theoretical Computer Sci- +ence +(ITCS). +LIPIcs. +Leibniz +Int. +Proc. +Inform., +vol. +94, +pp. +Art. +No. +25, 15. Schloss Dagstuhl. Leibniz-Zent. 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In: 44th +Annual ACM Symposium on Theory of Computing (STOC). pp. 887–898. ACM +(2012). https://doi.org/10.1145/2213977.2214056 + +34 +Matthew Anderson, Zongliang Ji, and Anthony Yang Xu +Solver +SAT +UNSAT +Total +Timeouts +cadical-hack-gb +17.51 +15.97 +33.48 +15 +cadical-less-UP +19.81 +16.14 +35.95 +15 +cadical-PriPro +19.49 +15.62 +35.11 +15 +cadical-PriPro no bin +16.55 +15.73 +32.28 +15 +cadical-rp +19.08 +15.78 +34.85 +15 +cadical-sc2021 +18.82 +16.80 +35.62 +16 +Cadical SCAVEL01 +33.49 +16.73 +50.23 +15 +Cadical SCAVEL02 +40.97 +27.28 +68.26 +15 +cleanmaple +30.44 +18.93 +49.37 +17 +CleanMaple PriPro +30.70 +20.18 +50.87 +18 +hCaD +19.70 +16.52 +36.22 +16 +hKis +13.15 +17.30 +30.45 +16 +kissat bonus +13.04 +16.59 +29.63 +15 +kissat cf +12.06 +16.19 +28.26 +14 +kissat gb +12.52 +17.27 +29.79 +17 +kissat-MAB +15.28 +16.07 +31.36 +15 +kissat-sat crvr gb +13.37 +16.64 +30.01 +16 +kissat-sc2021 +12.32 +16.08 +28.40 +14 +kissat-sc2021-sat +12.02 +16.06 +28.08 +14 +kissat-sc2021-sweep +12.82 +16.24 +29.07 +16 +lstech maple +15.13 +14.83 +29.96 +12 +Maple MBDR BJL6 Tier2 +19.46 +16.02 +35.47 +14 +Maple MBDR BJL7 Local +19.98 +15.49 +35.47 +13 +Maple MBDR Cent PERM 10K +25.20 +15.96 +41.16 +12 +Maple MBDR Cent PERM 75K +25.07 +16.00 +41.06 +12 +Maple simp21 +12.53 +16.72 +29.26 +15 +MapleSSV +15.56 +16.68 +32.24 +16 +parafrost-nomdm-sc2021 +18.11 +15.56 +33.67 +14 +parafrost-sc2021 +24.15 +15.61 +39.76 +14 +Relaxed LCFTP +12.80 +17.55 +30.35 +16 +Relaxed LCFTP V2 +13.97 +16.17 +30.14 +12 +Relaxed LCMDCBDL BLB +15.38 +15.95 +31.33 +14 +Relaxed LCMDCBDL SCAVEL01 +13.95 +16.08 +30.03 +15 +Relaxed LCMDCBDL SCAVEL02 +25.45 +79.43 +104.88 +17 +slime +17.26 +14.73 +31.99 +13 +MapleCOMSPS +12.98 +17.42 +30.40 +16 +Gurobi +30.20 +0.00 +30.20 +0 +Hybrid +0.00 +0.01 +0.01 +0 +Table 4: Scores for solvers on our SUSP verification benchmark. The SAT and UNSAT +score are out of 80, the total score and timeouts are out of 160. Lower scores are better +and minimum values for each SAT solver are bold in each column. The top part of the +table includes the SAT solvers we tested from the 2021 SAT Competition [6]. + +Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles +35 +Fig. 7: Plots of the sorted relative completion times for SAT and UNSAT instances on +the five best-scoring solvers for that instance type. + +Sorted Instance # vs Relative Completion Time +SAT +80 +kissat cf = 12.06 +kissat_gb = 12.52 +kissat-sc2021 = 12.32 +F OZ +kissat-sc2021-sat = 12.02 +MapleCOMSPS = 12.98 +60 +Maple simp21 = 12.53 +# +hybrid = 0.00 +50 +gurobi = 30.20 +40 +orted +30 +S +20 - +10 - +0 : +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Relative Completion Time +UNSAT +80 - +70 - +60 - +# +Instance +50 +40 +orted +lstech maple = 14.83 +30 +MapleCOMSPS = 17.42 +Maple MBDR BJL7 Local = 15.49 +20 - +parafrost-nomdm-sc2021 = 15.56 +parafrost-sc2021 = 15.61 +10 - +slime = 14.73 +hybrid = 0.01 +gurobi = 0.00 +0 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Relative Completion Time \ No newline at end of file diff --git a/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/load_file.txt b/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75466f16e7d04eb7f49c6612622cc82ed9681414 --- /dev/null +++ b/BdAyT4oBgHgl3EQfRvfl/content/tmp_files/load_file.txt @@ -0,0 +1,1347 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf,len=1346 +page_content='Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles⋆ Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Department of Computer Science Union College Schenectady, New York, USA {andersm2, jiz, xua}@union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In subsequent work with Kleinberg and Szegedy, they connected this to the search for combinatorial objects called strong uniquely solvable puzzles (strong USPs) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We begin a systematic computer-aided search for these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We develop and implement constraint-based algorithms build on reductions to SAT and IP to verify that puzzles are strong USPs, and to search for large strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We produce tight bounds on the maximum size of a strong USP for width k ≤ 5, construct puzzles of small width that are larger than previ- ous work, and improve the upper bounds on strong USP size for k ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although our work only deals with puzzles of small-constant width, the strong USPs we find imply matrix multiplication algorithms that run in O(nω) time with exponent ω ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' While our algorithms do not beat the fastest algorithms, our work provides evidence and, perhaps, a path to finding families of strong USPs that imply matrix multiplication algorithms that are more efficient than those currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Keywords: matrix multiplication · strong uniquely solvable puzzle · arithmetic complexity · integer programming · satisfiability · satisfiability benchmark · upper bounds · reduction · application 1 Introduction An optimal algorithm for matrix multiplication remains elusive despite substan- tial effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We focus on the square variant of the matrix multiplication problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', given two n-by-n matrices A and B over a field F, the goal is to com- pute the matrix product C = A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The outstanding open question is: How many field operations are required to compute C?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The long thought-optimal na¨ıve algorithm based on the definition of matrix product is O(n3) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The groundbreaking work of Strassen showed that it can be done in time O(n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='808) [30] using a divide-and-conquer approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A long sequence of work concluding with Coppersmith and Winograd’s algorithm (CW) reduced the running time ⋆ An extended abstract of this paper appeared in the Proceedings of SAT 2020 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00074v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='CC] 30 Dec 2022 2 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: The leftmost diagram is a width-4 size-5 puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The middle three diagrams are the three sets of subrows of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The rightmost diagram is the puzzle P ′ resulting from reordering the subrows of P as indicated by the arrows and then recombining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since P can be rearranged as P ′ ̸= P without overlap, P is not uniquely solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' to O(n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='376) [26,28,31,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Recent computer-aided refinements of CW by others reduced the exponent to ω ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3728639 [16,32,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Approach Cohn and Umans [12] introduced a framework for developing faster algorithms for matrix multiplication by reducing this to a search for groups with subsets that satisfy an algebraic property called the triple-product property, which allows matrix multiplication to be embedded in the group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Their approach takes inspiration from the O(n log n) algorithm for multiplying degree- n univariate polynomials by embedding into the group algebra of the fast Fourier transform, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [14, Chapter 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Subsequent work [11] elaborated on this idea and developed the notion of combinatorial objects called strong uniquely solvable puzzles (strong USPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' These objects imply a group algebra embedding for matrix multiplication, and hence give a matrix multiplication algorithm as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A width-k puzzle P is a subset of {1, 2, 3}k, and the cardinality of P is the puzzle’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Each element of P is called a row of P, and each row consists of three subrows that are elements of {1, ∗}k, {2, ∗}k, {3, ∗}k respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In- formally, a puzzle P is a uniquely solvable puzzle (USP) if there is no way to permute the subrows of P to form a distinct puzzle P ′ without cells with num- bers overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 1 demonstrates a puzzle that is not a USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A uniquely solvable puzzle is strong if a tighter condition for non-overlapping holds (see Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For a fixed width k, the larger the size of a strong USP, the faster matrix multiplication algorithm it gives [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In fact, Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' show that there exist an infinite family of strong USPs that achieves ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We follow Cohn et al.’s program by developing: (i) verification algorithms and heuristics to determine whether a puzzle is a strong USP, (ii) search algo- rithms to find large strong USPs, (iii) practical implementations1 of these 1 Source code available here: https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/paraphase/matmult 3232 2 2 3 3 3232 １132 11 2 3 1123 1213 1 1 2 3 1312 3113 11 3 3 3113 1 321 1 1 2 3 １231Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 3 algorithms, and (iv) new upper bounds on the size of strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The most successful of our verification algorithms work by reducing the problem through 3D matching to the satisfiability (SAT) and integer programming (IP) prob- lems that are then solved with existing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithms we develop are not efficient—they run in worst-case exponential time in the natural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, the goal is to find a sufficiently large strong USP that would provide a faster matrix multiplication algorithm, and the resulting algorithm’s running time is independent of the running time of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The inefficiency of our algorithms limit the search space that we can feasibly examine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Results Our theoretical results and implementation produces new bounds on the size of the largest strong USP for small-width puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For small-constant width, k ≤ 12, we beat the largest sizes of [11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our lower bounds on maximum size are witnessed by strong USPs we found via search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For k ≤ 5 we give tight upper bounds determined by exhaustively searching all puzzles after modding out common symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For k ≤ 12, we improve the upper bounds on the size of strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although our current results do not beat [11] for unbounded k, they give evidence that there may exist families of strong USPs that give matrix multiplication algorithms that are more efficient than those currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The best strong USP we can produce imply matrix multiplication algorithms with ω ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also create a benchmark data set of SAT/UNSAT instances based on our reductions from strong-USP verification and examine the performance of solvers from the 2021 SAT Competition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Related Work For background on algorithms matrix multiplication problem, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There are also a number of negative results known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Na¨ıvely, the dimensions of the output matrix C implies that the problem requires at least Ω(n2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Slightly better lower bounds are known in general and also for specialized models of computation, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [29,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There are also lower bounds known for a variety of algorithmic approaches to matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Ambainis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' showed that the laser method cannot alone achieve an algorithm with ω ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3078 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A recent breakthrough on arithmetic progressions in cap sets [15] combined with a conditional result on the Erd¨os-Szemeredi sunflower conjecture [3] imply that Cohn et al.’s strong USP approach cannot achieve ω = 2 + ϵ for some ϵ > 0 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Subsequent work has generalized this barrier [1,2] to a larger class of algorithmic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Despite this, we are unaware of a concrete lower bound on ϵ implied by these negative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There remains a substantial gap in our understanding between what has been achieved by the positive refinements of LeGall, Williams, and Stothers, and the impossibility of showing ω = 2 using the strong USP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Recently Fawzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' showed how reinforcement learning techniques can be used to develop new matrix multiplication algorithms [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Their work produces matrix multiplication algorithms with ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='77, which is faster than Strassen’s 4 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu original algorithm (ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='81), but far from the refinements of Coppersmith- Winograd (ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='372) or the results achieved in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Organization Section 2 begins with the formal definition of a strong USP and the Cohn-Umans framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Sections 3 & 4, respectively, discuss our algorithms and heuristics for verifying that and searching for a puzzle that is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Section 5 describes several upper bounds on the size of strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Sections 6 & 7 discuss our implementation and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 2 Preliminaries For an integer k, we use [k] to denote the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For a set Q, SymQ de- notes the symmetric group on the elements of Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', the group of permutations acting on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' introduced the idea of a puzzle [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Definition 1 (Puzzle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For s, k ∈ N, an (s, k)-puzzle is a subset P ⊆ [3]k with |P| = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We call s the size of P, and k the width of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We say that an (s, k)-puzzle has s rows and k columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The columns of a puzzle are inherently ordered and indexed by [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The rows of a puzzle have no inherent ordering, however, it is often convenient to assume that they are ordered and indexed by the set of natural numbers [s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' establish a particular combinatorial property of puzzles that allows one to derive group algebras that matrix multiplication can be efficiently embedded into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Such puzzles are called strong uniquely solvable puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, to give some intuition we first explain a simpler version of the property called uniquely solvable puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Definition 2 (Uniquely Solvable Puzzle (USP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' An (s, k)-puzzle P is uniquely solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or (ii) there exists r ∈ P and c ∈ [k] such that at least two of the following hold: (π1(r))c = 1, (π2(r))c = 2, (π3(r))c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Informally, a puzzle is not uniquely solvable if each row of the puzzle can be broken into ones, twos, and threes pieces and then the rows can be reassembled in a different way so that each new row is a combination of a ones, a twos, and a threes piece where there is exactly one element of [3] for each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that uniquely solvable puzzles can have at most 2k rows because each ones piece, twos piece, and threes piece must be unique, as otherwise the duplicate pieces can be swapped making the puzzle not uniquely solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The definition of strong uniquely solvable puzzle is below, it is nearly the same except that it requires that there be a collision on a column between exactly two pieces, not two or more pieces like in the original definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Definition 3 (Strong USP (SUSP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' An (s, k)-puzzle P is strong uniquely solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or (ii) there exists r ∈ P and c ∈ [k] such that exactly two of the following hold: (π1(r))c = 1, (π2(r))c = 2, (π3(r))c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 5 Finally, Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' defined a strengthening of SUSP which requires that every triple of rows witness the necessary overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Definition 4 (Local SUSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A local strong uniquely solvable puzzle is an (s, k)-puzzle where for each triple of rows u, v, w ∈ P with u, v, w not all equal, there exists c ∈ [k] such that (uc, vc, wc) is an element of L = {(1, 2, 1), (1, 2, 2), (1, 1, 3), (1, 3, 3), (2, 2, 3), (3, 2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Every SUSP P corresponds to a much larger local SUSP P ′, which, informally, is the result of concatenating and duplicating the rows of P to explicitly demon- strate the ∀π1, π2, π3 part of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proposition 1 ([11, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be a (s, k)-SUSP, then there is a local (s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', s · k)-SUSP P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that in all of the definitions, local, strong, uniquely solvability is invariant to the ordering of the rows of the puzzle, because P is a set—we use this fact implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' show the following connection between the existence of strong USPs and upper bounds on the exponent of matrix multiplication ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 1 ([11, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let ϵ > 0, if there is a strong uniquely solv- able (s, k)-puzzle, there is an algorithm for multiplying n-by-n matrices in time O(nω+ϵ) where ω ≤ min m∈N≥3 � 3 log m log(m − 1) − 3 log s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' s · k log(m − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This result motivates the search for large strong USPs that would result in faster algorithms for matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In the same article, the authors also demon- strate the existence of an infinite family of strong uniquely solvable puzzles, for width k divisible by three, that achieves a non-trivial bound on ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 2 ([11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There is an infinite family of strong uniquely solvable puzzles that achieves ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Finally, they conjecture that strong uniquely solvable puzzles provide a route to achieving quadratic-time matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Unfortunately, as mentioned in the introduction, this conjecture was shown to be false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 3 ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Strong uniquely solvable puzzles cannot show ω < 2 + ϵ, for some ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' That said, there remains hope that the uniquely solvable puzzle approach could beat the refinements of Coppersmith-Winograd even if it cannot reach ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Algorithm 1 : Brute Force Verification Input: An (s, k)-puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: YES, if P is a strong USP and NO otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function VerifyBruteForce(P) 2: for π2 ∈ SymP do 3: for π3 ∈ SymP do 4: if π2 ̸= 1 ∨ π3 ̸= 1 then 5: found = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: for r ∈ P do 7: for i ∈ [k] do 8: if δri,1 + δ(π2(r))i,2 + δ(π3(r))i,3 = 2 then found = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 9: if not found then return NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 10: return YES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3 Verifying Strong USPs The core focus of this article is the problem of verifying strong USPs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', given an (s, k)-puzzle P, output YES if P is a strong USP, and NO otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In this section we discuss the design of algorithms to solve this computational problem as a function of the natural parameters s and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' All of the exact algorithms we develop in this section have worst-case expo- nential running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, asymptotic worst-case running time is not the metric we are truly interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Rather we are interested in the practical per- formance of our algorithms and their capability for locating new large strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithm that we ultimately develop is a hybrid of a number of simpler algorithms and heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We begin by discussing a na¨ıve brute force algorithm based on the defini- tion of strong USP (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1), see how it motivations a reduction to the 3D matching problem (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2), and then how we might formulate a re- duction to the satisfiability and integer programming problems (Subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='4 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We then describe several verification heuristics based on properties of strong USP (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='6) and combine them with the verification algorithms to produce a hybrid algorithm Verify (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' As we discuss in Sub- section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2, our hybrid algorithm is quickly able to check whether a given puzzle is a strong USP and aid in the search for strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 Brute Force The obvious algorithm for verification comes directly from the definition of a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Informally, we consider all ways of permuting the twos and threes pieces relative to the ones pieces and check whether the non-overlapping con- dition of Definition 3 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A formal description of the algorithm is found in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 7 The ones in Line 4 of Algorithm 1 denote the identity in SymP , and δa,b is the Kronecker delta function which is one if a = b and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that Algorithm 1 does not refer to the π1 of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is because the strong USP property is invariant to permutations of the rows and so π1 can be thought of as an arbitrary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Hence, we fix π1 = 1 to simplify the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Seeing that |SymP | = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', we conclude that the algorithm runs in time O((s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' )2 · s · k · poly(s)) where the last factor accounts for the operations on permutations of s elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The dominant term in the running time is the contribution from iterating over all pairs of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Finally, notice that if P is a strong USP, then the algorithm runs in time Θ((s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' )2·s·k·poly(s)), and that if P is not a strong USP the algorithm terminates early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithm’s poor performance made it unusable in our implementation, however, its simplicity and direct connection to the definition made its implementation a valuable sanity check against later more elaborate algorithms (and it served as effective onboarding to the undergraduate students collaborating on this project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although Algorithm 1 performs poorly, examining the structure of a seem- ingly trivial optimization leads to substantially more effective algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Con- sider the following function on triples of rows a, b, c ∈ P: f(a, b, c) = ∨i∈[k](δai,0+ δbi,1+δci,2 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We can replace the innermost loop in Lines 7 & 8 of Algorithm 1 with the statement found = found ∨ f(r, π1(r), π2(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that f neither depends on P, r, nor the permutations, and that Algorithm 1 no longer depends directly on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To slightly speed up Algorithm 1 we can precompute and cache f before the algorithm starts and then look up values as the algorithm runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We precompute f specialized to the rows in the puzzle P, and call it fP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2 Strong USP Verification to 3D Matching It turns out to be more useful to work with fP than with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' It is convenient to think of fP as a function fP : P × P × P → {0, 1} that is the complement of the characteristic function of the relations of a tripartite hypergraph HP = ⟨P ⊔ P ⊔ P, ¯ fP ⟩ where the vertex set is the disjoint union of three copies of P and fP indicates the edges that are not present in HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let H = ⟨P ⊔ P ⊔ P, E ⊆ P 3⟩ be a tripartite 3-hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We say H has a 3D matching (3DM) iff there exists a subset M ⊆ E with |M| = |P| and for all distinct edges e1, e2 ∈ M, e1 and e2 are vertex disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e1 ∩ e2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Determining whether a hypergraph has a 3D matching is a well-known NP- complete problem (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We say that a 3D matching is non-trivial if it is not the set {(r, r, r) | r ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 2 demonstrates a 3-hypergraph with a non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The existence of non-trivial 3D matchings in HP is directly tied to whether P is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A puzzle P is a strong USP iff HP has no non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We first argue the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose that Hp has a non-trivial 3D matching M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We show that P is not a strong USP by using M to construct π1, π2, π3 ∈ 8 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 2: An example hypergraph G with edges E = {(r1, r1, r2), (r1, r3, r3), (r2, r2, r1), (r2, r3, r1), (r3, r2, r3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The highlighted edges are a non-trivial 3D matching M = {(r1, r1, r2), (r2, r3, r1), (r3, r2, r3)} of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' SymP that witness this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let π1 be the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For each r ∈ P, define π2(r) = q where (r, q, ∗) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that q is well defined and unique because M is 3D matching and so has vertex disjoint edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Similarly define π3(r) = q where (r, ∗, q) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that by construction M = {(π1(r), π2(r), π3(r)) | r ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since M is a matching of HP , M ⊆ ¯ fP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Because M is a non-trivial matching at least one edge in (a, b, c) ∈ M has either a ̸= b, a ̸= c, or b ̸= c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This implies, respectively, that as constructed π1 ̸= π2, π1 ̸= π3, or π2 ̸= π3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In each case we have determined that π1, π2, and π3 are not all identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Thus we determined permutations such that for all r ∈ P, f(π1(r), π2(r), π3(r)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This violates Condition (ii) of Definition 3, hence P is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The forward direction is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose that P is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We show that HP has a 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For P not to be a strong USP there must exist π1, π2, π3 ∈ SymP not all identical such that Condition (ii) of Definition 3 fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Define e(r) = (π1(r), π2(r), π3(r)) and M = {e(r) | r ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since Condition (ii) fails, we have that fP (e(r)) = false for all r ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This means that for all r ∈ P, e(r) ∈ ¯ fP and hence M ⊆ ¯ fP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since π1 is a permutation, |M| = |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that M is non-trivial because not all of the permutations are identical and there must be some r ∈ P with e(r) having non-identical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Thus M is a non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ As a consequence of Definition 3, strong-USP verification is in coNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that although 3D matching is an NP-complete problem, Lemma 4 does not im- mediately imply that verification of strong USPs is coNP-complete because HP is not an arbitrary hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' It remains open whether strong-USP verification is coNP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 4 implies that to verify P is a strong USP it suffices to determine whether HP has a non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In the subsequent subsec- tions we examine algorithms for the later problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We can, in retrospect, view Algorithm 1 as an algorithm for solving 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We note that the parameters s and k are not fully independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' First, s ≤ 3k because the maximum number of rows in a puzzle of width k is |[3]k| = 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Sec- ond, we eliminate the dependence on k entirely by transforming an (s, k)-puzzle GMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 9 Algorithm 2 : Bidirectional Dynamic Programming Verification Input: An (s, k)-puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: YES, if P is a strong USP and NO otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function VerifyDynamicProgramming(P) 2: Let T = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: Construct 3D matching instance HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4: function SearchHalf(ℓ, Q, ℓQ, R, ℓR, δ, t) 5: if ℓ = t then 6: if δ = 1 then ▷ Forward Base Case 7: Insert (Q, R) into T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8: return false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 9: else ▷ Reverse Base Case 10: if (P − Q, P − R) ∈ T then 11: return true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 12: else 13: return false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 14: res = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ▷ Recursive Case 15: for ℓ′ Q = ℓQ + 1 to s do 16: for ℓ′ R = ℓR + 1 to s do 17: if (pℓ, pℓ′ Q, pℓ′ R) ∈ HP ∧ ¬res then 18: res = SearchHalf(ℓ + δ, Q ∪ {pℓ′ Q}, ℓ′ Q, R ∪ {pℓ′ R}, ℓ′ R, δ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 19: return res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 20: SearchHalf(1, ∅, 0, ∅, 0, 1, ⌊s/2⌋ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 21: return SearchHalf(s, ∅, 0, ∅, 0, −1, ⌊s/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' into a 3D matching instance on the vertex set [s]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, this transformation is not without cost, because the size of HP is a function of the cube of s rather than linear in the size of the puzzle s · k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3 Dynamic Programming The realization that the verification of strong USPs is a specialization of 3D matching leads to a dynamic programming algorithm for verification that runs in linear-exponential time O(22spoly(s)+poly(s, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The reduction allows us to replace the permutations from SymP with subsets of P and effectively reduce the cost of the outer loops of Algorithm 1 from s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' = Θ(2s log s) to 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Algorithm 2 describes a recursive bidirectional dynamic programming al- gorithm for strong-USP verification that uses the 3D matching instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithm consists of two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let t = ⌊s/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The first phase determines all possible sets Q, R ⊆ P with |Q| = |R| = t such that there is 3D matching M1 of HP when restricted to the vertices {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , pt} ⊔ Q ⊔ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The sets Q, R satisfying the requirement are stored in a table T during the first phase on Line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The second phase determines all possible sets Q, R ⊆ P with |Q| = |R| = s−t 10 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu such that there is a 3D matching M2 of HP when restricted to the vertices {pt+1, pt+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , ps} ⊔ Q ⊔ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For each pair (Q, R) the algorithm considers in the second phase, it checks whether (P − Q, P − R) was inserted into T during the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If the pair is present, it means that there is a 3D matching of HP which is M = M1 ∪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This works because, by Line 10, M1 and M2 are partial 3D matchings on {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , pt} ⊔ (P − R) ⊔ (P − Q) and {pt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ps} ⊔ R ⊔ Q, respectively, which implies that M1 and M2 are vertex disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The first phase always returns false, which is ignored, and the second phase returns whether a complete matching could be found, and, hence, by Lemma 4, whether P is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The running time of this algorithm is dominated by the number of pairs of sets (Q, R) it examines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that rows of P are considered in order in Lines 15 & 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further, the algorithm tracks the index of the last elements added to Q and R in ℓQ and ℓR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithm only adds new elements to Q or R that have higher indexes than ones previously added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Altogether this implies that each pair of sets (Q, R) is only considered at most once during a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since Q, R ⊆ P, there are at most �t i=0 �s i � �s i � ≤ (�t i=0 �s i � )2 ≤ (2s)2 = 4s pairs (Q, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This means that SearchHalf is called at most 4s times during each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Hence the running time of the algorithm is O(4s·s2·poly(s)+T3DM(s, k)) where s2 factor comes from the inner loops, poly(s) time to manipulate the sets and track the contents of T as a hash table, and T3DM(s, k) accounts for the time to construct HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The memory requirements of Algorithm 2 are similarly high—the first phase uses O(4s · s) bits to store T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that Algorithm 2 does not early terminate on P that are strong USP, because it must search through all pairs before determining that none can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The algorithm could be modified to allow early termination when P is not a strong USP by causing the second phase of search to immediately return in Line 18 once the first 3D matching witness has been located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, this still requires the first phase to run to completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A remedy for this would be to run both phases in parallel and have them check against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We chose not to because it would substantially complicate the implementation and would be unlikely to ultimately improve the performance of our combined algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For comparison, more advanced techniques like those of Bj¨orklund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' can achieve a better asymptotic time of O(2spoly(s)) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We chose not to implement their algorithm, because we judged that it would not substantially increase the domain for which verification was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='4 3D Matching to Satisfiability By Lemma 4, one can determine whether a puzzle P is a strong USP by con- structing the graph HP and deciding whether it has a non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Here we reduce our 3D matching problem to the satisfiability (SAT) problem on conjunctive normal form (CNF) formulas and then use a state-of-the-art SAT solver to resolve the reduced problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To perform the reduction, we convert the graph HP into a CNF formula ΨP , a depth-2 formula that is the AND of Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 11 ORs of Boolean literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We construct ΨP so that ΨP is satisfiable iff HP has a non-trivial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let HP = ⟨V = P ⊔ P ⊔ P, E ⊆ P 3⟩ be the 3D matching instance associated with the puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our goal is to determine whether there is a non-trivial 3D matching M ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A na¨ıve reduction would be to have variables Mu,v,w indicating inclusion of each edge (u, v, w) ∈ P 3 in the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This results in a formula ΨP with s3 variables and size Θ(s5) because including an edge e ∈ P 3 excludes the Θ(s2) edges e′ with e∩e′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To decrease the size of ΨP we instead use sets of variables to indicate which vertices in the second and third part of V are matched with each vertex in the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In particular we have Boolean variables M 1 u,v and M 2 u,w for all u, v, w ∈ P, and these variable map to assignments in the na¨ıve scheme in the following way: M 1 u,v ∧ M 2 u,w ⇔ Mu,v,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We now write our CNF formula for 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' First, we have clauses that prevents non-edges from being in the matching: Ψ non-edge P = � (u,v,w)∈E (¬M 1 u,v ∨ ¬M 2 u,w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (1) Second, we add clauses require that every vertex in HP is matched with some edge: Ψ ≥1 P = � � u∈P (∨v∈P M 1 u,v) ∧ (∨w∈P M 2 u,w) � ∧ � � v∈P (∨u∈P M 1 u,v) � ∧ � � w∈P (∨u∈P M 2 u,w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (2) Third, we require that each vertex be matched with at most one edge and so have clauses that exclude matching edges that overlap on one or two coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Ψ ≤1 P = � i∈{1,2} � (u,v),(u′,v′)∈P 2 (u = u′ ∨ v = v′) ∧ (u, v ̸= u′, v′) ⇒ ¬M i u,v ∨ ¬M i u′,v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (3) Fourth, we exclude the trivial 3D matching by requiring that at least one of the diagonal edges not be used: Ψ non-trivial P = � u∈P ¬M 1 u,u∨¬M 2 u,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Finally, we AND these into the overall CNF formula: ΨP = Ψ non-edge P ∧ Ψ ≤1 P ∧ Ψ ≥1 P ∧ Ψ non-trivial P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The size of the CNF formula ΨP is Θ(s3), has 2s2 variables, and is a factor of s2 smaller than the na¨ıve approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Thus we reduce 3D matching to satisfiability by converting the instance HP into the CNF formula ΨP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='5 3D Matching to Integer Programming In parallel to the previous subsection, we use the connection between verifica- tion of strong USPs and 3D matching to reduce the former to integer program- ming, another well-known NP-complete problem (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [21]) and then apply 12 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu a state-of-the-art solver to resolve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Again, let HP = ⟨V, E⟩ be the 3D match- ing instance associated with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We construct an integer program QP over {0, 1} that is infeasible iff P is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Here the reduction is simpler than the previous one because linear constraints naturally capture matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We use Mu,v,w to denote a variable with values in {0, 1} to indicate whether the edge (u, v, w) ∈ P 3 is present in the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To ensure that M is a subset of E we add the following edge constraints to QP : ∀u, v, w ∈ P, ∀(u, v, w) ̸∈ E, Mu,v,w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also require that each vertex in each of the three parts of the graph is incident to exactly one edge in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is captured by the following vertex constraints in QP : ∀w ∈ P, � u,v∈P Mu,v,w = � u,v∈P Mu,w,v = � u,v∈P Mw,u,v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lastly, since we need that the 3D matching be non-trivial we add the constraint: � u∈P Mu,u,u < |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To check whether P is a strong USP we determine whether QP is not feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', that no assignment to the variables M satisfy all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We note that reduction from 3D matching to IP is polynomial time and that there are s3 variables in QP , and that the total size of the constraints is s3 · Θ(1) + 3s · Θ(s2) + 1 · Θ(s3) = Θ(s3), similar to size of ΨP in the SAT reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='6 Heuristics Although the exact algorithms presented in the previous sections make sub- stantial improvements over the brute force approach, the resulting performance remains impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To resolve this, we also develop several fast verification heuristics that may produce the non-definitive answer MAYBE in place of YES or NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Then, to verify a puzzle P we run this battery of fast heuristics and return early if any of the heuristics produce a definitive YES or NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' When all of the heuristics result in MAYBE, we then run one of the slower exact algorithms that were previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The heuristics have different forms, but all rely on the structural properties of strong uniquely solvable puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Downward Closure The simplest heuristics we consider is based on the fact that strong USPs are downward closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If P is a strong USP, then so is every subpuzzle P ′ ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be a strong USP and P ′ ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' By Definition 3, for every (π1, π2, π3) ∈ Sym3 P not all identity, there exist r ∈ P and i ∈ [k] such that exactly two of the following hold: (π1(r))i = 1, (π2(r))i = 2, (π3(r))i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Consider restricting the permutations to those that fix the elements of P\\P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For these permutations it must be the case that r ∈ P ′ because otherwise r ∈ P\\P ′ and there is exactly one j ∈ [3] for which (πj(r))i = j holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Thus we can drop the elements of P\\P ′ and conclude that for every tuple of permutations in SymP ′ the conditions of Definition 3 hold for P ′, and hence that P ′ is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ This leads to a polynomial-time heuristic that can determine that a puzzle is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Informally, the algorithm takes an (s, k)-puzzle P and s′ ≤ s, Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 13 Algorithm 3 : Downward-Closure Heuristic Input: An (s, k)-puzzle P, and size s′ ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: NO, if P has a set of s′ rows that do not form a strong USP, and MAYBE otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function HeuristicDownwardClosed(P, s′) 2: for P ′ ⊆ P, |P ′| = s′ do 3: if P ′ is not a strong USP then return NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4: return MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' and verifies that all subsets P ′ ⊆ P with size |P ′| = s′ are strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If any subset P ′ is not a strong USP, the heuristic returns NO, and otherwise it returns MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For completeness, this algorithm is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This algorithm runs in time O( � s s′ � T(s′, k)) where T(s′, k) is the runtime for verifying an (s′, k)-puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In practice we did not apply this heuristic for s′ larger than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' When s′ is some constant d, the running time becomes O(sd · T(d, k)) = O(sdk) using the brute force algorithm (Algorithm 1) for verification of the puzzle P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Unique Pieces Every strong uniquely solvable puzzle is a uniquely solvable puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A necessary condition for a puzzle to be a USP is that for each element in [3], the collection of subrows contains no duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 6 (Implicit in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If P is a USP, then for all e ∈ [3], and distinct rows r1, r2 ∈ P, there is a column c ∈ [k] were one of the rows r1 or r2 has an e and the other one does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that this is not the case, and dis- tinct rows r1, r2 ∈ P have e in exactly the same columns for some e ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We show that P is not a USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Choose πe = (r1r2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', the permutations that trans- poses the subrows for e in rows r1 and r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Choose the other two permutations for the elements of [3]\\{e} to be the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since the permutations are not all the identity, the second half of Definition 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, the puzzle that results from the permutations is identical to P and for all c ∈ [k] and each row r ∈ P there exists exactly on i ∈ [3] where (πi(r))c = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Hence the definition of uniquely solvable is not satisfied and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ Note that the reverse direction of Lemma 6 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The puzzle in Figure 1 is an example of this: It is not uniquely solvable, but the subrows for each element are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We can make Lemma 6 effective as via a linear-time heuristic capable of ruling out puzzles that are not (strong) USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although straightforward, for completeness we formalize our approach in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' When the sets are implemented as hash tables, the expected running time of this algorithm is O(s· k) time, which is linear in the size of the puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' An alternative worst-case O(s · k) time implementation uses radix sort to sort the characteristic sequences 14 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Algorithm 4 : Unique Pieces Heuristic Input: An (s, k)-puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: NO, if a witness is found for P not being a (strong) USP, and MAYBE otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function HeuristicUniquePieces(P) 2: Initialize empty sets S1, S2, S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: for r ∈ P do 4: for e ∈ [3] do 5: Let h = {c ∈ [k] | rc = e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: if h ∈ Se then return NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7: Se = Se ∪ {h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8: return MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' of the subrows as binary numbers and then scans adjacent rows to to detect duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The unique pieces heuristic is equivalent to the downward-closure heuristic for subpuzzles of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be an (s, k)-puzzle, then HeuristicUniquePieces(P) = HeuristicDownwardClosed(P, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We show both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose that P fails the unique pieces heuristic for, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e = 1, then there are distinct rows r1, r2 ∈ P where the cells that contain 1 are all in the same columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This means we can swap those 1’s subrows without causing overlap or changing the puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This implies that P ′ = {r1, r2} is not a (strong) USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since |P ′| = 2 and P ′ ⊆ P, the downward closure heuristic for s′ = 2 will also conclude that P is not a (strong) USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose that P fails the downward-closure heuristic for s′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Then there is a pair of distinct rows r1, r2 ∈ P for which P ′ = {r1, r2} is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Suppose there is no columns were r1 and r2 differ, then the subrows of r1, r2 are the same for all elements, and so P fails the unique pieces heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For the other case, suppose there is a least one column c ∈ [k] where r1 and r2 differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', let that column be ((r1)c, (r2)c) = (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Because P ′ is not an SUSP and this column is (1, 2), there can be other no columns that are in from the set {(1, 3), (2, 3), (3, 2), (3, 1)} otherwise they would form an SUSP with the column (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This means the only columns that P ′ contains are from the set {(1, 2), (2, 1), (1, 1), (2, 2), (3, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Therefore, the columns which contain 2 must match and the subrows for 2 in r1 and r2 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Thus, P ′, and so P, fails the unique pieces heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ A corollary of this proof is that for size-two puzzles, every USP is also a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be a (2, k)-puzzle, if P is a uniquely solvable puzzle, then P is a strong uniquely solvable puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 15 Since the unique pieces heuristic is equivalent to the downward-closure heuristic for s′ = 2 and the running time of unique pieces is linear in the puzzle size, O(s·k), and the running time of downward closed is O(s2 ·k), we use the unique pieces heuristic in place of downward closed for s′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Greedy This heuristic attempts take advantage of Lemma 4 and greedily search for a 3D matching for the instance HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The heuristic proceeds iteratively, de- termining the vertex of the first part of the 3D matching instance with the least edges and randomly selecting an edge of that vertex to put into the 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If the heuristic successfully constructs a 3D matching it returns NO indicating that the input puzzle P is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If the heuristic reaches a point were prior commitments have made the matching infeasible, the heuristic starts again from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This process is repeated some number of times be- fore it gives up and returns MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In our implementation we use s2 attempts because it is similar to the running time of the reductions and it empirically re- duced the number of instances requiring full verification in the domain of puzzles with k = 6, 7, 8 while not increasing the running time by too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The greedy heuristic is formalized in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The array cts is used to store the number of edges cts[u] that remain associ- ated with vertex u along the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Much of the algorithm is devoted to maintaining this invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The sets U, V, W store the vertices along the three coordinates, respectively, that have already been incorporated into the partial 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Like in Algorithm 2 we do not store the matching itself, only the vertices involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The break at Line 10 triggers when the partial 3D matching is a dead end and cannot be extended into a full 3D matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The condition of Line 23 is true when a full 3D matching has been constructed and causes the algorithm to return that P is not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The running time of this algorithm is O(s3t+T3DM(s, k)), where T3DM(s, k) is the time required to construct 3D matching instances from (s, k)-puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This algorithm has the potential to be considerably slower than the downward- closure heuristic, and in practice we set t = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, the main loop can terminate early at Line 10 when it fails to extend the 3D matching, this permits the expected time to much less than the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For a puzzle P that is a strong USP, the heuristic takes the full Ω(s3t + T3DM(s, k)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Compared to the downward-closure and unique pieces heuristics this heuristic is much less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' As a result we only run it when when the other heuristics have failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' See Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2 for a comparison of effectiveness these heuristics in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='7 Hybrid Algorithm Our final verification algorithm (Algorithm 6) is a hybrid of several exact al- gorithms and heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The size thresholds for which algorithm and heuristic to apply were determined experimentally for small k and are focused on the values where our strong USP search algorithms are tractable k ≤ 6 (or nearly 16 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Algorithm 5 : Greedy Heuristic Input: An (s, k)-puzzle P, and iteration bound t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: NO, if a witness is found for P not being a strong USP, and MAYBE otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function HeuristicGreedy(P) 2: Construct 3D matching instance HP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: for i = 1 to t do 4: for u ∈ P do 5: cts[u] = � v,w∈P HP (u, v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ▷ Number of edges incident vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: Let U, V, W = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7: Let m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ▷ Number of edges in matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8: while m < s do 9: Select u ∈ {w ∈ ¯U | cts[w] = maxv∈ ¯U cts[v]} uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 10: if cts[u] = 0 then break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 11: Let D = {(v, w) ∈ ¯V × ¯W | HP (u, v, w) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 12: Select (v, w) ∈ D uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 13: for v′ ∈ P do ▷ Update edge counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 14: for w′ ∈ P do 15: if (v′, w′) ∈ ¯V × ¯W and HP (u, v′, w′) = 1 then 16: cts[u]--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 17: if (v′, w′) ∈ ¯U × ¯W and HP (v′, v, w′) = 1 and v′ ̸= u then 18: cts[v′]--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 19: if (v′, w′) ∈ ¯U × ¯V and HP (v′, w′, w) = 1 and v′ ̸∈ {u, v} then 20: cts[v′]--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 21: U, V, W = U ∪ {u}, V ∪ {v}, W ∪ {w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ▷ Add edge to matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 22: m = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 23: if m ≥ s then return NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ▷ 3D matching found so not SUSP, halt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 24: return MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' tractable k ≤ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We decide to run both of the reductions to SAT and IP in parallel because it is not clear which algorithm performs better in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since verification halts when either algorithm completes, the wasted effort is within a factor of two of what the better algorithm could have done alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also chose to do this because we experimentally observed that there were many instances that one of the algorithms struggled with that the other did not—this resulted in a hybrid algorithm that out performed the individual exact algorithms on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We show in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2 that our hybrid algorithm and heuristics perform well in practice at quickly verifying strong USPs for small width k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Fur- ther, Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3 contains a discussion of the relative performance of the SAT and IP approaches on different instance types from our benchmark experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 17 Algorithm 6 : Hybrid Verification Input: An (s, k)-puzzle P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: YES, if P is a strong USP, and NO otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function Verify(P) 2: if s ≤ 2 then return VerifyBruteForce(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: Return result if HeuristicUniquePieces(P) is not MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4: if s ≤ 7 then return VerifyDynamicProgramming(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 5: Return result if HeuristicDownwardClosed(P, 3) is not MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: Return result if HeuristicGreedy(P) is not MAYBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7: Run VerifySAT(P) and VerifyIP(P) in parallel and return first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4 Searching for Strong USPs With a practical verification algorithm in hand, we consider the problem of searching for large strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Because the set of strong USPs is downward closed, a natural search strategy is: Start with the empty set and repeatedly consider adding rows while maintaining the strong-USP property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, while this strategy will lead to a maximal-size strong USP, it is not guaranteed to produce a maximum-size strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is because the set of strong USPs does not form a matroid, rather it is only an independence system (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In particular, (i) the empty puzzle is a strong USP and (ii) the set of strong USP are downward closed by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The final property required to be a matroid, the augmentation property, requires that for every pair of strong USPs P1, P2 with |P1| ≤ |P2| there is a row of r ∈ P2\\P1 such that P1 ∪ {r} is also a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For a simple counterexample consider the strong USPs P1 = {32} and P2 = {12, 23}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Using Lemma 6, we see that neither P1 ∪ {12} = {12, 32} nor P1 ∪{23} = {23, 32} are strong USPs, and hence the augmentation property fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' One consequence is that na¨ıve greedy algorithms will likely be ineffective for finding maximum-size strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Furthermore, we do not currently know of an efficient algorithm that can take a strong USP P and determine a row r such that P ∪ {r} is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Despite that, we have had some success in applying general-purpose tree- search techniques with pruning based on the symmetries of strong USPs together with our practical verification algorithm to construct maximum-size strong USPs for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 Puzzle Symmetry Since puzzles are defined as sets of rows, the ordering of the rows of a puzzle P does not affect the SUSP property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Similarly, but slightly less obviously, the SUSP property is invariant to reordering the columns of the puzzle, because the required existential condition ∃c ∈ [k] st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=') from Definition 3 is independent 18 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu of the ordering of the columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lastly, the alphabet [3] typically used to repre- sent the elements of a puzzle is completely arbitrary, any set of three distinct values would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' These values are not interpreted mathematically, aside from their convenience in expressing the SUSP definition concisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This logic can be formalized into the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let ρ ∈ Sym[k], δ ∈ Sym[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A (s, k)-puzzle P is a strong USP iff {(δ(rρ(c)))c∈[k] | r ∈ P} is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Follows immediately from Definition 1 and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ This lemma implies that the SUSP property is invariant with respect to these kinds of puzzle transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We call two puzzles P, P ′ that are related in this way isomorphic, and use the notation P ∼= P ′ to denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The relation ∼= is an equivalence relation, because permutations are invertable, and so it partitions the set of puzzles into equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This notion of isomorphism is naturally related to the same notion in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For each (s, k)-puzzle P we can define a colored, undirected graph GP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This graph consists of vertices that are partitioned into four sets of different colors: V = {rowr}r∈[s]⊔{colc}c∈[k]⊔{ei}i∈[3]⊔{vr,c}(r,c)∈[s]×[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There are s+k+3+s·k vertices in GP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The first three parts are vertices representing the rows and columns of P, and the elements of [3], respectively, and the fourth part are vertices for each of the s·k cells in the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The edge relation of GP is straightfor- ward: Each vertex vr,c is connected to three vertices corresponding to the row, columns and element that the cell indexed (r, c) contains in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In particular, the three edges attached to vr,c are (vr,c, rowr), (vr,c, colc), (vr,c, eltP (r,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In total, GP has 3·s·k edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Because the vertex sets for rows, columns, and elements are each uniquely colored and each cell of P is connected to vertices representing its row, column, and element, the automorphisms of GP are in 1-1 correspondence to the automorphisms of P under permutations of rows, columns, and elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This implies that for two (s, k)-puzzles P, P ′, if GP ∼= GP ′ then there exists per- mutations of the rows, columns, and elements of P which results in P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further by Lemma 8, if GP ∼= GP ′, then P ∼= P ′, and P is an SUSP iff P ′ is an SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2 Symmetry-Pruned Tree Search A natural way to search for strong USPs is based on breadth-first search and uses the fact that strong USP are downward closed (Lemma 5): To find the largest possible width-k strong USP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (i) start with all possible first rows – the 3k (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' k)-puzzles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (ii) attempt to extend the resulting puzzles with all possible rows keeping only the puzzles that are strong USPs and which are not isomorphic to the strong USPs that have been seen before to form the new search frontier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' and (iii) repeat Step (ii) until the search frontier is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To ensure the algorithm does not revisit isomorphic puzzles, we use canonical graph representations [Gp] of the puzzle graphs GP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' A canonical graph represen- tation is a binary encoding of a graph with the property that for any two graphs G1, G2, [G1] = [G2] iff G1 ∼= G2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' As the search algorithm runs we Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 19 Algorithm 7 : Symmetry-Pruned Breadth-First Search Input: An integer k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Output: The number b, which is the size of the largest width-k strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 1: function SP-BFS(k) 2: Let Q be an empty queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: Let I be an empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4: Let b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 5: enqueue(Q, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: while Q is not empty do 7: P = dequeue(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8: for r ∈ [3]k\\P do 9: Let P ′ = P ∪ {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 10: if Verify(P ′) and [G′ P ] ̸∈ I then 11: enqueue(Q, P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 12: I = I ∪ {[G′ P ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 13: b = |P ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 14: return b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' record the set I of canonical graph representations [GP ] of each distinct puzzle P that has been added to the search frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Each time a puzzle P ′ is consid- ered for being added to the search frontier we first check whether its canonical graph representation [GP ′] ∈ I, if it is, we do not add P ′ to the frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The use of canonical representations of puzzles dramatically shrinks the search space by searching from [P] rather than every P ′ ∼= P and by not allowing duplicates of [P] to be enqueued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This algorithm SP-BFS is formalized in Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We argue the correctness of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For k ∈ N, SP-BFS(k) returns the maximum integer s for which there exists an (s, k)-SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Ignoring the pruning that I performs for a moment, it is routine to argue that SP-BFS behaves like a generic breadth-first search algorithm over the tree of all strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is because of the downward-closure property of strong USP (Lemma 5), which makes any strong USP P reachable from the trivial strong USP ∅ using a series of row inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' SP-BFS(k) results in an exhaustive search of all strong USPs of width k and return the maximum size b of such SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We argue that when considering the pruning that I contributes to, SP- BFS(k) enqueues exactly one element of each equivalence class of puzzles that are SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Then, as a consequence of Lemma 8, the algorithm must explore ev- ery equivalence class of width-k SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Hence, it explores an equivalence class with SUSPs of maximum size and subsequently returns that size, which is the expected output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To complete the argument and show that the symmetry pruned search covers the entire search space of equivalence classes, suppose, for the sake of contradic- 20 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu tion, that there is some smallest s such that there is an (s, k)-puzzle P that does not have its equivalence class [P] searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We know that s > 1, because the algorithm starts by considering all possible (1, k)-puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P ′ be the (s−1, k)- puzzle created from P by removing one of its rows r, P ′ has as least one row because s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' By hypothesis, the equivalence class of [P ′] has been visited by SP-BFS because P ′’s size is s − 1 < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Consider [P] and remove the row that corresponded to r to form [P]′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' It must be the case that [P ′] ∼= [P]′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This isomor- phism extends to [P] in that there must be a row r′ such that ([P ′]∪{r′}) ∼= [P], where r′ is replaces the row remove from [P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Therefore, since [P ′] is searched, the algorithm must consider all possible rows to extend by, including r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is means that the equivalence class of [P] is searched, a contradicting our assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Therefore every equivalence class of SUSPs is searched by SP-BFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ This approach reduces the size of the search space, improving both the run- ning time of the search and the space required to keep track of the frontier puz- zles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The worst case running time of SP-BFS is O(3k·#EQUIV (k)·(TVerify(sk+ 1, k)+TCanonize(sk, k)), where #EQUIV (k) is the number equivalence classes of strong USP of width k, TVerify(sk + 1, k) is the time to verify the maximum size (sk +1, k)-puzzles examined by the algorithm, and TCanonize(sk, k) is the time to compute the canonical graph representation of each puzzle P considered by the algorithm (assuming TVerify and TCanonize are monotone in their parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' See Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 for the experimental results of running SP-BFS and a discussion of implementation issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 5 Upper Bounds Although the main focus of this research line is to construct sufficiently large strong USP that would imply faster matrix multiplication algorithms, our tech- niques and approach can also be applied to search for tighter upper bounds on the size of strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We describe several SUSP-size upper bounds in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ω Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Prior work explicitly discusses bounds on the capacity of infinite families of USP (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since every SUSP is a USP, these bounds also apply to SUSP and can be restated to apply to individual puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The first bound, which we denote as the “ω bound”, results from (i) Lemma 1, which is monotone non-increasing for fixed k, and (ii) the fact that ω ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To compute this bound we evaluate the inequality of Lemma 1 on increasingly large s until just before the consequence implies ω < 2 which is in contradiction with ω ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Unique Pieces Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The second bound, which we denote as the “unique pieces bound”, following directly from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Since that lemma requires that each row of a (strong) USP have a unique ones, twos, and threes piece, the total number of rows in a strong USP cannot be more than 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 21 USP Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The third bound, which we denote as the “USP bound”, results from the proof of [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although not spelled out in that article, the proof relies on the following subclaim that directly bounds s as a function of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be a (s, k)-USP, then s ≤ k � c1=0 k−c1 � c2=0 min �� k c1 � , � k c2 � , � k k − (c1 + c2) �� = O � k2 · � 3 22/3 �k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that the USP bound is asymptotically tighter than the unique pieces bound as 3 22/3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='8899 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Clique Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The fourth bound, which we denote as the “clique bound”, results from the fact that SUSPs are downward closed (Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In particular if P is an SUSP, then for every P ′ ⊆ P with 2 rows must also be an SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Fix k ∈ N and consider a graph Gk whose vertices correspond to the possible rows of a width-k puzzle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', strings in [3]k, and where there is an edge between r1, r2 ∈ [3]k if {r1, r2} is an SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that by downward closure, each (s, k)-SUSP corresponds to a clique of size s in Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This approach naturally generalizes from the Clique problem to h-HypergraphClique problem where the graph Gh k consists the same 3k vertices as Gk = G2 k, but instead has the arity-h edges {r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , rh} which are (h, k)-SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be an (s, k)-SUSP and 2 ≤ h ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Then for Gh k = ⟨V = [3]k, E = {P ′ ⊆ V | P ′ is a strong USP and |P ′| = h}⟩, (Gh k, s) ∈ h-HypergraphClique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Therefore, the size of a maximum hypergraph clique in Gh k is an upper bound of size of width-k SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We use “clique bound” to denote the specific instantiation of this bound for h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Exhaustive Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For fifth bound, which we denote as the “exhaustive bound”, we consider the results of Algorithm 7 when run in the domain of k where the full search space can be feasibly explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Because these bounds are based on exhaustive search they are inherently tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Downward-Closure Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The final bound we consider follows from the downward- closure property of SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P be an (s, k)-SUSP with k > 1, then there exists an (⌈ s 3⌉, k − 1)-SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Fix any c ∈ [k] and consider the cth column of P, then, by averaging, there must be an element of e ∈ [3] that appears at least ⌈ s 3⌉ times in that column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Let P ′ ⊂ P be the subpuzzle of P whose rows have e in the cth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' P ′ is a strong USP, because P is a strong USP and strong USPs are downward 22 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu closed (Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Form P ′′ by removing the cth column of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' P ′′ is a strong USP, because P ′ is a strong USP and the strong-USP property is invariant to addition or removal of constant columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' By construction, P ′′ is a (⌈ s 3⌉, k − 1)- SUSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ⊓⊔ This bound is not as independently applicable like the others, but it can lift upper bounds of s ≤ u at k to s ≤ 3u at k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' See Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 for the results of evaluating the above bounds for small width and a discussion of issues involved in concretely calculating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6 Implementation We implemented our verification algorithms, heuristics, and search algorithms, along with various utilities and appropriate datastructures to represent under- lying information such as puzzles in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The source code for our implementa- tion is available under a MIT License at https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/paraphase/ matmult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We use a number of external libraries with subroutines that are key to the functioning of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our IP-based verifier and Clique bound calcula- tor both use the commercial, closed-source mixed-integer programming solver Gurobi to solve the integer programs produced by our reductions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our SAT- based verifier uses, by default, the kissat-sc2021-sat solver from the 2021 SAT Competition by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Biere, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Fleury, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Heisinger [6, page 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that the conference version of this article used the MapleCOMSPS solver—see Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3 for a discussion of solver benchmarks, comparisons, and choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We implemented Algorithm 7 using our hybrid verifier, and the graph automor- phism library Nauty [24] as a subroutine to perform the required graph canon- ization on GP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The original versions of our SP-BFS implementation targeted a high-performance computing cluster environment, because our brute force and dynamic programming implementations were not efficient enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Subsequent improvements to our verification algorithms made this unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Despite this, our SP-BFS implementation is still in MPI and uses a MapReduce framework [27] to maintain a distributed search frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our code base also contains multiple implementations of depth-first-search- inspired algorithms for locating strong USPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' These algorithms use our hybrid verification implementation and puzzle symmetry pruning technique discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For brevity and to keep this article focused on strong-USP verifi- cation, we elect not to discuss these algorithms and defer them to a subsequent article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' That said, some of the concrete puzzles we found and report in the next section were generated by such algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' These puzzles once found were ex- perimentally verified as strong USPs using the techniques discussed in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 23 k 1 2 3 4 5 6 7 8 9 10 11 12 [11] s ≥ 1 2 3 4 4 10 10 16 36 36 36 136 ω ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='70 This work s ≥ 1 2 3 5 8 14 21 30 42 64 112 196 ω ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='66 Table 1: Comparison with [11] of lower bounds on the maximum of size of width-k strong USPs and the upper bounds on ω they imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Bold font indicates tight results for that k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7 Experimental Results Our experimental results come in several flavors for small-constant width k: (i) constructive lower bounds on the maximum size of width-k strong USPs witnessed by found puzzles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (ii) upper bounds on the maximum size of width-k strong USPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (iii) the number of SUSPs and SUSP equivalence classes for width k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' (iv) experimental data comparing the run times of our verification algorithms and distinguishing likelihood of our heuristics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' and (v) a benchmark data set of SAT/UNSAT instances that we use to compare the effectiveness of competitive SAT solvers as subroutines for the SAT-based part of our verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' All of the results in this section were produced by running our algorithm implementations on the same Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='04 PC with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 GHz Intel Core i9-10980XE CPU and 128 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 New Upper and Lower Bounds on the Size of Strong USPs New Lower Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Table 1 summarizes new lower bounds for maximum SUSP size in comparison with [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The lower bounds of [11] are from the constructions in their Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='8, which give families of strong USPs for even k or k divisible by three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For k’s which are not divisible by two or three, we extrapolate their construction by adding a new column, this preserves the SUSP property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The upper bounds on ω in this table are computed by plugging s and k into Lemma 1 and optimizing over m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For clarity we omit ω’s that would be larger than previous columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our results in this table we produced by running SP-BFS and other search algorithms which verify that the final result is a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our bounds are tight for all k ≤ 5, because of the exhaustive nature of SP-BFS, and constructively improve the known lower bounds for 4 ≤ k ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 3 contains representative examples of maximal-size strong USPs we found for k ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The strong uniquely solvable (14, 6)-puzzles we found represent the greatest improvement in ω versus the construction of [11] for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further, our puzzle for k = 12 is the result of taking the Cartesian product of two copies of a strong uniquely solvable (14, 6)-puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='8 of [11] 24 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3: Representative maximal-size strong USPs found for width k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' gives an infinite family of strong USPs that achieves ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='48 as k goes to infinity, which is stronger than our results are directly able to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' New Upper Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Table 2 summarizes the results of evaluating the bounds from Section 5 for puzzles of width k ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The calculations were routine except for the clique bound that required constructing Gk, converting it into a mixed integer program, and solving that program using Gurobi [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This was feasible on our test system up to k = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also experimented with calculating the upper bounds for the 3-HypergraphClique bound, but found it infeasible to compute for k ≥ 5 and so have omitted the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The final row of the table contains the best upper bounds we achieved, including applying the downward-closure bound to lift adjacent bounds at k = 6 and k = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' These upper bounds are stronger than those immediately implied by [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that exhaustive search produced the best and tightest bounds, and that the clique bound is considerably stronger than the unique pieces, USP, and ω bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The unique pieces bounds appears to be stronger than the USP bound, but we know that that is an artifact of the small value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' As k increase, the USP bound will become tighter than the unique pieces bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Based on the processing time we spent on k = 6, we conjecture that s = 14 is tight for k = 6 and that our lower bounds for k > 6 are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Our results suggests there is considerable room for improvement in the construction of strong USPs, and that it is possible that there exist large puzzles for k = 7, 8, 9 that would beat [11]’s constructions and perhaps come close to the Coppersmith-Winograd refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' That said, it seems that new insights into the SUSP search problem are required to proceed for k > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Counting Strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Table 3 shows the number of strong USPs and equiva- lence classes of SUSP exhaustively calculated using SP-BFS with and without symmetric pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that the number of strong USPs is many orders of magnitude more than the number of equivalence classes of strong USPs, even for (3, 3)-SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Exhaustive search became infeasible even with puzzle symmetry 312 331213 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 25 k Bound 1 2 3 4 5 6 7 8 9 10 11 12 ω 3 7 15 31 62 120 230 438 831 1,575 2,890 5,637 Unique 2 4 8 16 32 64 128 256 512 1,024 2,048 4,096 USP 3 6 12 24 45 87 168 312 597 1,140 2,112 4,023 Clique 1 3 5 9 17 30 55 105 186 348 654 Exhaustive 1 2 3 5 8 Best 1 2 3 5 8 24 55 105 186 348 654 1,962 Table 2: Upper bounds on the size of SUSPs for widths k ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Bold font indicates the bound is tight, and blanks indicate the calculation for this puzzle width was infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' pruning for k ≥ 6 as the memory usage of Algorithm 7 for storing the search frontier exceeds the 128GB available on our test system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='2 Algorithm Performance To measure the performance of our verification algorithms and heuristics we ran them on 10,000 random puzzles at each point on a sweep through parameter space for widths k = 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 12 and sizes s = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We chose to test performance via random sampling because we do not have access to a large set of solved instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This domain coincides with the frontier of our search space, and we tuned the parameters of the heuristics and algorithms in the hybrid algorithm to perform well in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We did not deeply investigate performance charac- teristics outside of this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In Figures 4, 5, & 6 we plot results, for brevity, that are representative of the parameter space only for k ∈ {6, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Running Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 4 shows the average running times of our verification algorithms in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The brute force and dynamic programming algorithms perform poorly except for very small size, s ≤ 8, and their curves loosely match the exponential-time bounds we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The plots for the two reduction-based algorithms (SAT and IP) behave similarly to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' They are slower than brute force and dynamic programming for small values of s, and their behavior for large s is quite a bit faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We speculate that the former is due to the cost of constructing the reduced instance and overhead of the third party tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further observe that the SAT reduction handily beats the IP reduction on large size for k = 6, but as k increases, the gap decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also note that across the settings of k the IP reduction has effectively the same running time and is independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is likely because the size of the IP instance depends only on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The hybrid algorithm generally performs best or close to best at small values of s and is clearly faster for large values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Notice that it matches the dynamic programming algorithm closely for small values of s and then diverges when the 26 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu k s 1 2 3 4 5 6 1 1 3 2 9 3 27 4 81 5 243 7 729 2 2 24 9 408 33 4,848 91 50,160 229 486,024 3 9 1,800 240 182,304 2,429 8,361,000 16,971 291,347,280 4 728 2,445,120 59,149 992,377,400 1,611,648 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 5 190 3,248,640 707,029 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6 2,337,715 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7 1,359,649 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8 89,196 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 9 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Table 3: Number of equivalence classes (bold face, left) versus total number of encoded SUSPs (normal face, right) by (s, k)-puzzle dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Computed using Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Empty cells indicate that the number of SUSPs and equivalence classes is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='s indicate unknown values that were infeasible to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' reduction-based algorithms and heuristics are activated at larger s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that the hybrid algorithm is effectively constant time for large s, though the size for which this happens increases as a function of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We expect this is because the density of strong USPs decreases rapidly with s, and that the randomly selected puzzles are likely far from satisfying Definition 3 and, hence, they are quickly rejected by the unique pieces heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further evidence of this is that running time of the hybrid algorithm converges to the running time of the unique pieces heuristic for large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Heuristic Effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 5 shows the probability that each individual heuristic distinguishes a random puzzle in our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that the distinguishing power of the downward closure heuristic for s′ = 2 and unique pieces heuristics coincide, demonstrating experiment consistency with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further, and for the same reason, the downward closure heuristic for s′ = 3 has at least as high a distinguishing likelihood as the unique pieces heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In the plots, these three heuristics achieve almost 100% probability of distinguishing random puzzles by size s = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The greedy heuristic perform less well than the others and get substantially worse as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We do not plot the running times of the heuristics here, but they behave as expected by the earlier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' As we noted earlier, unique pieces is linear time in the size of the puzzle and the fastest of the heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 4 shows how the running time of the hybrid algorithm and unique pieces converges as essentially all random puzzles of large size, which the benchmark examined, are verified as non-SUSPs by this heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Variation in Running Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Finally, we look at the variation in the running times of the hybrid algorithm in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For small s, the running time dis- tribution is far from a normal distribution–the average is far above the median Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 4: Log plots of the average running times for verifying 10,000 random (s, k)-puzzles for each s ∈ [50], k ∈ {6, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The plots describe the behavior of five verification algo- rithms brute force (BF), dynamic programming (DP), reduction to satisfiability (SAT), reduction to integer programming (IP), and our hybrid algorithm (Hybrid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The run- ning time of the unique pieces heuristic is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' and middle 50% of running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This effect becomes even more pronounced as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' However, we find that as s increases, the median running time converges with the median running time of the unique pieces heuristic, and then for larger s, the average running time converges as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is a consequence of the hybrid algorithm having to run the orders of magnitude slower reduction- based algorithms when the fast heuristics fail to resolve the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although not plotted here, we found that the range of the distribution of running times for the SAT-based verifier was larger than for the IP-based verifier, even though the IP-based verifier was slower on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Overall, our hybrid verification algorithm performs reasonably well in prac- tice on random instances, despite reductions through NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='3 Choice of SAT Solver In the conference version of this article we examined only one SAT solver for use in our implementation, MapleCOMSPS, a conflict-driven solver that uses a learning rate branching heuristic, and that was a top performer at the 2016 SAT Competition [7,23,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In this article we create a set of benchmark satisfiability instances, using the SUSP verification reduction on a variety of puzzles (recall Average Verification Time (sec) vs Puzzle Size k=6 k=9 100 1 0 4 A 10-1 口 (sec) 00 10-2 A Time Hybrid 10-3 BF 10-4 DP 8 SAT 10-5 IP Unique 10-6 胰*★* ★★ 10 20 30 40 50 10 20 30 40 50 Puzzle size Puzzle size28 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 5: Plots of the likelihood that each of the heuristics produces a definitive results on 10,000 random (s, k)-puzzles for each size s ∈ [50] and width k ∈ {6, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Here “row pairs” is HeuristicDownwardClosed(P, 2) and “row triples” is HeuristicDownwardClosed(P, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The row pairs points are plotted, but are hard to see, because the unique pieces points coincides with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='4), and examined the performance of 352 solvers submitted to the main track of the 2021 SAT Competition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We select benchmark instances consisting of (s, k)-puzzle with sizes from the set {(2, 2), (3, 3), (5, 4), (8, 5), (14, 6), (21, 7), (30, 8), (42, 9)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We choose these sizes, because we want positive and negative instances and these sizes represent the largest strong USPs of each width we have been able to locate through search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For each size we created ten puzzles that are strong USPs and ten puzzles that are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To create the ten non-SUSPs we randomly generated a puzzle of that size and verified it was not a strong USP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To create the ten strong USPs we for each size we used the results of our search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Then we ran all of the puzzles through our SAT reduction to create .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='dimacs files for each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Note that the SUSPs correspond to UNSAT instances and non-SUSPs correspond to SAT instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In total there are 160 instances in this benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We then ran each of the 35 solvers on each the 160 instance files and check the output of each run against the expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For each trial, we record the user CPU time reported by the Linux time command, or a timeout if the program runs more than 5000 seconds without halting (mimicking the rules of the real SAT competition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For comparison, we also run the MapleCOMSPS solver (from 2 There were 39 SAT solvers submitted to the main track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We use the default build configuration for each submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We were unable to build three of them, and one that builds repeatedly crashed on all benchmarks without producing a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We tested the remaining 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Heuristic Definitive Result Likelihood vs Puzzle Size k=6 k=9 100 Definitive Result 80 60 40 Greedy Row Pairs % 20 Row Triples Unique 0 5 10 15 20 25 30 5 10 15 20 25 30 Puzzle size Puzzle sizeMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 29 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 6: Log box plots of the distribution of the running times of the hybrid verification algorithm on 10,000 random (s, k)-puzzles for each s ∈ [50], k ∈ {6, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The blue circles denote the average running times of the hybrid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The dark blue blocks indicates the median times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The thick vertical lines indicate the middle 50% of times, and the thin vertical lines indicate the full range of running times at each s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' earlier version of this article), our MIP-based verifier (recall Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='5) and our final hybrid verification algorithm on the same set of benchmark puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' To compare the results of each solver we calculate the maximum time to complete each instance across all of the runs, which is 5000 seconds if a run timed out, and then divide by that maximum time to normalize all of the running times to the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We calculate a benchmark score for each solver by summing their relative running times across all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Table 4 contains the benchmark scores for each solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' MapleCOMSPS, the solver we used in the conference version of this article, performs similarly to the best scoring solvers from the 2021 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The recorded timeouts across all solvers come almost exclusively from the UNSAT instances derived from (30, 8)-SUSPs and (42, 9)-SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The Gurobi-based ver- ifier performs substantially worse than the best performing satisfiability solvers on SAT instances (non-SUSPs), but dramatically better on UNSAT instances (SUSPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Figure 7 shows the performance of the Gurobi-based verifier against the five solvers with the best SAT scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In this plot the instance completion times for each solver are sorted in increasing order, so that curves further to the left are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' If this were not a log-plot, the area to the left of the curve would be proportional to the benchmark scores from Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Observe that for SAT instances, the SAT solvers, including MapleCOMSPS, follow similar trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Gurobi performs an order of magnitude worse across all SAT instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The hybrid algorithm, although plotted, is not visible because of how effective the heuristics are at identifying random SAT (non-SUSP) instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For UNSAT Hybrid Running Time (sec) vs Puzzle Size k=6 k=9 10 10-3 10-5 10-6 10 20 30 40 50 10 20 30 40 50 Puzzle size Puzzle size30 Matthew Anderson, Zongliang Ji, and Anthony Yang Xu instances, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Gurobi performs relatively more slowly for small, easier instances, but substantially better than the SAT solvers for larger, harder instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The performance of the solvers on easier UNSAT instances is more varied than the corresponding case for SAT instances, but this does not translate into much of a difference in benchmark score because the magnitude of the relative completion time is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For UNSAT instances, the benchmark score is dominated by the number of timeouts, each of which effectively adds one to the score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Indeed, the plots for the SAT solver cut off between instance numbers 60 to 70, because the remaining instances cause timeouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Finally, notice that hybrid algorithm out performs the others for small UNSAT instances – these are instances of the sort where the brute force and bi-directional search algorithms are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' For larger instances the hybrid algorithm tracks an order of magnitude worse than the Gurobi-based verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' This is because our algorithm is tuned to encounter many more SAT instances (non-SUSPs) than UNSAT instances (SUSPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Further, because the one-sided heuristics rule out SAT instances quickly in practice, on UNSAT in- stances the hybrid algorithm runs these heuristics first, but then has to fall back on the Gurobi-based verifier causing some overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Ultimately, the results of these benchmarking experiments suggest that there is not a substantial difference between using the 2016 MapleCOMSPS and the best solvers from the 2021 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Even so, we choose kissat-sc20221-sat as the default solver in our implementation, because it performed the best on our benchmark of SAT instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Using our current approach, Gurobi is essential to the feasible verification of SUSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The benchmark instances and puzzles, and the entirety of the raw timing data can be found in our repository3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 8 Conclusions We initiated the first study of the verification of strong USPs and developed practical software for both verifying and searching for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We give tight results on the maximum size of width-k strong USPs for k ≤ 5 and improved upper and lower bounds on maximum strong-USP size for k ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We prove a number of properties of strong USPs related the verification and search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' We also produce a new set of benchmark instances for SAT solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Although our results do not produce a new upper bound on the running time of matrix multiplication, they demonstrate there is promise in this ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' There are a number of open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Is strong-USP verification coNP- complete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' What is the maximum strong-USP capacity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Is there a way to bridge the apparent gap between the values of ω implied by single SUSPs and the values implied by infinite families of SUSPs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' What are tight bounds on maximum-size strong USPs for k ≥ 6 and do these bound lead to asymptotically faster algo- rithms for matrix multiplication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 3 https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/paraphase/matmult/src/main/data_set/ Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 31 The main bottleneck in our work is the size of the search space—new insights seem to be required to substantially reduce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Are there subclasses of strong USPs that can be more effectively searched?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Are there search strategies that would be more effective on this space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Acknowledgments The authors thank the anonymous reviewers for their detailed and thoughtful suggestions for improving this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The second and third authors thank Union College for the Undergraduate Summer Research Fellowships funding their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The first author thanks the many undergraduate students that have contributed in some form to this project over the years, including: Jonathan Kimber, Akriti Dhasmana, Jingyu Yao, Kyle Doney, Quoc An, Harper Lyon, Zachary Dubinsky, Talha Mushtaq, Jing Chin, Diep Vu, Hung Duong, Vu Le, Siddhant Deka, Baibhav Barwal, Aavasna Ru- pakheti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Alman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=', Williams, V.' 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not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Numerische mathematik 13(4), 354–356 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='1007/BF02165411 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Strassen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=': The asymptotic spectrum of tensors and the exponent of matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In: 27th Annual Symposium on Foundations of Computer Science (FOCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 49–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' IEEE (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' https://doi.' metadata={'source': 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Coppersmith-Winograd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' In: 44th Annual ACM Symposium on Theory of Computing (STOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 887–898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' ACM (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='org/10.' metadata={'source': 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+page_content='95 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='08 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='03 15 Relaxed LCMDCBDL SCAVEL02 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='45 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='43 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='88 17 slime 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='26 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='73 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='99 13 MapleCOMSPS 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='98 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='42 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='40 16 Gurobi 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='20 0 Hybrid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='01 0 Table 4: Scores for solvers on our SUSP verification benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The SAT and UNSAT score are out of 80, the total score and timeouts are out of 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Lower scores are better and minimum values for each SAT solver are bold in each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' The top part of the table includes the SAT solvers we tested from the 2021 SAT Competition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' 7: Plots of the sorted relative completion times for SAT and UNSAT instances on the five best-scoring solvers for that instance type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content=' Sorted Instance # vs Relative Completion Time SAT 80 kissat cf = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='06 kissat_gb = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='52 kissat-sc2021 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='32 F OZ kissat-sc2021-sat = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='02 MapleCOMSPS = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='98 60 Maple simp21 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='53 # hybrid = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 50 gurobi = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='20 40 orted 30 S 20 - 10 - 0 : 10-5 10-4 10-3 10-2 10-1 100 Relative Completion Time UNSAT 80 - 70 - 60 - # Instance 50 40 orted lstech maple = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='83 30 MapleCOMSPS = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='42 Maple MBDR BJL7 Local = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='49 20 - parafrost-nomdm-sc2021 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='56 parafrost-sc2021 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='61 10 - slime = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='73 hybrid = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='01 gurobi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} +page_content='00 0 10-5 10-4 10-3 10-2 10-1 100 Relative Completion Time' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAyT4oBgHgl3EQfRvfl/content/2301.00074v1.pdf'} diff --git a/BtE2T4oBgHgl3EQfngj6/content/tmp_files/2301.04010v1.pdf.txt b/BtE2T4oBgHgl3EQfngj6/content/tmp_files/2301.04010v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d877c0cbb889ee2a006a0d0068de21b62af29101 --- /dev/null +++ b/BtE2T4oBgHgl3EQfngj6/content/tmp_files/2301.04010v1.pdf.txt @@ -0,0 +1,1390 @@ +Energy deposition and formation of nanostructures in the interaction of highly charged +xenon ions with gold nanolayers +I. Stabrawaa, D. Bana´sa,∗, A. Kubala-Kuku´sa, Ł. Jabło´nskia, P. Jagodzi´nskia, D. Sobotaa, K. Szarya, M. Pajeka, K. Skrzypiecb, E. +Mendykb, M. Borysiewiczc, M. D. Majki´cd, N. N. Nedeljkovi´ce +aInstitute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland +bDepartament of Chemistry, Maria Curie-Skłodowska University, Plac M. Curie-Skłodowskiej 3, 20-031 Lublin, Poland +cInstitute of Electron Technology, aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland +dFaculty of Technical Sciences, University of Priˇstina in Kosovska Mitrovica, Knjaza Miloˇsa 7, 38220 Kosovska Mitrovica, Serbia +eFaculty of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia +Abstract +The effect of the deposition of kinetic energy and neutralization energy of slow highly charged xenon ions on the process of the +nanostructures creation at the surface of gold nanolayers is investigated. The nanolayers of thickness of 100 nm were prepared by +e-beam evaporation of gold on crystalline silicon Si(100) substrate. The samples were irradiated at the Kielce EBIS facility of the +Jan Kochanowski University (Kielce, Poland), under high vacuum conditions. The irradiations were performed for constant kinetic +energy 280 keV and different ions charge states (Xeq+, q = 25, 30, 35, 36 and 40) and for constant charge state Xe35+ and different +kinetic energies: 280 keV, 360 keV, 420 keV and 480 keV. The fluence of the ions was on the level of 1010 ions/cm2. Before and +after irradiation the nanolayer surfaces were investigated using the atomic force microscope. +As the result, well pronounced modifications of the nanolayer surfaces in the form of craters have been observed. A systematic +analysis of the crater sizes (diameter on the surface and depth) allowed us to determine the influence of the deposited kinetic and the +neutralization energy on the size of the obtained nanostructures. The results are theoretically interpreted within the micro-staircase +model based on the quantum two-state vector model of the ionic Rydberg states population. The charge dependent ion-atom +interaction potential inside the solid is used for the calculation of the nuclear stopping power. According to the model the formation +of the nanostructures is governed by the processes of the ionic neutralization in front of the surface and the kinetic energy loss +inside the solid. The interplay of these two types of processes in the surface structure creation is described by the critical velocity. +Using the proposed theoretical model, the neutralization energy, deposited kinetic energy and critical velocities were calculated and +compared qualitatively with the experimental results. The results are consistent (after normalization) with previous experimental +data and molecular dynamics simulations for single ionized Xe and crystalline gold surface. +1. Introduction +Modification of metal, semiconductor and insulator surfaces +by the ion irradiation is of great importance for developing +new technologies for manufacturing a small functional elec- +tronics systems with nanometer dimensions, and has the po- +tential to introduce novel nanostructures and material proper- +ties not achievable by any other material processing methods +[1]. Modification of materials by swift (high kinetic energy) +heavy ion (SHI) irradiation is already used in many industrial +processes, such as: the generation of nanopores in polymers [2], +controlled drug delivery in biomedicine [3], precise band gaps +modification [4, 5], modification of high temperature supercon- +ductors [6], and others [7, 8]. It has also been demonstrated that +with SHI beams regular patterns (usually in amorphic form [9]) +of lateral dimensions in the order of several tens of nanometers +can be created. +One of the promising alternatives for creation of surface +nanostructures is modification of surface by an impact of a +∗Corresponding author +Email address: d.banas@ujk.edu.pl (D. Bana´s ) +single (i.e. each ion creates nanostructure) low-energy (slow) +highly charged ions (HCI). The term slow HCI usually refers +to impact velocities v ≪ 1 a.u., corresponding to 25 keV/amu +(nuclear stopping power regime). HCI are characterized by an +additional (to the kinetic energy) high potential energy, result- +ing from the removal of many of the electrons from the neutral +atom. For example, for Xe50+ ion the potential energy is around +100 keV, i.e. 8400 times higher than that of a single charged +xenon Xe+ ion. The neutralization energy of the HCI in the +interaction with solid surface is also large for very slow ions +(in keV energy range). As a consequence, the interaction of +slow single HCI with a surface is also governed by the potential +(neutralization) energy of the ion [10, 11, 12]. This energy is +deposited on a small surface area along the first few nanometers +below the target surface [13]. Recent research on 2D materials +shows [14], that potential energy deposition of highly charged +ion (Xe38+) is limited to only up to two layers within multilayer +MoS2 (on graphene). For very low ionic velocities (down to +v = 0.03 a.u.) the deposited potential energy (close to the ionic +neutralization energy) can lead [15] to creation of various sur- +face nanostructures, so far mainly observed on insulators such +Preprint submitted to Vacuum +January 11, 2023 +arXiv:2301.04010v1 [physics.atom-ph] 10 Jan 2023 + +as alkali and alkaline earth halides, oxides and polymers, but +also on highly oriented pyrolytic graphite (HOPG), sapphire +and gold crystals, and silicon semiconductor [16]. On the other +hand, for moderate ionic velocities (v ≈ 0.25 a.u.) both the +neutralization and the deposited kinetic energy participate in +the surface modification [10, 12]. Nanostructures created us- +ing HCI can have a form of hillocks, craters (called also pits) +or caldera-like structures, with diameter of about 5-20 nm and +a few nanometers vertical extension [15, 16, 11]. It is known +from experiments, that different parameters of ion beams, type +of irradiated materials and processing conditions lead to differ- +ent characteristic of the modifications obtained on a material +surface, including defect production, sputtering of material and +changes in material surface topology. +The recent studies of nanostructures formation on surfaces by +HCI concentrate mainly on the basic characterization of nanos- +tructures and fundamental understanding of the mechanisms re- +sponsible for the surface modifications [15, 10, 12]. Moreover, +most of the experimental observations were performed for in- +sulator while for semiconductors (pure Si) and metals (Ti, Au) +only single experiments were carried out, which due to the lack +of the systematic studies did not allow for a detailed exami- +nation of the mechanism of nanostructures production on such +surfaces [16]. The reason for the small interest in this type +of studies were the earlier experiments with swift heavy ions +(SHI), which suggested that in the interaction of such ions with +materials of high thermal conductivity, the production of nanos- +tructures is unlikely due to the rapid outflow of energy from the +area of impact. However, the results of experiments performed +by Pomeroy et al. [17] and our recent results [18] showed that +different nanostructures can be produced by slow single HCI +also on metallic surfaces. Unfortunately in both of these exper- +iments potential and kinetic energies of the ions were simulta- +neously changed, which made it difficult to separate their influ- +ence on the produced nanostructures. +Systematic experimental studies of the interaction mecha- +nism are very important also from the theoretical point of view +because there is still no unified picture of the nanostructure +creation process. Up to now, the proposed theoretical models +of the nanostructure production by slow single HCI, includ- +ing Coulomb explosion [19, 20, 21], molecular dynamics sim- +ulations [22, 23], inelastic thermal spike model [24, 25], and +plasma model [26, 27] describe the mechanism only in a qual- +itative manner and agree quantitatively only with the results +of selected experiments, mainly for insulators. For the metal- +lic surface modifications, the micro-staircase model of the HCI +neutralization accompanied by the charge dependent model of +the kinetic energy loss has been proposed [28, 12]. +The aim of the present study is the systematic experimental +and theoretical investigation of the mechanism of energy depo- +sition and nanostructures creation in collisions of a single HCI +with metallic surfaces. We consider the moderate ionic veloc- +ity region, characterized by the interplay of the neutralization +energy and the deposited kinetic energy. We performed the ex- +periment with Xeq+ ions (q = 25, 30, 35, 36 and 40) impinging +upon a gold nanolayer at kinetic energy 280-290 keV and with +Xe35+ ions at kinetic energies: 280 keV, 360 keV, 420 keV and +480 keV. As the results, we obtained the well pronounced mod- +ification of the surface in the form of craters. In the present +paper, the results are interpreted within the prediction of the +micro-staircase model [28, 12] and molecular dynamics simu- +lations for single ionized xenon hitting crystalline gold surface +[29]. According to the micro-staircase model, simultaneously +with the ion cascade neutralization above the surface, the neu- +tralization energy deposits into the solid inducing the first desta- +bilization of the target as a consequence of the high free elec- +tron density characteristic for conducting surfaces. Below the +surface, the kinetic energy loss is governed by the elastic colli- +sions between the ion (carrying the information about the ionic +initial charge and velocity) and target atoms. +This article is organized as follows. In Section 2 we dis- +cuss the energy deposition process during the interaction of HCI +with surface and current status of the experimental and theoret- +ical studies for single and highly ionized xenon atoms interact- +ing with metallic (gold) surface. In this section we also intro- +duce the micro-staircase model of the HCI-metal interaction. +Section 3 is devoted to the present experiment. We characterize +samples, describe the experimental conditions and the atomic +force microscope (AFM) system used for the sample imaging. +In Section 4 we present examples of the AFM images and ex- +tracted diameters and depths of the observed craters. In Sec- +tion 5 we discuss the results and compare them with theoretical +predictions and available experimental data for single ionized +xenon [30]. The concluding remarks are given in Section 6. +2. HCI - surface interaction +2.1. Overview of the HCI interaction with metallic surfaces +Up to now, nanometer-sized structures produced by individ- +ual HCI impact on conductive surfaces were reported for a crys- +talline Au(111) by Pomeroy et al. [17]. In this experiment, +the samples were irradiated with 200 keV Xe25+ and 350 keV +Xe44+ ions, which have significantly different potential ener- +gies, 8 keV and 51 keV, respectively. After irradiation the sam- +ples were analyzed in situ with scanning tunneling microscope +(STM). The STM images showed many different features on +the gold surface, such as isolated hexagons, hexagonal rings +with craters in the center and hexagonal islands with pits, with +the features density approximately equal to the ions fluence. +It is worth to note that previous sputtering measurements [31] +with gold did not report a measurable increase in sputter yield +with increasing of the HCI charge and thus probability for a +nanostructure formation on gold was assumed to be negligible. +Finally, Pomeroy et al. concluded that the primary formation +mechanism of the features they observed on Au(111), is related +to the kinetic energy (nuclear energy loss) and seems weakly +dependent on the potential energy of the HCI, they emphasized +the simultaneous change in the potential and kinetic energy of +the ions used in the experiment, which complicated to isolate +their contribution to the created nanostructures. Subsequent at- +tempt to repeat Pomeroy et al. experiment using 440 keV Xe44+ +ions has failed, probably due to too high surface roughness [32]. +A similar experiment, but at lower velocities, was also carried +out by our group [18]. In this experiment, nanolayers of gold +2 + +Q=Q +Rmin +D +Q R +( ) +q-1 +micro-staircase model of the cascade neutralization +and surface destabilization +final charge +state +elastic collisions +in solid +nanocrater formation +Q = q +fin +q-2 +e +- +macro steps +e +- +e +- +intermediate Rydberg state population +Q = q +fin +Z +q+ +initial charge +state +surface +Figure 1: Schematic description of the micro-staircase model of the cascade neutralization with intermediate Rydberg state population followed by rapid deexcitation +(both presented by dashed curves) and the nanocrater formation processes during the interaction of HCI with solid surface [12]. +and titanium, were irradiated with low-energy (50-120 keV) +highly charged xenon ions. The samples were prepared at In- +stitute of Electronic Materials Technology, Warsaw, Poland, by +sputtering of gold (50 nm) and titanium (75 nm) nanolayers on +polished crystalline quartz SiO2(100) 4-inch diameter wafers, +and titanium (25 nm - 75 nm) nanolayers on crystalline sili- +con Si (100) wafers. The samples were irradiated at the Kielce +EBIS facility (Institute of Physics, Jan Kochanowski Univer- +sity, Kielce, Poland) [33]. As a result of irradiation, we were +able to create nanohillocks on both titanium and gold surfaces +and perform statistical analysis of their heights and volumes us- +ing AFM images [18]. In this experiment the kinetic energy of +the ions have been charge dependent (because of the ion source +configuration) and thus it was difficult to extract separately the +potential or the kinetic energy influence. +A systematic analysis of the Xeq+ ion interaction with gold +nanolayers at moderate velocities (craters formation) will be +presented in Section 3. +2.2. Overview of the single charged ions interactions with +metallic surfaces +A many of scientists have discovered small craters on metal- +lic surfaces bombarded with single ionized high-energy heavy +ions which they attribute to the effect of spikes. The concept of +thermal spikes resulting from single ion impacts was discussed +for the first time in the literature in the 1950s by researchers +such as Brinkmann [34], Seeger [35], and Seitz and Koehler +[36, 37]. In particular, Merkle and J¨ager used transmission elec- +tron microscopy (TEM) to examine Au surfaces irradiated with +single ionized Bi and Au ions, in the energy range of 10-500 +keV and discovered craters on the irradiated surfaces for the +ion energies above 50 keV with fewer than 1% of collisions +causing the crater formation [38]. Average crater sizes were +typically about 5 nm. Although the authors conclude that spike +effects were responsible for the crater formation they attribute +the effect mainly to sublimation of surface atoms from the sur- +face [38]. Following this experiment, Birtcher and Donelly ir- +radiated Au(110) films with Xe+ ions at energies of 50 keV, +200 keV or 400 keV. They found, using in situ TEM, that single +xenon ion impacting on gold forms crater with size as large as +12 nm and that approximately 2-5% of impinging ions produce +craters [39, 30]. Authors concluded that crater formation re- +sults from ion-induced sudden melting (and volume expansion) +of the material associated with localized energy deposition (sur- +face energy spikes) and explosive outflow of material from the +hot molten core. The later experiments of Donelly and Birtcher +on surfaces of Ag, In, and Pb led them to the same conclusions +[40]. +The results of Donelly and Birtcher experiment [30] were ex- +amined using classical molecular-dynamics (MD) simulations +by Bringa et al. [29]. They performed simulations of crater for- +mation during 0.4-100 keV single charged Xe+ bombardment +of Au target. The simulations confirmed that the craters are +built by liquid flow of atoms from the interaction zone. They +also found that energy density needed for crater production +strongly depends on the heat spike lifetime and that for xenon +energies higher than 50 keV cratering can results from lower +energy densities due to long lifetime of the heat spike. MD +simulated cratering probability was always higher than 50% in +the studied energy range [29]. +2.3. Nanostructure formation on metallic surface: +micro- +staircase model +Recently, in the article [12] we discussed the nanohillocks +formation by the impact of Xeq+ ions on titanium and gold +nanolayers [18] using the micro-staircase model for the cascade +neutralization based on the quantum two-state vector model +(TVM). The model takes into account both the ionic neutraliza- +tion energy and the kinetic energy deposition inside the solid +3 + +[28, 12]. The similar model can be used for the analysis od the +craters formation. +According to the model, the process of the cascade neutral- +ization of the ion, Q = q → q−1 → ...Q(R) → ...qfin, is mainly +localized in front of the surface (see Fig. 1). At ion-surface +distance R, the electron is captured from the metal into the in- +termediate high-n (Rydberg) state of the ion almost in a ground +state. The population of each macro step consists of several +micro-steps (population of the low-l Rydberg states nQ with the +probabilities PnQ). For example, considering the Xe25+ ion im- +pinging upon the metal surface at moderate velocity v = 0.25 +a.u. we have the following populate scheme [28]: at ion-surface +distances R, in the range from R = 28 a.u. to R = 10 a.u., the +Rydberg states corresponding to n = 23 to n = 15 of the ion +with charge Q = q − 1 = 24 (core charge 25) are populated +with probabilities Pn25 = 0.01 → 0.2, � Pn25 = 1. After the first +macro-step is finished at R = 10 a.u, the population of the ion +of the charge Q = q − 2 with core charge 24 begins in the range +from R = 9 a.u. to R = 6 a.u. The states n = 14 to n = 12 are +populated with probabilities Pn24 = 0.3 → 0.35, � Pn24 = 1, and +so on. In Fig. 1) we present the final stages of the macrosteps +Q = q, Q = q − 1, ... Q = qfin. At each macro step, the rapid +deexcitation could be via radiative process and closer to the sur- +face via Auger type processes with secondary electron emission +in interplay with the described population process [28]. The +neutralization cascade finishes when the HCI arrives into the in- +teraction region at minimal ion-surface distance R = Rmin [12] +with the final charge qfin (Rmin is the distance from the jellium +edge [41]). The corresponding neutralization energy W(q,nMV) +within the nanolayer-metal-vacuum (nMV) system is deposited +into the first nanometers of the surface [42] very fast (few fs for +metal targets [11, 17]), increasing the energy density in the im- +pact region [43] and inducing the destabilization of the surface +[12]. +The neutralization energy W(q,nMV) is defined as a difference +between the potential energy Ep ≡ Wq,pot (which describes the +state of the ion before the beginning of the neutralization) and +the potential energy in front of the solid surface WqnMV +fin ,pot [41, +44, 12]: +W(q,nMV) = Wq,pot − WqnMV +fin ,pot. +(1) +The energy W(q,nMV) can be calculated using the results valid +for the metal-vacuum MV-system [12], which is supported by +the experimental fact that the lattice structure of the nanolayer +is very similar to the bulk material for the layers thickness [45] +considered in the present article. +Although many elementary charge exchange processes are +possible at solid surface, we assume that the ion with Q = qfin +penetrates the surface. Within the framework of the model, we +also assume that neutralization inside the solid in the process +of the nanostructure formation can be neglected (has negligible +influence on the total deposited energy). The last assumption +is based on the fact that the nanocraters are formed in the nar- +row region of the depth ∆x smaller compared to the penetration +depth necessary for the ionic charge to be significantly changed +[12]. Therefore, we use Q ≈ qfin as the ionic charge for the +analysis of the ionic motion and the corresponding processes +inside the target (see Fig.2). For more accurate description of +the overall neutralization process, the analysis of the neutraliza- +tion below the surface can be added to our model. +Below the surface, the ions constantly lose their kinetic en- +ergy due to the elastic collisions with the target nuclei (nu- +clear stopping power dEn/dx) [11, 46, 47] and the inelastic in- +teraction with the target electrons (electronic stopping power +dEe/dx). Simultaneously, the damage of the local atomic struc- +ture and the surface modification due to the ionic kinetic energy +loss take place. On the overall ionic trajectory the ionic kinetic +energy is deposited into the solid. However, analyzing the ex- +perimentally obtained surface nanostructures, relevant is only +the near surface region of the length ∆x [12], so that +Ek,dep = (dE/dx) · ∆x. +(2) +For low to moderate ionic velocities electronic stopping power +can be neglected, i.e., dE/dx = dEn/dx = NS n, where N is the +atomic density of the target. The corresponding nuclear stop- +ping cross section S n we calculate using the classical scattering +theory with charge dependent ion-target atom interaction poten- +tial [48, 12]: +Vint(r) = (Z1 − Q)Z2 +r +ϕ( r +au +) + QZ2 +r ϕ( r +as +), +(3) +where Z1 and Z2 are the nuclear charges of the projectile and the +target atom, respectively. Other quantities are explicitly given +in [12]. We note that, the charge dependence of the energy +loss has been firstly theoretically introduced by Biersack [49]. +Further, the charge dependent kinetic energy transfer for HCI +interacting with C nanomembrane and C foil target was elabo- +rated in [50]. The time-dependent interatomic potential energy +was used to get the more accurate model for the calculation of +the kinetic energy loss in [51]. +3. Experiment +The studies presented in this paper are continuation of our +research [18, 52] related to the nanostructure formation in in- +teractions of highly charged xenon ions with metallic surfaces. +The aim of the experiments carried out for the purposes of cur- +rent work is to separate, as far as it is possible, the influence of +kinetic and potential energies of the HCI xenon ions on the pro- +duced surface nanostructures. In the measurements we use gold +nanolayers of the thickness of 100 nm deposited on Si (110) +wafers. The structure and properties of such nanolayers were +expected to be similar to the bulk metal, but with possible lower +density due to nanolayer structure [45] and an influence of the +substrate cannot be completely excluded [10]. +3.1. Samples +The 100 nm Au nanolayers used in this experiment were +prepared at Institute of Electron Technology (Warsaw, Poland) +using high vacuum (13·10−8 hPa) e-beam evaporation VST +TFDS-462U deposition system. The metallic nanolayers were +evaporated on Topsil (Warsaw, Poland) Si (110) polished prime +4 + +wafers type N 4-inch diameter. The thickness of the silicon +wafer was 0.635 mm ± 0.015 mm. The deposition rate was +0.4 nm/s. The thickness of the gold was set and controlled by +a crystal oscillator. Just after preparing, the wafers were cut +into rectangles of dimensions of 0.5 cm x 1 cm. The rough- +ness of the samples surface was checked by AFM technique. +Root-mean-squared (RMS) roughness determined by the AFM +technique (UMCS, Lublin, Poland) for a few randomly selected +(1 µm x 1 µm) areas of the gold nanolayers using NanoScope +Analysis ver.1.40 (Veeco, USA) program were on the level of +0.45±0.1 nanometers. The crystalline structure of the substrate +was confirmed based on the measurements carried out using +the XRD technique. Using the GIXRD technique, it was deter- +mined that the 100 nm gold nanolayers have a polycrystalline +structure homogeneous in depth. Additionally using the XRR +technique, the thickness and density of the nanolayers were +measured. Thickness turned to be consistent with the declared +one (100 nm), but the density was slightly lower (but within +the uncertainties) than the bulk density. The XRR, XRD and +GIXRD measurements were performed with X’Pert Pro MPD +reflectometer/diffractometer (for details see [53, 54]), placed at +Institute of Physics of UJK (Kielce, Poland). The 100 nm Au +nanolayers were irradiated at the Kielce EBIS facility of the +Jan Kochanowski University (Kielce, Poland) [33], under high +vacuum conditions. After irradiation the samples were again +checked with AFM technique (UMCS, Lublin). +3.2. Kielce EBIS facility +The Kielce EBIS facility, built by the Dreebit (Dresden, Ger- +many), is equipped with electron beam ion trap (EBIS-A) [55]. +The source supplies a wide range of slow HCI from bare ions of +light elements to Ne-like and Ar-like ions of high-Z elements. +The maximum electron energy and current available for ioniza- +tion of the trapped ions are equal 25 keV and 200 mA, respec- +tively. The ions produced in the EBIS-A source can be extracted +both in a pulse mode (pulse width from 2 µs up to 40 µs) and +leaky mode (DC mode) by applying an acceleration voltage up +to 30 kV. Highly charged ions extracted from the EBIS-A ion +source are guided by ion beam optical elements (einzel lens and +X-Y deflectors) of the first straight section of the facility to the +double focusing analyzing magnet separating the ions accord- +ing to their mass to charge ratio. The first section of the beam- +line includes a quadrupole section with pressure gauge, 4-jaw- +slit collimation system and a Faraday cup. The ions separated +in the analyzing magnet are directed to the second straight sec- +tion of the EBIS-A facility. In this section, a pressure gauge, +X-Y deflectors, a Faraday cup and an einzel lens are mounted. +Finally, the highly charged ions collide with a sample mounted +on a 5-axis universal manipulator placed in the experimental +chamber. The manipulator allows for x, y, z linear movements, +polar and azimuthal rotations of a sample and variation of its +temperature in the range of 100-1000 K. The beamline can be +biased with positive or negative high voltage allowing ion ac- +celeration or deceleration. For current facility configuration the +ion energies can be set from 2.5 keV x q up to 30 keV x q, with +q denoting the ion charge state. All components of the EBIS-A +facility fulfill the UHV standards and after baking of the sys- +tem at 150◦C the pressure is in the few 10−10 mbar range (in +the beamline). One of the unique features of the EBIS facility +is the ability to prepare, irradiate by highly charged ions and +characterize the studied samples in the UHV conditions. +3.3. Measurements +In the measurements isotopically pure highly charged Xeq+ +ions were extracted from the EBIS-A and, after selecting given +ion charge state in the dipole magnet, were used to irradiate the +nanolayers. The ion beam current was measured with a mov- +able Faraday cup mounted in front of the sample. The spot ra- +dius of the ion beam on the sample was around 1.5 mm ± 15% +as it was determined by moving the Faraday cup across the ion +beam (from the beam profile). The ion fluence was estimated +on the level 1010 ions/cm2 (with uncertainty of the 10-15%). +The samples were first placed in a loading chamber pumped +to about 10−7 mbar, and then transmitted to the experimental +chamber. The vacuum in the experimental chamber was around +(2 − 5) × 10−8 mbar. After irradiation, the sample was trans- +ferred back to the loading chamber and was stored there until it +was removed for atomic force microscopy investigations, which +were performed in the air. The measurements were performed +for two configuration: constant kinetic energy of the ions equal +to 280-290 keV and different charge states of the xenon ions +Xeq+, where q = 25, 30, 35, 36, 40 and constant charge state +(Xe35+) of the ions and different kinetic energies 280 keV, 360 +keV, 420 keV and 480 keV. +3.4. AFM system +The topographic modifications of the samples surface in- +duced by Xeq+ ions were investigated using atomic force mi- +croscopy in the Analytical Laboratory of Faculty of Chem- +istry, UMCS, Lublin, Poland. AFM measurements of the stud- +ied samples were performed using Multimode 8 (Bruker) AFM +equipped with NanoScope software (Bruker-Veeco, USA). The +AFM was operated in SCANASYST-HR fast scanning mode +using SCANASYST-AIR-HR probe (Silicon Tip on Nitride +Lever) (Bruker) with the cantilever of force constant k = 0.4 +N/m. The lateral and vertical resolutions were 4 nm and 0.1 +nm for the 1 µm x 1 µm, and 2 nm and 0.1 nm for the 500 nm +x 500 nm images. The obtained images were analyzed with +Nanoscope Analysis ver. 1.40 software (Veeco, USA). +4. Results +4.1. AFM images +The AFM images of the nanolayers before irradiation (left +panel) and after irradiation with 280-290 keV Xe30+, Xe36+, +Xe40+, and Xe35+ of different kinetic energies are presented in +the Fig. 2. The images were analyzed using the NanoScope +Analysis software. The size of presented area is 500 nm x 500 +nm. In the images of the irradiated samples, we can clearly see +the modifications caused by the ion impact. We would like to +stress here, that such excellent images of metallic surface mod- +ification caused by HCI impact, to our knowledge, have never +5 + +Xe36+ Ekin= 290 keV +Xe30+ Ekin= 280 keV +Xe40+ Ekin= 290 keV +Xe35+ Ekin= 280 keV +Xe35+ Ekin= 360 keV +Xe35+ Ekin= 420 keV +Xe35+ Ekin= 480 keV +before irradiation +Figure 2: Topographic AFM 3D images of Au 100 nm nanolayer deposited on Si surface before and after irradiation with HCI Xeq+. Top row: images of the +nanolayers before irradiation (left panel) and after irradiation with 280-290 keV Xe30+, Xe35+, Xe40+. Bottom row: images of the nanolayers after irradiation with +Xe35+ of different kinetic energies. The images were analysed using the software NanoScope Analysis ver. 1.40 (Veeco, USA). +been registered. The same modifications were observed for all +irradiated samples, with surface density of the nanostructures +approximately equal to the ion fluence, i.e. one nanostructure +per one HCl ion impact. Analogous efficiency of the nanostruc- +ture creation was observed by Pomeroy [17]. +The measured modifications have the form of craters, which +is confirmed by enlarged AFM 3D images of the individual +Figure 3: Upper panel: examples of the 3D AFM images of the nanostructures +on Au 100 nm/Si nanolayer surface irradiated by 280-290 keV Xe30+, Xe35+ +and Xe40+. Lower panel: upside-down 3D AFM images of the nanostructures +on Au 100 nm/Si nanolayer surface irradiated by 360, 420 and 480 keV Xe35+. +The images were analysed using the software NanoScope Analysis ver. 1.40 +(Veeco, USA). +nanostructures which are presented in the Fig. 3. In the upper +panel the example of the 3D AFM images of the nanostructures +on Au 100 nm/Si nanolayer surface irradiated by 280-290 keV +Xe30+, Xe35+, Xe40+ are presented. All observed structures had +a similar crater-like shape, i.e. a cavity, sometimes with a ring +around it (check the middle image). Merkle and Jager [38] and +Bringa et al. [29] postulate that these rings around the cavity +arise from the sputtering (or rather an outflow) of the original +atoms being at the place of the structure formation. This was +confirmed by MD simulations presented in the article of Bringa +et al. [29]. Similar shape of the crater formed on a Si(100) +surface by bombardment of a Xe44+ HCI was also observed in +the simulations performed by Insepov et al. [27] using plasma +model of space charge neutralization based on impact ioniza- +tion of semiconductors at high electric fields. In the lower panel +of the Fig. 3 upside-down 3D AFM images of the nanostruc- +tures on Au 100 nm/Si nanolayer surface irradiated by 360, 420 +and 480 keV Xe35+ are presented to confirm crater like shape of +the nanostructures. +4.2. Analysis of the AFM images +In many surface studies, a common data analysis strategy is +to correlate the mean size of nanostructures (parameters like: +diameter, depth, volume) with different ion parameters, e.g. ki- +netic energy (ionic velocity) and potential energy (ionic charge +state), nuclear and electronic stopping powers, etc. Following +this strategy we have performed size analysis of the observed +nanostructures. +For this purpose, all observed images were +6 + +Xe30+ +Xe35+ +Xe40+ +20 nm +360 keV +420 keV +480keV +20 nm0.0 +1:Height +500.0nm0.0 +1:Height +500.0nm0.0 +1:Height +500.0nm0 +1:Height +500.0nm0.0 +1:Height +500.0nm0.0 +1:Height +500.0nm0.0 +1:Height +500.0nm0.0 +1:Height +500.0nmFigure 4: Example of the individual crater profile (black points) with fitted +Gaussian curve (solid line). The sigma σ (standard deviation), FWHM (full +width in the half of maximum) and crater depth d quantities are Gaussian distri- +bution parameters. The crater diameter on the surface is assumed as 2FWHM. +first carefully checked and optimized with NanoScope Anal- +ysis software and, after extracting of the data from the AFM +images (using STEP function of the software), analysed with +Origin Pro data analysis software. From the individual profile +of the craters we have extracted their size parameters, includ- +ing the diameter on the surface. All unambiguously identified +structures on the surface of the samples were analyzed indepen- +dently. Example of the individual crater profile (black points) +is presented in the Figure 4. +The profiles were fitted by Gaussian curve (solid line), which +reflected very well the shape of the crater. At this point, we note +that the needle used for the AFM analysis of all samples had a +tip curvature radius r = 2 nm, and the structures after HCI mod- +ification were characterized by diameters of about 10-25 nm, +so an incorrect tip contact was considered unlikely, especially +in the diameters of nanostructures on the sample surface. The +crater diameter on the surface was defined as double FWHM (2 +× FWHM). In the example presented in the Figure 4, the crater +diameter at surface was fitted as 15.16 nm, while its depth as +0.93 nm. An alternative way to determine the diameter is to +take values of four standard deviations (4×σ). In presented ex- +amples, it gives 4σ = 12.88 nm. In general, it was observed +that crater diameter defined as 2 × FWHM was about 10-15% +higher than 4×σ quantity. +For the Au nanolayer irradiated by Xe ions in given charge +state, for each sample around 50 to 150 craters were analyzed in +the way described above. Finally, the mean values of the crater +depth and crater diameters for each irradiated Au nanolayer +were calculated. Based on the statistical analysis of the crater +profiles, the dependence of the crater depth and crater diameter +on the Xe ions potential and kinetic energy were studied. In +the case of the crater depth no dependence on the ions poten- +tial energy was observed. The crater depth was on the constant +level of about 0.9 nm ± 0.15 nm. The linear fit to the data, in- +cluding uncertainties defined by the standard deviation of mean +value, gave a very week dependence (the slope is equal to 0.001 +nm/keV). +The obtained mean values of craters diameter in function of +the potential energy of the Xeq+ ions are plotted in the Fig. +5. The uncertainties marked for experimental points were cal- +culated as the sum of the mean value standard deviation and +10% of the mean value (compensation of the difference be- +tween 2FWHM and 4σ quantities within uncertainty). The Xe +ions charge states marked by (*) denoted a slightly different ki- +netic energy (290 keV), caused by difficulties in setting a given +charge state and kinetic energy. As one can see from the fig- +ure we have observed clear influence of the ionic charge state +(expressed via initial potential energy) on the nanocrater diam- +eter. For the lowest ion charge state (25+) the mean nanocrater +diameter is 12.0 nm, next this parameter systematically grows, +reaching for the highest charge state the value 23.4 nm. +The results of the study of the crater diameter in function of +the ions kinetic energy are shown in the Figure 6 for Xe35+. The +nanocrater diameter is in the range 13-15 nm. The linear func- +tion fitted to the experimental points showed a weak alteration +of the dependence. In the Figure 6 the results of Donnelly and +Birtcher experiment [39] for Xe+ ions are also presented which +confirm small dependence of the created nanocrater diameter +on the ions kinetic energy in the considered energy range. On +the other hand, the nanocrater diameters for HCI xenon ions are +much higher than for single ionized xenon. +Figure 5: Dependence of the craters diameter on the Xeq+ ions potential energy. +The Xe ions charge states marked by (*) denoted a slightly different kinetic +energy (290 keV). +7 + +40 +Au 100 nm on Si +35 +f craters diameter (nm) +E,= 280 keV +30 +Xe40+ (*) +25 +Xe36+(*) +Xe35+ +20 +T +Xe30+ +15 +Mean of +10 +2FWHM +5 +0 +0 +5 +10 +15 +¥20 +25 +30 +35 +40　45 +Ep (keV)2FWHM +-0.2 +-0.4 +Depth (nm) +-0.6 +FWHM +-0.8 +o = 3.22 nm +W +FWHM = 7.58 nm +d = 0.93 nm +-1.0 +- +-1.2 +-3g +-2g +a +2g +3g +(×c,c) +experiment +-1.4 +fit +0 +10 +20 +30 +40 +50 +60 +Distance (nm)Figure 6: Dependence of the nanocrater diameter created by the Xe35+ ions +impinging on Au surface on the kinetic energy (this experiment). For compari- +son, the results of Donnelly and Birtcher experiment [39] for Xe+ ions are also +presented. In the figure, also the nuclear stopping power S1/2 (solid line) is +presented. +5. Discussion +5.1. Theoretical model of the crater formation +In order to interpret the present experiments performed with +Xeq+ ions of initial charges q = 25, 30, 35, 36 and 40 in the +interaction with a gold 100 nm nanolayer deposited on Si (110) +wafers (nMV-system) at velocity v = 0.29 a.u., as well as to +examine the velocity dependence studied in the case of Xe35+ +for v = 0.29, 0.33, 0.36 and 0.38 a.u. we use the micro-staircase +model. +The formation of nanocraters we discuss from the stand- +point of the energy dissipation into the surface, which consists +both of the neutralization energy and the deposited kinetic en- +ergy [41, 44, 12]. For velocities characteristic for the crater +formation the neutralization is incomplete so that the corre- +sponding neutralization energy represents only a part of the +ionic initial potential energy. The remaining potential energy +Ep − W(q,nMV) contributes to the charge dependent potential in- +teraction (Eq.(3)) between the ion and the target atoms and thus +it is converted into kinetic energy of the target atoms (stop- +ping power calculated in micro-staircase model is charge de- +pendent). We calculate the neutralization energy according to +Eq. (1) for W(q,nMV) = W(q,MV); taking into account that the neu- +tralization energy is weakly dependent on the solid work func- +tion φ, we consider the neutralization energy for φ = 5 eV (work +function of Au is 5.47 eV). For the calculation of the kinetic +energy loss we employ Eq. (2) for the active interaction length +∆x ≈ 5¯c ≈ 38.5 a.u., where ¯c is the mean lattice constant for +Au-target; we note that the crater depth dmax ≈ 1nm = 18.9 a.u. +To define the ion-atom interaction in solid we use the charge of +the projectile Q = q fin(q, v) obtained in [12]. +Figure 7: Upper panel: neutralization energy W(q,nMV) and deposited kinetic +energy Ek,dep versus ionic velocity v for Xeq+ ions, q = 25, 30, 35 and 40, +impinging on the Au nanolayers (formation of the crater in the present exper- +iment). Lower panel: the critical ionic velocity vc versus initial ionic charge +q. +In Fig. 7, at upper panel, we present the neutralization energy +W(q,nMV) and the deposited kinetic energy Ek,dep relevant for the +surface nanocrater creation by the impact of Xeq+ ions with core +charges q = 25, 30, 35 and 40 on 100 nm Au nanolayer on Si +(110) wafers as a function of the ionic velocity v. The neutral- +ization energy W(q,nMV) decreases with increasing of the ionic +velocity v; on the other hand, the deposited kinetic energy Ek,dep +increases with increasing of v. The results indicate the interplay +of these two energies in the process of the surface nanocrater +formation. That is, we define [12] the critical velocity vc by the +relation: +W(q)(vc) = Ek,dep(vc). +(4) +For velocities v ≪ vc (very low ionic velocities) dominant +8 + +1800 +100 nm Au on Si +V +1600 +c +V +exp +1400 +.40+ +△x = 38.5 a.u +(a.u.) +Xe +1200 +1000 +.35+ +Xe +800 +Xe40+ +600 +30+ +Xe25+ +Xe +400 + 25+ +Xe +200 +0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +v (a.u.)35+ +25 +Xe +E = 25.5 keV +p +Mean of craters diameter (nm) +present experiment (Xe35+ +351 +20 +Donnelly and Birtcher (Xet) +15 +10 +5 +/2 +n +0 +0 +100 +200 +300 +400 +500 +600 +Ek (keV)100 nm Au on Si +Vexp +0.25 +C +exp +0.20 +(a.u.) +x = 38.5 a.u. +7 +0.15 +0.10 +≤ v. nanohillocks formation +V +0.05 +exp +V. craters formation +V +exp +0.00 +25 +30 +35 +40 +ionic charge (a.u.)Table 1: +Critical velocities vc in the case of the surface nanocrater formation +in the nMV-system by the impact of Xeq+ ions. +100 nm Au nanolayer +q +25 +30 +35 +40 +Ek (keV) +280 +280 +280 +280 +vexp (a.u.) +0.29 +0.29 +0.29 +0.29 +vc (a.u.) +0.07 +0.17 +0.22 +0.23 +role in the energy participation in the solid has the neutraliza- +tion energy W(q,nMV), while for v ≫ vc (swift heavy ions) the +deposited kinetic energy Ek,dep completely determines the pro- +cess of the nanostructure formation [12]. The quantity vc we +present in Fig. 7 at lower panel as a function of the initial ionic +charge q. The values of the critical velocities are also given in +Table 1. +For all considered ionic charges the critical velocities vc are +lower compared to the experimental value vexp = 0.29 a.u. +(Ek = Mv2 +exp/2, M = 131 · 1836 a.u. for Xeq+ ions, where +Ek denotes the initial ionic kinetic energy). For charges q = 35 +and q = 40, the critical ionic velocities vc are close to the ex- +perimental one (see Table 1), indicating that both energies con- +tribute to the crater formation. The values of vc for Xe25+ and +for Xe30+ are much smaller than the experimental ones, so that +the main contribution in the nanostructure formation gives the +deposited kinetic energy Ek,dep. Concerning the type (shape) +of the nanostructures, the appearance of the nanocraters in ex- +periment is in accord with the prediction of the micro-staircase +model. On the other hand, the hillocks have been obtained in +experiment with Xe35+ ions [18] impinging upon the surface of +the 50 nm gold layer at velocity 0.19 a.u., while the critical one +is 0.22 a.u. [12]. In the case of 25 nm titanium nanolayers the +experimental velocities for q = 20, 25, 30 and 35 were 0.144, +0.16, 0.176 and 0.19, in a.u., respectively. The corresponding +critical velocities are 0.06, 0.16, 0.22 and 0.24, in a.u. [12]. The +results of the present experiment and the results of the previous +ones [18, 52, 17] confirm a common conclusion: for the ionic +velocities v < vc or v ≈ vc the surface modification leads to the +nanohillocks formation [18, 12], while for v > vc the predomi- +nant surface structures are the craters (rings) [52, 17, 12]. +The neutralization energy W(q,nMV) and the deposited kinetic +energy Ek,dep can be also connected to the size of the formed +nanostructures. The experimental results for the crater diame- +ters show the significant increasing from q = 25 to q = 40, see +5. For vexp = 0.29 a.u. and Xe25+ ion the diameter D = 12 +nm (226.8 a.u.) +and for Xe40+ ion diameter D = 23.4 nm +(442.3 a.u.). The ionic neutralization energy (and also the ini- +tial ionic potential energy) exhibits the same increasing behav- +ior (W(25,nMV) = 47 a.u. and W(40,nMV) = 162 a.u.), see Table 2. +The q dependence of the deposited kinetic energy, obtained +on the base of Eq. 3, is less pronounced (Ek,dep for Xe25+ is 557 +a.u. and for Xe40+, Ek,dep = 587 a.u.), see Table 2. For these +reasons, it is convenient to present the experimentally obtained +crater diameters as a function of the potential (or the neutraliza- +tion) energy. +The velocity effect on the crater diameter D for Xe35+ ions +we study for the experimental values vexp = 0.29, 0.33, 0.36 +and 0.38 a.u. From the experimental results one recognize the +weak decreasing of the quantity D with increasing of the ionic +velocity (kinetic energy) see Fig. 6; (for v =0.29 a.u. diameter +D = 15 nm (283.5 a.u.) and for v =0.38 a.u. diameter D = 12 +nm (226.8 a.u.)). On the other hand, the deposited kinetic en- +ergies Ek,dep increase slightly with increasing of v (for v=0.29 +a.u. Ek,dep= 577 a.u. and for v =0.38 a.u. Ek,dep =635 a.u.), +while the neutralization energy W(35,nMV) show a noticeable de- +creasing character (for v =0.29 a.u. W(35,nMV) =127 a.u. and for +v =0.38 a.u. W(35,nMV) =35.5 a.u.), see Tab. 3. +The role of the neutralization energy W(q,nMV) and the de- +posited kinetic energy Ek,dep can be more precisely discussed +from the relation between the crater diameter and the total de- +posited energy: Etot,dep = Ek,dep + W(q,nMV). Assuming that +the energy Etot,dep is localized in the cylindrical region od the +diameter D and depth ∆x, we get the relation: +D = f +� +Ek,dep + W(q,nMV), +(5) +where the factor f reflects the target properties. Both energies +Ek,dep and W(q,nMV) are charge dependent, so that the nanocrater +size will express the same behavior. From Eq. 5 and Tables 2 +and 3 one conclude that the main contribution to the diameter D +gives the deposited kinetic energy. However, the neutralization +energy term in Eq. 5 must be taken into account in order to ob- +tain the experimentally observed behavior of the crater diameter +D discussed in Tables Tab. 2 and Tab. 3: pronounced increas- +ing od D with increasing of q and weak decreasing of D with +increasing of the ionic velocity v. The discussed significance +of the deposited kinetic energy and the role of the neutraliza- +tion energy is characteristic for the moderate velocity case used +in the experiment. We note that for the very low velocities, +the neutralization energy (close to the potential energy) plays +a dominant role. The increasing of D with increasing of q and +the v-dependence of the crater diameter obtained in the present +experiment is in a qualitative agreement with the prediction of +the proposed model. +Within the framework of micro-staircase model, the mecha- +nism of the nanocraters and nanohillocks formation at metallic +surfaces is different. At velocities v < vc characteristic for the +hillock formation, a dominant role has the neutralization pro- +cess: the strength of the bonds between atoms decreases in- +ducing their stretching. The rearrangement of atoms leads to +Table 2: Neutralization energy W(q,nMV), deposited kinetic energy Ek,dep and +crater diameter D in the case of the surface nanocrater formation for v = vexp = +0.29 a.u. in the nMV-system by the impact of Xeq+ ions, for ∆x ≈ 38.5 a.u. +100 nm Au nanolayer +q +25 +30 +35 +40 +W(q,nMV) (a.u.) +47 +88 +127 +162 +Ek,dep (a.u.) +557 +569 +577 +587 +D (nm) +12 +13 +15 +23.4 +9 + +Table 3: Neutralization energy W(q,nMV), deposited kinetic energy Ek,dep and +crater diameter D in the case of the surface nanocrater formation for vexp = +0.29, 0.33, 0.36 and 0.38 a.u. in the nMV-system by the impact of Xe35+ ions, +for ∆x ≈ 38.5 a.u. +100 nm Au nanolayer +vexp (a.u.) +0.29 +0.33 +0.36 +0.38 +W(q,nMV) (a.u.) +127 +66.8 +47.2 +35.5 +Ek,dep (a.u.) +577 +600 +608 +612 +D (nm) +15 +12.9 +13.9 +12.15 +the rise of the volume above the surface and hillock forma- +tion. The deposited energy is insufficient for melting the ma- +terial (the hillocks are formed without melting). The predicted +mechanism of the nanohillock formation on metal surface [12] +is different in comparison to the thermal spike model used in +the case of nanohillock formation on insulator [56]. In the case +of crater creation (for v > vc) considered in the present paper, +the neutralization (above the surface) induces the lattice vibra- +tion and the first destabilization of the target. Inside the solid, +the elastic collisions of the charged projectile with target atoms +and produced recoils lead to the disordering of the target atoms +generating the highly disturbed near surface area V. A large +amount of kinetic energy deposits into the solid, resulting in a +significant decrease in the target cohesive energy. The strength +of the bounds between the target atoms inside the crater vol- +ume tends to be zero and a number of atoms are ejected from +the surface. In the intermediate stages of the craters formation, +in the centre of the active volume V, it is possible that the tem- +perature far exceeds the melting temperature. The deposited +neutralization energy during the process above the surface has +a small contribution to the nanocrater formation in comparison +to the deposited kinetic energy during the collision cascade be- +low the surface; however, the main q and v dependence of the +crater size are governed by the neutralization energy. +5.2. MD simulations +We also compare the present experimental results with +molecular dynamics (MD) simulations presented in [29]. We +compare our results for HCI xenon ion with Xe+ ion after fit- +ting the data on Fig. 5, and further normalization to the poten- +tial energy equal to the potential energy of Xe+. The results +of the comparison we present in Fig. 8. In the figure, the nu- +clear stopping power S1/2 (solid line) and the ion energy E1/3 +(dashed line) curves are also presented. We obtain very good +agreement with the experimental data of Donnelly and Birtcher +[39] and MD simulations [29], which confirms the validity of +our experimental procedure. +It is important to mention that, very recently, molecular dy- +namics methodology coupled with two-temperature model (2T- +MD) [57], was used by Khara et al. to simulate the structural +evolution of bcc metals (Fe and W) and fcc metals (Cu and +Ni) following irradiation by SHI (electronic stopping power +regime) [23]. +They found that number of material parame- +ters (melting temperature, electronic thermal conductivity and +Figure 8: Comparison of the results of crater radius, as a function of the ion +kinetic energy, obtained by our group with the results of the experiment (Don- +nelly and Birtcher) for single charged Xe+ ion impinging on Au surface [39] +and MD simulations [29]. In the figure, the nuclear stopping power S1/2 (solid +line) and ion energy E1/3 (dashed line) curves are also presented. +electron-phonon coupling strength), and their electronic proper- +ties temperature dependence, have a strong influence on the re- +sistance of metals to damage induced by SHI irradiation. They +also showed that high thermal conductivity and relatively low +electron-phonon coupling of fcc metals render them relatively +insensitive to damage, in spite of their relatively low melting +temperatures. The strong electron-phonon coupling of the bcc +metals (Fe and W) is primarily responsible for the sensitivity +of these metals to damage [23]. The cited calculations are in +contradiction with the experimental results for Au (fcc metal) +- HCI systems, for which we obtain the surface nanocraters in +the velocity range v ∈ [0.29, 0.38] a.u. and the nanohillocks +for lower ionic velocities v ∈ [0.144, 0.19] a.u. [18]. In the +case of nanocrater formation, both the deposited kinetic en- +ergy and the neutralization energy participate in the process; +the nanohillocks are formed predominantly by the participation +of the neutralization energy. The calculations [23] showed a +significantly different response of bcc and fcc metals to the de- +position of energy in the interaction of SHI ions with surfaces +and encouraged us to undertake such tests for HCI. At the mo- +ment, similar calculations does not exist for HCI, where it is +necessary to take into account the neutralization process. The +model proposed here represents a theoretical approach of that +kind, stimulated by the experimental findings. +6. Conclusions +Understanding of mechanism of the nanostructures creation +on metallic surfaces is very important both from the theoret- +ical and possible application point of view. In this paper we +10 + +10 +Xe->Au +.1/3 +this experiment (extrapolated +Donnelly and Birtcher +Crater radius (nm) +MD simulations +H +1/2 +8 +1 +1 +10 +100 +1000 +Ek (keV)have studied Au nanolayers surfaces irradiated by slow highly +charged Xeq+ ions (q = 25, 30, 35, 36 and 40). For the first +time, for such systems, well pronounced modifications of the +nanolayers surfaces, due to impact of the HCI ions, in the form +of nanocraters have been observed. This allowed for systemati- +cal study of dependence of the size of nanostructures on poten- +tial and kinetic energy of the ions. Analysis of the crater diam- +eter D for different initial charge states q of the Xe ions showed +a significant dependence of the quantity q (expressed via poten- +tial energy in Fig. 5). Additionally, for interaction of the Xe35+ +ions with Au nanolayers the dependence of the structure forma- +tion on the ion kinetic energy (280 keV, 360 keV, 420 keV and +480 keV) was studied. Week alteration of the crater diameter +(Fig. 6) with the ion kinetic energy was observed in the ana- +lyzed energy range. Our results were qualitatively interpreted +within the micro-staircase model for the neutralization energy +combined by the charge dependent kinetic energy deposition. +The experimental results are also compared with the available +simulations and the previous experimental data. The results will +be potentially of great importance for further development of +modern technologies (e.g. single HCI nano-pattering [58], role +of the HCI impurities in tokamak plasma-metallic wall interac- +tion [59]) and will open up many application possibilities (e.g. +DNA sequencing or water desalination [60]). +Acknowledgments +The equipment was purchased thanks to the financial support +of the European Regional Development Fund in the framework +of the Polish Innovative Economy Operational Program (con- +tract no. WNP-POIG.02.02.00-26-023/08), the Development +of Eastern Poland Program (contract no. +POPW .01.01.00- +26-013/09-04) and Polish Ministry of Education and Science +(project 28/ 489259/SPUB/SP/2021). N. N. Nedeljkovi´c and +M. D. Majki´c are grateful for the support of the Ministry of +Education, Science and Technological Development of the Re- +public of Serbia (projects 171016, 171029). +References +[1] J. V. Barth, G. Costantini, K. Kern, Engineering atomic and molecular +nanostructures at surfaces, Nature 437 (2005) 671–679. doi:10.1038/ +nature04166. +[2] P. Apel, Swift ion effects in polymers: industrial applications, Nuclear +Instruments and Methods in Physics Research Section B: Beam Inter- +actions with Materials and Atoms 208 (2003) 11–20. doi:10.1016/ +S0168-583X(03)00634-7. +[3] V. Rao, J. Amar, D. Avasthi, R. N. 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Lett. 10 (5) (2019) 904–910. doi: +10.1021/acs.jpclett.8b03666. +12 + diff --git a/C9AzT4oBgHgl3EQfiP2U/vector_store/index.pkl b/C9AzT4oBgHgl3EQfiP2U/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d30733eca6d6c8c5de2ea6c9fb0d1dca256c726f --- /dev/null +++ b/C9AzT4oBgHgl3EQfiP2U/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0740d309b77b02f3297e7d171ed6aec2497ff29b2cd00fd0e621246a9612df48 +size 192326 diff --git a/C9E4T4oBgHgl3EQfeg2i/vector_store/index.pkl b/C9E4T4oBgHgl3EQfeg2i/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..05b69b20fb75a0e275d1640836d0ae6db72bd480 --- /dev/null +++ b/C9E4T4oBgHgl3EQfeg2i/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:33aafc7695c82b0c1967a10fb2066765d6479bd98643fd06ae0dc3ff371a1fc7 +size 256675 diff --git a/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/2301.11798v1.pdf.txt b/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/2301.11798v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..127ff646228d91216133a5d044f654610ac0632d --- /dev/null +++ b/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/2301.11798v1.pdf.txt @@ -0,0 +1,592 @@ +MedSegDiff-V2: Diffusion based Medical Image +Segmentation with Transformer +Junde Wu1, Rao Fu2, Huihui Fang1, Yu Zhang2, and Yanwu Xu1 +1 Baidu Research +2 Mind Vogue Lab +Abstract. The Diffusion Probabilistic Model (DPM) has recently gained +popularity in the field of computer vision, thanks to its image genera- +tion applications, such as Imagen, Latent Diffusion Models, and Stable +Diffusion, which have demonstrated impressive capabilities and sparked +much discussion within the community. Recent studies have also found +DPM to be useful in the field of medical image analysis, as evidenced by +the strong performance of the medical image segmentation model Med- +SegDiff in various tasks. While these models were originally designed +with a UNet backbone, they may also potentially benefit from the in- +corporation of vision transformer techniques. However, we discovered +that simply combining these two approaches resulted in subpar perfor- +mance. In this paper, we propose a novel transformer-based conditional +UNet framework, as well as a new Spectrum-Space Transformer (SS- +Former) to model the interaction between noise and semantic features. +This architectural improvement leads to a new diffusion-based medical +image segmentation method called MedSegDiff-V2, which significantly +improves the performance of MedSegDiff. We have verified the effective- +ness of MedSegDiff-V2 on eighteen organs of five segmentation datasets +with different image modalities. Our experimental results demonstrate +that MedSegDiff-V2 outperforms state-of-the-art (SOTA) methods by a +considerable margin, further proving the generalizability and effective- +ness of the proposed model. +Keywords: Multi-rater learning · Optic disc/cup segmentation · Glau- +coma diagnosis +1 +Introduction +Medical image segmentation is the process of dividing a medical image into +distinct regions of interest. It is a crucial step in many medical image analysis +applications, such as diagnosis, surgical planning, and image-guided surgery. The +ability to better understand and track changes over time in these images is vital +for medical professionals. In recent years, there has been a growing interest in +automated medical image segmentation methods, as they have the potential to +improve the consistency and accuracy of results. With the advancement of deep +learning techniques, several studies have successfully applied neural network- +based models, including classical convolutional neural networks (CNNs) [11] and +arXiv:2301.11798v1 [eess.IV] 19 Jan 2023 + +2 +J. Wu et al. +the recently popular vision transformers (ViTs) [2,22], to medical image segmen- +tation tasks. +Very recently, the Diffusion Probabilistic Model (DPM) [9] has gained popu- +larity as a powerful class of generative models, capable of generating high-quality +and diverse images [18–20]. Inspired by its success, some researchers have at- +tempted to apply DPM in the field of medical image segmentation [6,13,16,23, +25]. One such method, called MedSegDiff [25], achieved great success and outper- +formed previous state-of-the-art (SOTA) segmentation methods, such as nnUNet +and TransUNet. However, these methods are all based on classical UNet back- +bones. In a separate line of research, vision transformers, which have shown out- +standing performance in vision representation learning on natural images, have +also brought success in medical image segmentation and have quickly become +a popular approach. Among them, transformer-convolution hybrid architectures +have attracted the most attention and achieved the best performance. +A natural next step is to combine the transformer-based UNet, such as Tran- +sUNet, with DPM. However, we found that this straightforward strategy leads +to subpar performance. One issue is that the transformer-abstracted conditional +feature is not compatible with the feature of the backbone. The transformer +learns deep semantic features from the raw image, while the diffusion backbone +abstracts features from a corrupted, noisy mask. Additionally, the dynamic and +global nature of the transformer makes it more sensitive than CNNs. Thus, the +adaptive condition strategy used in MedSegDiff causes larger variance in the +outputs in the transformer setting. This requires running the model more times +for ensemble and makes it harder to converge during training. +To overcome the aforementioned challenges, we have designed a novel transformer- +based conditional UNet architecture for the diffusion process. The main idea is +to use two different conditioning techniques to condition the backbone model +with the source image segmentation features in the diffusion process. One is the +anchor condition, which integrates the conditional segmentation features into +the diffusion model encoder to reduce the diffusion variance. The other is the +semantic condition that integrates the conditional segmentation embedding into +the diffusion embedding. To effectively bridge the gap between diffusion noise +embedding and conditional semantic features, we propose a novel transformer +mechanism called the Spectrum-Space Transformer (SS-Former) that learns the +interaction between them. This allows the model to have a smaller diffusion +variance while also benefiting from the global and dynamic representation capa- +bilities provided by the transformer. +More specifically, in the anchor condition, we integrate the decoded segmen- +tation feature of the condition model into the encoded features of the diffusion +model. We design a novel Gaussian Spatial Attention mechanism to implement +this integration. It relaxes the conditional segmentation feature with more uncer- +tainty, thus providing the diffusion process more flexibility to further calibrate +the predictions. In the semantic condition, we integrate the semantic segmenta- +tion embedding into the diffusion model embedding using our novel SS-Former. +SS-Former is an interlaced cross-attention chain with one part that enhances the + +Title Suppressed Due to Excessive Length +3 +semantic embedding using the noise embedding and another part that enhances +the noise embedding using the semantic embedding. We design a novel cross- +attention mechanism over the frequency domain to eliminate the high-frequency +noises in the noise embedding, thus aligning the noise and semantic features. +We have verified MedSegDiff-V2 on a wide range of medical segmentation tasks, +such as optic-cup segmentation, brain tumor segmentation, abdominal organs +segmentation, and thyroid nodule segmentation. The images used in these tasks +have different modalities, such as MRI, CT, and ultrasonography. MedSegDiff-V2 +outperforms the previous state-of-the-art (SOTA) on all the tasks with different +modalities, which showcases the generalization and effectiveness of the proposed +method. In brief, the contributions of this paper are: +– The first to integrate transformer into a diffusion-based model for general +medical image segmentation. +– An anchor condition with Gaussian Spatial Attention to mitigate the diffu- +sion variance and speed up the ensemble. +– A semantic condition with SS-Former to model the segmentation noise and +semantic feature interaction. +– SOTA performance on sixteen medical segmentation tasks with different +image modalities. +2 +Method +2.1 +Overall architecture +The overall flow of MedSegDiff-V2 is shown in Figure 1. To introduce the pro- +cess, consider a single step t of the diffusion process. The noisy mask xt is first +inputted to a UNet with conditional integration, called the Diffusion Model. +The condition sources are the segmentation features extracted from the raw im- +ages through another standard UNet, called the Condition Model. Two different +conditioning manners are applied to the Diffusion Model: anchor condition and +semantic condition. Following the flow of the input, the anchor condition is first +imposed on the encoder of the Diffusion Model. It integrates the anchor segmen- +tation features, which are the decoded segmentation features of the Condition +Model, into the encoded features of the Diffusion Model. This allows the diffusion +model to be initialized by a rough but static reference, which helps to reduce the +diffusion variances. The semantic condition is then imposed on the embedding +of the Diffusion Model. This integrates the semantic segmentation embedding of +the Condition Model into the embedding of the Diffusion Model. This conditional +integration is implemented by the SS-Former, which bridges the gap between the +noise and semantic embedding, and abstracts a stronger representation with the +advantage of the global and dynamic nature of transformer. +MedSegDiff is trained using a standard noise prediction loss Lnoise following +DPM [9] and an anchor loss Lanchor. Lanchor is a combination of soft dice loss +and cross-entropy loss. Specifically, the total loss function is represented as: +Lt +total = Lt +noise + (t ≡ 0 +(mod α))(Ldice + βLce) +(1) + +4 +J. Wu et al. +Fig. 1: An illustration of MedSegDiff. For the clarity, the time step encoding and +skip connection in UNet are omitted in the figure. +where t ≡ 0 (mod α) control the times of supervision over Condition Model +through hyper-parameter α, β is another empirical hyper-parameter to weight +the cross-entropy loss. +2.2 +Anchor Condition with Gaussian Spatial Attention +Without the inductive bias of convolution layer, transformer blocks have stronger +representation but also to be more sensitive to the input variance. Directly +adding the transformer block to the Diffusion Model will cause the large vari- +ance on each times’ outputs, as we show the experimental results in Section 3. +To overcome this negative effect, we introduce the anchor condition operation +to the Diffusion Model. +The anchor condition integrates the anchor, which is the decoded segmenta- +tion features of the Condition Model into the encoder features of the Diffusion +Model. We propose a Gaussian Spatial Attention to represent the uncertainty +nature of the given segmentation features from the Condition Model. Formally, +consider we integrate the last conditional segmentation feature f −1 +c +into the first +diffusion feature f 0 +d. Gaussian Spatial Attention can be expressed as: +fanc = Max(f −1 +c +∗ kGauss, f −1 +c +), +(2) +f +′0 +d = Sigmoid(fanc ∗ kConv1×1) · f 0 +d + f 0 +d, +(3) + +SS-Former +Timestep t +Scale & Shift +X +Scale & Shift +MLP +C +FFT +MLP +W +Timestep tTitle Suppressed Due to Excessive Length +5 +where ∗ denotes slide-window kernel manipulation, · denotes general element- +wise manipulation. In Eqn. 2, we first apply a Gaussian kernel kG over f −1 +c +to +smooth the activation, as f −1 +c +serves as an anchor but may not be completely +accurate. The mean and variance of. The mean and variance of kG are learnable. +We then select the maximum value between the smoothed map and the original +feature map to preserve the most relevant information, resulting in a smoothed +anchor feature fanc. In Eqn. 3, we integrate fanc into f 0 +d to obtain an enhanced +feature f +′0 +d . Specifically, we first apply a 1×1 convolution k1×1conv to reduce the +number of channels in the anchor feature to 1. Then, we use a sigmoid activation +function on the anchor feature and add it to each channel of f 0 +d, similar to the +implementation of spatial attention [24]. Gaussian Spatial Attention extracts +a rough anchor feature from the Condition Model and integrates it into the +Diffusion Model. This provides the Diffusion Model with a correct range for +predictions while also allowing it to further refine the results. +2.3 +Semantic Condition with SS-Former +We propose a novel transformer architecture, called Spectrum-Space Trans- +former (SS-Former), to effectively integrate the conditional segmentation em- +bedding into the diffusion embedding. SS-Former is composed of several blocks +that share the same architecture. Each block consists of two cross-attention- +like modules. The first encodes the diffusion noise embedding into the condition +semantic embedding, and the next module encodes the noise-blended semantic +embedding into the diffusion noise embedding. This allows the model to learn +the interaction between noise and semantic features and achieve a stronger rep- +resentation. +Since the Diffusion Model predicts the redundant noise from the noisy mask +input, it will have a domain gap between its embedding and that of the condi- +tional segmentation semantic embedding. This gap can lead to confusion when +using matrix manipulations in a stranded transformer. To address this chal- +lenge, we propose a novel spectrum-space attention mechanism. The key idea +is to merge semantic and noise information in Fourier space, rather than Eu- +clidean space. This allows for the separation and blending of components based +on frequency-affinity in different spectrums. Formally, consider c0 is the deep- +est feature embedding of Condition Model and e is that of Diffusion Model. +We first transfer c0 and e to the Fourier space, denoted as F(c0) and F(e), re- +spectively. Note that the feature maps are all patchlized and liner projected in +accordance with the standard vision transformer method. Then we compute an +affinity weight map over Fourier space taking e as the query and c0 as the key, +as represented by the following equation: +M = a sin(w(F(c0)Wq)(F(e)Wk)T ), +(4) +where Wq and Wk are the learnable query and key weights in Fourier space. We +then employ a periodic active function to limit the representation spectrum in +Fourier space, as a substitute of standard activation applied in Euclidean space + +6 +J. Wu et al. +in standard self-attention. In the implementation, we use a sine function sin +with learnable amplitude a and frequency w as the constraint. +The affinity map is then transferred back to Euclidean space using inverse +fast Fourier transform (IFFT) and applied to condition features in value, +f = F −1(M)(c0wv), +(5) +where W v is the learnable value weights. We then apply the time embedding +to an AdaIN normalization following the classic diffusion implementation [15], +which normalizes the feature and then expands it using scale and shift parame- +ters learned from the time embedding. This makes the transformer adaptive to +the step information. We also use a Multi-layer Perceptron (MLP) to further +refine the attention result, obtaining the final feature ˜c0. The following attention +module is symmetric to the first one, using the combined feature ˜c0 as the query +and noise embedding e as the key and value, in order to transform the segmen- +tation features to the noise domain. The transformed feature c1 will serve as the +condition embedding for the next block. +3 +Experiments +3.1 +Dataset +We conduct the experiments on total four different medical image segmentation +datasets. One dataset is used to verify the general segmentation performance, +which is Multi-Organ Segmentation in Abdominal CT Images. We use public +AMOS2022 [12] dataset, which we employ 200 multi-contrast abdominal CT +from AMOS 2022 with sixteen anatomies manually annotated for abdominal +multi-organ segmentation. The other three datasets are used to verify the model +performance on multi-modal images, which are the optic-cup segmentation from +fundus images, the brain tumor segmentation from MRI images, and the thyroid +nodule segmentation from ultrasound images. The experiments of glaucoma, +thyroid cancer and melanoma diagnosis are conducted on REFUGE-2 dataset +[4], BraTs-2021 dataset [1] and DDTI dataset [17], which contain 1200, 2000, +8046 samples, respectively. The datasets are publicly available with segmentation +labels. Train/validation/test sets are split following the default settings of the +dataset. +3.2 +Implementation Details +The primary architecture of MedSegDiff is a modified ResUNet [26], which we +implement using a ResNet encoder followed by a UNet decoder. The specific +network configuration can be found in [25]. All experiments were conducted +using the PyTorch platform and trained/tested on 4 NVIDIA A100 GPUs. All +images were uniformly resized to a resolution of 256×256 pixels. The networks +were trained in an end-to-end manner using the AdamW [14] optimizer with a +batch size of 32. The initial learning rate was set to 1 ×10−4. + +Title Suppressed Due to Excessive Length +7 +3.3 +Main Results +To verify the general medical image segmentation performance, we compare +MedSegDiff-V2 with SOTA segmentation methods on multi-organ segmentation +dataset AMOS2022. The quantitative results are shown in Table 1. In the table, +we compare with the segmentation methods which are widely-used and well- +recognized in the community, including the CNN-based method nnUNet [10], +the transformer-based methods TransUNet [2], UNetr [8], Swin-UNetr [7] and +the diffusion based method EnsDiff [23], MedSegDiff [25]. We also compare with +a simple combination of diffusion and transformer model. We replace the UNet +model in MedSegDiff to TransUNet and denoted it as ’MedSegDiff + Tran- +sUNet’ in the table. We evaluate the segmentation performance by Dice score. +The compared methods are all implemented with their default setting. +Table 1: The comparison of MedSegDiff-V2 with SOTA segmentation methods +over AMOS dataset evaluated by Dice Score. Best results are denoted as bold. +Methods +Spleen R.Kid L.Kid Gall. +Eso. +Liver Stom. Aorta IVC +Panc. RAG LAG +Duo. +Blad. Pros. Avg +TransUNet +0.881 0.928 0.919 0.813 0.740 0.973 0.832 0.919 0.841 0.713 0.638 0.565 0.685 0.748 0.692 0.792 +Baseline +(EnsDiff) +0.905 0.918 0.904 0.732 0.723 0.947 0.738 0.915 0.838 0.704 0.677 0.618 0.715 0.673 0.680 0.779 +UNetr +0.926 0.936 0.918 0.785 0.702 0.969 0.788 0.893 0.828 0.732 0.717 0.554 0.658 0.683 0.722 0.784 +Swin-UNetr +0.959 0.960 0.949 0.894 0.827 0.979 0.899 0.944 0.899 0.828 0.791 0.745 0.817 0.875 0.841 0.880 +nnUNet +0.951 0.962 0.939 0.889 0.843 0.962 0.870 0.958 0.865 0.835 0.801 0.768 0.835 0.832 0.836 0.876 +MedSegDiff +0.963 0.965 0.953 0.917 0.846 0.971 0.906 0.952 0.918 0.854 0.803 0.751 0.819 0.868 0.855 0.889 +MedSegDiff ++ TransUNet +0.941 0.932 0.921 0.934 0.813 0.946 0.867 0.921 0.880 0.821 0.793 0.528 0.788 0.813 0.837 0.762 +MedSegDiff-V2 0.971 0.969 0.964 0.932 0.864 0.976 0.934 0.968 0.925 0.871 0.815 0.762 0.827 0.873 0.871 0.901 +As seen in Table 1, advanced network architectures and sophisticated designs +are crucial for achieving good performance. With regards to network architec- +ture, well-designed transformer-based models such as Swin-UNetr outperform +the carefully designed CNN-based model, nnUNet. The diffusion-based model +MedSegDiff again outperforms the transformer-based models on most of the or- +gans. However, network architecture alone is not the sole determining factor for +performance. For example, the well-designed CNN-based model nnUNet consid- +erably outperforms the transformer-based model TransUNet and UNetr in the +table. This is also true for diffusion-based models. We can see that a straight- +forward adoption of the diffusion model for medical image segmentation, i.e., +EnsDiff, achieves an unsatisfied performance. A simple combination of trans- +former and diffusion model, i.e., MedSegDiff+TransUNet, obtains even worse +performance than the standard MedSegDiff. This is because the transformer is +more sensitive to adaptive conditions and extracts more delicate semantic fea- +tures that diverge from the diffusion backbone. By introducing anchor condition +and SS-Former in MedSegDiff-V2, the diffusion + transformer model overcomes +these challenges and shows superior performance. We compare it with diffusion- +based models, i.e., EnsDiff and MedSegDiff, using the same ensemble times (all +set to five times), and it produces more stable and accurate results as shown in +the table. + +8 +J. Wu et al. +Fig. 2: The visual comparison with SOTA segmentation models. +Figure 2 presents a qualitative comparison of MedSegDiff-V2 and other com- +petitive methods. It can be observed that MedSegDiff-V2 segments more ac- +curately on parts that are difficult to recognize by the human eye. Due to its +ability to benefit from the superior generation capability of the diffusion model +and the semantic representation capability of the transformer, it can generate +segmentation maps with precise and accurate details, even in low-contrast or +ambiguous areas. +We also compare our method to state-of-the-art (SOTA) segmentation meth- +ods proposed for three specific tasks with different image modalities. The main +results are presented in Table 2. In the table, ResUnet [26] and BEAL [21] are +used for optic disc and cup segmentation, TransBTS [22] and EnsemDiff [23] +are used for brain tumor segmentation, and MTSeg [5] and UltraUNet [3] are +used for thyroid nodule segmentation. We also compare to general medical image +segmentation methods on these three datasets. The segmentation performance +is evaluated using the Dice score and IoU. +As seen in Table 2, MedSegDiff-V2 outperforms all other methods on three +different tasks, showcasing its ability to generalize to various medical segmen- +tation tasks and image modalities. Compared to the UNet-based MedSegDiff, +it improves by 2.0% on Optic-Cup, 1.9% on Brain-Tumor, and 3.9% on Thy- +roid Nodule in terms of the Dice score, illustrating the effectiveness of the +transformer-based backbone. Additionally, when compared to MedSegDiff with +TransUNet, it overcomes compatibility issues and significantly improves perfor- +mance on all three tasks, demonstrating the effectiveness of the proposed anchor +condition and SS-Former. + +Title Suppressed Due to Excessive Length +9 +Table 2: The comparison of MedSegDiff with SOTA segmentation methods. Best +results are denoted as bold. +Optic-Cup Brain-Turmor Thyroid Nodule +Dice IoU Dice +IoU +Dice +IoU +ResUnet +80.1 72.3 +- +- +- +- +BEAL +83.5 74.1 +- +- +- +- +TransBTS +- +- +87.6 +78.3 +- +- +EnsemDiff +- +- +88.7 +80.9 +- +- +MTSeg +- +- +- +- +82.3 +75.2 +UltraUNet +- +- +- +- +84.5 +76.2 +UNetr +83.2 73.3 87.3 +80.6 +81.7 +73.5 +Swin-UNetr +84.3 74.5 88.4 +81.8 +83.5 +74.8 +nnUNet +84.9 75.1 88.2 +80.4 +84.2 +76.2 +TransUNet +85.6 75.9 86.6 +79.0 +83.5 +75.1 +MedsegDiff +85.9 76.2 88.9 +81.2 +84.8 +76.4 +MedsegDiff+TransUNet 82.1 72.6 86.1 +78.0 +79.2 +71.4 +MedSegDiff-v2 +87.9 80.3 90.8 +83.4 +88.7 +81.5 +3.4 +Ablation Study +We conducted a comprehensive ablation study to verify the effectiveness of the +proposed anchor conditioning and SS-Former. The results are shown in Table +3, where Anc.Cond. denotes anchor conditioning. We evaluate the performance +using the Dice score (%) on all three tasks. The models were run five times +for ensemble. From the table, we can see that Anc.Cond. significantly improves +the vanilla diffusion model, with an improvement of 2.4% on thyroid nodule +segmentation, 1.6% and 1.8% respectively. SS-Former learns the interaction be- +tween noise and semantic features with a vision transformer-based architecture, +further improving the segmentation results. It promotes MedSegDiff-V2 by over +1% on all three tasks and achieves new state-of-the-art performance. +Table 3: An ablation study on anchor conditioning and SS-Former. Dice score(%) +is used as the metric. +Anc.Cond. SS-Former OpticCup BrainTumor ThyroidNodule +84.6 +88.2 +84.1 +✓ +86.2 +89.4 +86.5 +✓ +✓ +87.9 +90.8 +88.7 +4 +Conclusion +In this paper, we enhance the diffusion-based medical image segmentation frame- +work by incorporating the transformer mechanism into the original UNet back- + +10 +J. Wu et al. +bone, called MedSegDiff-V2. We propose an anchor condition to ensure the sta- +bility of the model and a novel SS-Former architecture to learn the interaction +between noise and semantic features. The comparative experiments were con- +ducted on 18 organs and 4 medical image segmentation datasets with different +image modalities and our model outperformed previous state-of-the-art methods. +As the first transformer-based diffusion model for medical image segmentation, +we believe MedSegDiff-V2 will serve as a benchmark for future research. + +Title Suppressed Due to Excessive Length +11 +References +1. Baid, U., Ghodasara, S., Mohan, S., Bilello, M., Calabrese, E., Colak, E., Farahani, +K., Kalpathy-Cramer, J., Kitamura, F.C., Pati, S., et al.: The rsna-asnr-miccai +brats 2021 benchmark on brain tumor segmentation and radiogenomic classifica- +tion. arXiv preprint arXiv:2107.02314 (2021) +2. 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Computerized Medical Imaging +and Graphics 74, 61–71 (2019) + diff --git a/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/load_file.txt b/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85975f5c94cee84880e58a4190f93dfe08bd9517 --- /dev/null +++ b/CtFKT4oBgHgl3EQfYS5g/content/tmp_files/load_file.txt @@ -0,0 +1,684 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf,len=683 +page_content='MedSegDiff-V2: Diffusion based Medical Image Segmentation with Transformer Junde Wu1, Rao Fu2, Huihui Fang1, Yu Zhang2, and Yanwu Xu1 1 Baidu Research 2 Mind Vogue Lab Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The Diffusion Probabilistic Model (DPM) has recently gained popularity in the field of computer vision, thanks to its image genera- tion applications, such as Imagen, Latent Diffusion Models, and Stable Diffusion, which have demonstrated impressive capabilities and sparked much discussion within the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Recent studies have also found DPM to be useful in the field of medical image analysis, as evidenced by the strong performance of the medical image segmentation model Med- SegDiff in various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' While these models were originally designed with a UNet backbone, they may also potentially benefit from the in- corporation of vision transformer techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' However, we discovered that simply combining these two approaches resulted in subpar perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In this paper, we propose a novel transformer-based conditional UNet framework, as well as a new Spectrum-Space Transformer (SS- Former) to model the interaction between noise and semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This architectural improvement leads to a new diffusion-based medical image segmentation method called MedSegDiff-V2, which significantly improves the performance of MedSegDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We have verified the effective- ness of MedSegDiff-V2 on eighteen organs of five segmentation datasets with different image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Our experimental results demonstrate that MedSegDiff-V2 outperforms state-of-the-art (SOTA) methods by a considerable margin, further proving the generalizability and effective- ness of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Keywords: Multi-rater learning · Optic disc/cup segmentation · Glau- coma diagnosis 1 Introduction Medical image segmentation is the process of dividing a medical image into distinct regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' It is a crucial step in many medical image analysis applications, such as diagnosis, surgical planning, and image-guided surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The ability to better understand and track changes over time in these images is vital for medical professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In recent years, there has been a growing interest in automated medical image segmentation methods, as they have the potential to improve the consistency and accuracy of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' With the advancement of deep learning techniques, several studies have successfully applied neural network- based models, including classical convolutional neural networks (CNNs) [11] and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='11798v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='IV] 19 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' the recently popular vision transformers (ViTs) [2,22], to medical image segmen- tation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Very recently, the Diffusion Probabilistic Model (DPM) [9] has gained popu- larity as a powerful class of generative models, capable of generating high-quality and diverse images [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Inspired by its success, some researchers have at- tempted to apply DPM in the field of medical image segmentation [6,13,16,23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' One such method, called MedSegDiff [25], achieved great success and outper- formed previous state-of-the-art (SOTA) segmentation methods, such as nnUNet and TransUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' However, these methods are all based on classical UNet back- bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In a separate line of research, vision transformers, which have shown out- standing performance in vision representation learning on natural images, have also brought success in medical image segmentation and have quickly become a popular approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Among them, transformer-convolution hybrid architectures have attracted the most attention and achieved the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' A natural next step is to combine the transformer-based UNet, such as Tran- sUNet, with DPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' However, we found that this straightforward strategy leads to subpar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' One issue is that the transformer-abstracted conditional feature is not compatible with the feature of the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The transformer learns deep semantic features from the raw image, while the diffusion backbone abstracts features from a corrupted, noisy mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Additionally, the dynamic and global nature of the transformer makes it more sensitive than CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Thus, the adaptive condition strategy used in MedSegDiff causes larger variance in the outputs in the transformer setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This requires running the model more times for ensemble and makes it harder to converge during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' To overcome the aforementioned challenges, we have designed a novel transformer- based conditional UNet architecture for the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The main idea is to use two different conditioning techniques to condition the backbone model with the source image segmentation features in the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' One is the anchor condition, which integrates the conditional segmentation features into the diffusion model encoder to reduce the diffusion variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The other is the semantic condition that integrates the conditional segmentation embedding into the diffusion embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' To effectively bridge the gap between diffusion noise embedding and conditional semantic features, we propose a novel transformer mechanism called the Spectrum-Space Transformer (SS-Former) that learns the interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This allows the model to have a smaller diffusion variance while also benefiting from the global and dynamic representation capa- bilities provided by the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' More specifically, in the anchor condition, we integrate the decoded segmen- tation feature of the condition model into the encoded features of the diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We design a novel Gaussian Spatial Attention mechanism to implement this integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' It relaxes the conditional segmentation feature with more uncer- tainty, thus providing the diffusion process more flexibility to further calibrate the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In the semantic condition, we integrate the semantic segmenta- tion embedding into the diffusion model embedding using our novel SS-Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' SS-Former is an interlaced cross-attention chain with one part that enhances the Title Suppressed Due to Excessive Length 3 semantic embedding using the noise embedding and another part that enhances the noise embedding using the semantic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We design a novel cross- attention mechanism over the frequency domain to eliminate the high-frequency noises in the noise embedding, thus aligning the noise and semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We have verified MedSegDiff-V2 on a wide range of medical segmentation tasks, such as optic-cup segmentation, brain tumor segmentation, abdominal organs segmentation, and thyroid nodule segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The images used in these tasks have different modalities, such as MRI, CT, and ultrasonography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' MedSegDiff-V2 outperforms the previous state-of-the-art (SOTA) on all the tasks with different modalities, which showcases the generalization and effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In brief, the contributions of this paper are: – The first to integrate transformer into a diffusion-based model for general medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' – An anchor condition with Gaussian Spatial Attention to mitigate the diffu- sion variance and speed up the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' – A semantic condition with SS-Former to model the segmentation noise and semantic feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' – SOTA performance on sixteen medical segmentation tasks with different image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 2 Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 Overall architecture The overall flow of MedSegDiff-V2 is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' To introduce the pro- cess, consider a single step t of the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The noisy mask xt is first inputted to a UNet with conditional integration, called the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The condition sources are the segmentation features extracted from the raw im- ages through another standard UNet, called the Condition Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Two different conditioning manners are applied to the Diffusion Model: anchor condition and semantic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Following the flow of the input, the anchor condition is first imposed on the encoder of the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' It integrates the anchor segmen- tation features, which are the decoded segmentation features of the Condition Model, into the encoded features of the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This allows the diffusion model to be initialized by a rough but static reference, which helps to reduce the diffusion variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The semantic condition is then imposed on the embedding of the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This integrates the semantic segmentation embedding of the Condition Model into the embedding of the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This conditional integration is implemented by the SS-Former, which bridges the gap between the noise and semantic embedding, and abstracts a stronger representation with the advantage of the global and dynamic nature of transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' MedSegDiff is trained using a standard noise prediction loss Lnoise following DPM [9] and an anchor loss Lanchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Lanchor is a combination of soft dice loss and cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Specifically, the total loss function is represented as: Lt total = Lt noise + (t ≡ 0 (mod α))(Ldice + βLce) (1) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 1: An illustration of MedSegDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' For the clarity, the time step encoding and skip connection in UNet are omitted in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' where t ≡ 0 (mod α) control the times of supervision over Condition Model through hyper-parameter α, β is another empirical hyper-parameter to weight the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 Anchor Condition with Gaussian Spatial Attention Without the inductive bias of convolution layer, transformer blocks have stronger representation but also to be more sensitive to the input variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Directly adding the transformer block to the Diffusion Model will cause the large vari- ance on each times’ outputs, as we show the experimental results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' To overcome this negative effect, we introduce the anchor condition operation to the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The anchor condition integrates the anchor, which is the decoded segmenta- tion features of the Condition Model into the encoder features of the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We propose a Gaussian Spatial Attention to represent the uncertainty nature of the given segmentation features from the Condition Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Formally, consider we integrate the last conditional segmentation feature f −1 c into the first diffusion feature f 0 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Gaussian Spatial Attention can be expressed as: fanc = Max(f −1 c ∗ kGauss, f −1 c ), (2) f ′0 d = Sigmoid(fanc ∗ kConv1×1) · f 0 d + f 0 d, (3) SS-Former Timestep t Scale & Shift X Scale & Shift MLP C FFT MLP W Timestep tTitle Suppressed Due to Excessive Length 5 where ∗ denotes slide-window kernel manipulation, · denotes general element- wise manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 2, we first apply a Gaussian kernel kG over f −1 c to smooth the activation, as f −1 c serves as an anchor but may not be completely accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The mean and variance of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The mean and variance of kG are learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We then select the maximum value between the smoothed map and the original feature map to preserve the most relevant information, resulting in a smoothed anchor feature fanc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 3, we integrate fanc into f 0 d to obtain an enhanced feature f ′0 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Specifically, we first apply a 1×1 convolution k1×1conv to reduce the number of channels in the anchor feature to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Then, we use a sigmoid activation function on the anchor feature and add it to each channel of f 0 d, similar to the implementation of spatial attention [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Gaussian Spatial Attention extracts a rough anchor feature from the Condition Model and integrates it into the Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This provides the Diffusion Model with a correct range for predictions while also allowing it to further refine the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 Semantic Condition with SS-Former We propose a novel transformer architecture, called Spectrum-Space Trans- former (SS-Former), to effectively integrate the conditional segmentation em- bedding into the diffusion embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' SS-Former is composed of several blocks that share the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Each block consists of two cross-attention- like modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The first encodes the diffusion noise embedding into the condition semantic embedding, and the next module encodes the noise-blended semantic embedding into the diffusion noise embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This allows the model to learn the interaction between noise and semantic features and achieve a stronger rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Since the Diffusion Model predicts the redundant noise from the noisy mask input, it will have a domain gap between its embedding and that of the condi- tional segmentation semantic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This gap can lead to confusion when using matrix manipulations in a stranded transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' To address this chal- lenge, we propose a novel spectrum-space attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The key idea is to merge semantic and noise information in Fourier space, rather than Eu- clidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This allows for the separation and blending of components based on frequency-affinity in different spectrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Formally, consider c0 is the deep- est feature embedding of Condition Model and e is that of Diffusion Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We first transfer c0 and e to the Fourier space, denoted as F(c0) and F(e), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Note that the feature maps are all patchlized and liner projected in accordance with the standard vision transformer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Then we compute an affinity weight map over Fourier space taking e as the query and c0 as the key, as represented by the following equation: M = a sin(w(F(c0)Wq)(F(e)Wk)T ), (4) where Wq and Wk are the learnable query and key weights in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We then employ a periodic active function to limit the representation spectrum in Fourier space, as a substitute of standard activation applied in Euclidean space 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' in standard self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In the implementation, we use a sine function sin with learnable amplitude a and frequency w as the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The affinity map is then transferred back to Euclidean space using inverse fast Fourier transform (IFFT) and applied to condition features in value, f = F −1(M)(c0wv), (5) where W v is the learnable value weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We then apply the time embedding to an AdaIN normalization following the classic diffusion implementation [15], which normalizes the feature and then expands it using scale and shift parame- ters learned from the time embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This makes the transformer adaptive to the step information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We also use a Multi-layer Perceptron (MLP) to further refine the attention result, obtaining the final feature ˜c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The following attention module is symmetric to the first one, using the combined feature ˜c0 as the query and noise embedding e as the key and value, in order to transform the segmen- tation features to the noise domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The transformed feature c1 will serve as the condition embedding for the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 Dataset We conduct the experiments on total four different medical image segmentation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' One dataset is used to verify the general segmentation performance, which is Multi-Organ Segmentation in Abdominal CT Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We use public AMOS2022 [12] dataset, which we employ 200 multi-contrast abdominal CT from AMOS 2022 with sixteen anatomies manually annotated for abdominal multi-organ segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The other three datasets are used to verify the model performance on multi-modal images, which are the optic-cup segmentation from fundus images, the brain tumor segmentation from MRI images, and the thyroid nodule segmentation from ultrasound images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The experiments of glaucoma, thyroid cancer and melanoma diagnosis are conducted on REFUGE-2 dataset [4], BraTs-2021 dataset [1] and DDTI dataset [17], which contain 1200, 2000, 8046 samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The datasets are publicly available with segmentation labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Train/validation/test sets are split following the default settings of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 Implementation Details The primary architecture of MedSegDiff is a modified ResUNet [26], which we implement using a ResNet encoder followed by a UNet decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The specific network configuration can be found in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' All experiments were conducted using the PyTorch platform and trained/tested on 4 NVIDIA A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' All images were uniformly resized to a resolution of 256×256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The networks were trained in an end-to-end manner using the AdamW [14] optimizer with a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The initial learning rate was set to 1 ×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 Main Results To verify the general medical image segmentation performance, we compare MedSegDiff-V2 with SOTA segmentation methods on multi-organ segmentation dataset AMOS2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The quantitative results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In the table, we compare with the segmentation methods which are widely-used and well- recognized in the community, including the CNN-based method nnUNet [10], the transformer-based methods TransUNet [2], UNetr [8], Swin-UNetr [7] and the diffusion based method EnsDiff [23], MedSegDiff [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We also compare with a simple combination of diffusion and transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We replace the UNet model in MedSegDiff to TransUNet and denoted it as ’MedSegDiff + Tran- sUNet’ in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We evaluate the segmentation performance by Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The compared methods are all implemented with their default setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Table 1: The comparison of MedSegDiff-V2 with SOTA segmentation methods over AMOS dataset evaluated by Dice Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Best results are denoted as bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Methods Spleen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='Kid L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='Kid Gall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Liver Stom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Aorta IVC Panc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' RAG LAG Duo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Blad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Pros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Avg TransUNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='901 As seen in Table 1, advanced network architectures and sophisticated designs are crucial for achieving good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' With regards to network architec- ture, well-designed transformer-based models such as Swin-UNetr outperform the carefully designed CNN-based model, nnUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The diffusion-based model MedSegDiff again outperforms the transformer-based models on most of the or- gans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' However, network architecture alone is not the sole determining factor for performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' For example, the well-designed CNN-based model nnUNet consid- erably outperforms the transformer-based model TransUNet and UNetr in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This is also true for diffusion-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We can see that a straight- forward adoption of the diffusion model for medical image segmentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=', EnsDiff, achieves an unsatisfied performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' A simple combination of trans- former and diffusion model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=', MedSegDiff+TransUNet, obtains even worse performance than the standard MedSegDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' This is because the transformer is more sensitive to adaptive conditions and extracts more delicate semantic fea- tures that diverge from the diffusion backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' By introducing anchor condition and SS-Former in MedSegDiff-V2, the diffusion + transformer model overcomes these challenges and shows superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We compare it with diffusion- based models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=', EnsDiff and MedSegDiff, using the same ensemble times (all set to five times), and it produces more stable and accurate results as shown in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' 2: The visual comparison with SOTA segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Figure 2 presents a qualitative comparison of MedSegDiff-V2 and other com- petitive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' It can be observed that MedSegDiff-V2 segments more ac- curately on parts that are difficult to recognize by the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Due to its ability to benefit from the superior generation capability of the diffusion model and the semantic representation capability of the transformer, it can generate segmentation maps with precise and accurate details, even in low-contrast or ambiguous areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We also compare our method to state-of-the-art (SOTA) segmentation meth- ods proposed for three specific tasks with different image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The main results are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' In the table, ResUnet [26] and BEAL [21] are used for optic disc and cup segmentation, TransBTS [22] and EnsemDiff [23] are used for brain tumor segmentation, and MTSeg [5] and UltraUNet [3] are used for thyroid nodule segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We also compare to general medical image segmentation methods on these three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The segmentation performance is evaluated using the Dice score and IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' As seen in Table 2, MedSegDiff-V2 outperforms all other methods on three different tasks, showcasing its ability to generalize to various medical segmen- tation tasks and image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Compared to the UNet-based MedSegDiff, it improves by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='0% on Optic-Cup, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9% on Brain-Tumor, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9% on Thy- roid Nodule in terms of the Dice score, illustrating the effectiveness of the transformer-based backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Additionally, when compared to MedSegDiff with TransUNet, it overcomes compatibility issues and significantly improves perfor- mance on all three tasks, demonstrating the effectiveness of the proposed anchor condition and SS-Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 9 Table 2: The comparison of MedSegDiff with SOTA segmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Best results are denoted as bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Optic-Cup Brain-Turmor Thyroid Nodule Dice IoU Dice IoU Dice IoU ResUnet 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 BEAL 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 TransBTS 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 EnsemDiff 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 MTSeg 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 UltraUNet 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 UNetr 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 Swin-UNetr 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8 nnUNet 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 TransUNet 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 MedsegDiff 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 MedsegDiff+TransUNet 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 MedSegDiff-v2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 Ablation Study We conducted a comprehensive ablation study to verify the effectiveness of the proposed anchor conditioning and SS-Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The results are shown in Table 3, where Anc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' denotes anchor conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We evaluate the performance using the Dice score (%) on all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The models were run five times for ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' From the table, we can see that Anc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' significantly improves the vanilla diffusion model, with an improvement of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4% on thyroid nodule segmentation, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' SS-Former learns the interaction be- tween noise and semantic features with a vision transformer-based architecture, further improving the segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' It promotes MedSegDiff-V2 by over 1% on all three tasks and achieves new state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Table 3: An ablation study on anchor conditioning and SS-Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Dice score(%) is used as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Anc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' SS-Former OpticCup BrainTumor ThyroidNodule 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='1 ✓ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='5 ✓ ✓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content='7 4 Conclusion In this paper, we enhance the diffusion-based medical image segmentation frame- work by incorporating the transformer mechanism into the original UNet back- 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' bone, called MedSegDiff-V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' We propose an anchor condition to ensure the sta- bility of the model and a novel SS-Former architecture to learn the interaction between noise and semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' The comparative experiments were con- ducted on 18 organs and 4 medical image segmentation datasets with different image modalities and our model outperformed previous state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' As the first transformer-based diffusion model for medical image segmentation, we believe MedSegDiff-V2 will serve as a benchmark for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 11 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=' Baid, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=', Ghodasara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFKT4oBgHgl3EQfYS5g/content/2301.11798v1.pdf'} +page_content=', Mohan, S.' 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0000000000000000000000000000000000000000..fc28f1f54437ca3082b11b33f06f778a8e2b324a --- /dev/null +++ b/DNE4T4oBgHgl3EQf6A5m/content/tmp_files/2301.05328v1.pdf.txt @@ -0,0 +1,524 @@ +TETRA-PENTA-DECA-HEXAGONAL-GRAPHENE +(TPDH-GRAPHENE) HYDROGENATION PATTERNS: DYNAMICS +AND ELECTRONIC STRUCTURE +Caique Campos, Matheus Medina, Pedro Alves da Silva Autreto +Center for Natural and Human Sciences (CCNH) +Federal University of ABC (UFABC) +Santo André - SP, 09210-170, Brazil. +pedro.autreto@ufabc.edu.br +Douglas Soares Galvao +Physics Institute Gleb Wataghin (IFGW) +State University of Campinas (UNICAMP) +Campinas/SP, Brazil +galvao@ifi.unicamp.br +ABSTRACT +The advent of graphene has renewed the interest in other 2D carbon-based materials. Bhattacharya +and Jana have proposed a new carbon allotrope, composed of different polygonal carbon rings +containing 4, 5, 6, and 10 atoms, named Tetra-Penta-Deca-Hexagonal-graphene (TPDH-graphene). +This unusual topology created material with interesting mechanical, electronic, and optical properties +and several potential applications, including UV protection. Like other 2D carbon structures, chemical +functionalizations can be used to tune their TPDH-graphene properties. In this work, we investigated +the hydrogenation dynamics of TPDH-graphene and its effects on its electronic structure, combining +DFT and fully atomistic reactive molecular dynamics simulations. Our results show that H atoms +are mainly incorporated on tetragonal ring sites (up to 80% at% at 300 K), leading to the appearance +of well-delimited pentagonal carbon stripes. The electronic structure of the hydrogenated structures +shows the formation of narrow bandgaps with the presence of Dirac cone-like structures, indicative +of anisotropic transport properties. +1 +Introduction +The versatility in chemical bonding (different hybridizations) of carbon atoms allows the existence of a wide variety of +different structures (allotropes) [1], such as fullerenes [2], nanotubes [3], and graphene [4]. Graphene is a 2D allotrope +of sp2 carbon atoms tightly packed into a hexagonal honeycomb lattice. It presents high carrier mobility (5000cm2/V.s +)[4, 5], high thermal conductivity (5000WmK−1) [6], and Young modulus value of 1 TPa [7], one of the highest +values ever measured. It has unveiled new and unique physics phenomena, including the quantum Hall effect [8], the +ambipolar electric field effect [4], and the massless charge carriers of Dirac fermions [9]. These remarkable properties +have made graphene the subject of a large number of theoretical and experimental studies in different areas, such as +catalysis [10], electronics [11], spintronics [12], twistronics [13], and gas sensors [14], to name just a few. +However, graphene is a null electronic gap material, even exhibiting extraordinary electronic properties, which limits its +use in some applications [4]. Chemical functionalizations, such as hydrogenation, are one viable mechanism for altering +graphene-like structures’ properties (including opening the gap [15–17] or changing the Fermi level [18]). Structural +and electronic changes are introduced when the chemical species form covalent bonds. The partial hydrogenation of +graphene introduces unsaturated sp3 carbon atoms that can be used to attach additional functional groups. +arXiv:2301.05328v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +Running Title for Header +Figure 1: (a) Schematics of the unit cell of tetra-penta-deca-hexagonal-graphene (TPDH) and the corresponding +carbon-carbon bond-length values. The different colors indicate non-equivalent carbon atoms. (b) A 2 × 2 supercell +illustrating the TPDHG rings and the pores of the structure. The corresponding Unit cell vector values are indicated +in the highlighted red rectangle. (c) The structural setup simulation used in the simulations. A TPDH membrane +(indicated in blue) is deposited on a graphene frame (gray), and the TPDHG/graphene structure is immersed in a +hydrogen atmosphere (yellow). See text for discussions. +Despite these limitations, the advent of graphene created a revolution in materials science and renewed the interest in +2D carbon allotropes. Among these structures, it is worth mentioning graphynes and biphenylene carbon networks [19]. +Graphynes are the generic name for families of 2D carbon porous structures containing hexagon rings connected +by acetylenic groups and with sp and sp2 hybridized carbon atoms in the same lattice [19]. Graphdyines refer +to the structural families where two acetylenic groups connect the hexagons [20]. They can exhibit metallic and +semiconducting behaviors [21] and have been exploited in different technological applications [22]. +Biphenylene carbon networks (including biphenylene carbon and graphenylenes) are families of porous structures +composed of mixed carbon rings (pentagons, hexagons, heptagons, octagons, etc.) [19, 23, 24]. Similarly to graphynes, +they can be metallic, or semiconductors and have potential applications in catalysis [23], gas sensors [25], batteries +[26], and energy storage applications [27]. Recently, new synthetic routes for graphynes [28, 29] and biphenylene +carbon networks [30] have been reported increasing the interest in these materials. Bhattacharya and Jana [31] have +proposed a new structure composed of two pentagons and a tetragonal ring called tetra-penta-octogonal graphene +(TPO-graphene). It is metallic with a Dirac cone at 3.7 eV above the Fermi level. More recently, they proposed +another structure belonging to the tetra-pentagonal graphene family composed of sp2 carbon rings with 4, 5, 12, and 6 +atoms (Fig. 1) named tetra-penta-deca-hexagonal graphene (TPDH-graphene). It possesses thermal and dynamical +stability and exhibits elastic anisotropy with Young’s modulus value larger than that of graphene in a specific direction. +Depending on the morphology, TPDH-graphene nanoribbons can exhibit metallic, or semiconductor behavior [32]. +In this work, we have investigated the effects of hydrogenation on the structural and electronic properties of TPDH- +graphene (TPDH-gr). The hydrogenation of TPDH-gr sheets was investigated through reactive molecular dynamics +simulations. Structural optimization, energy, and electronic properties were further analyzed using ab initio (DFT) +calculations. +In spite of graphene’s extraordinary properties, it is a null gap material, which Chemical functionalization is one viable +mechanism to introduce specific modifications into graphene-like structures. Structural and electronic changes are +introduced when the chemical species being introduced form a covalent bond. For example, graphite oxides can form +oxygen groups in graphene sheets dispersed in water and organic solvents [33]. Stankovich et al. prepared graphite +oxides functionalized with isocyanates that were later exfoliated into graphene oxides dispersed in an aprotic polar +solvent [34] in a stable manner. Partial hydrogenation of graphene sheets introduces unsaturated carbon atoms sp3 that +neighbor unpaired with electrons that can be used to attach additional functional groups. Chemical functionalization +also allows one to change the electronic properties of the structure by opening a bandgap [15–17] or changing the Fermi +level [18]. +2 + +a) +b +c) +C4 +C3 +C2 +1.41A +1.50A +C1 +1.43A +A += 6.97 +-b +4.94 +aRunning Title for Header +Figure 2: Adsorption energies for TPDH-gr in a) the non-equivalent sites, b) with an H atom adsorbed in the C1 +site, c) two H atoms adsorbed in the C1 and C7’ sites, and d) tree H atoms adsorbed in the C1, C7’ and C5 sites. e) +Non-equivalent sites in TPDH-gr. f) remaining sites in the tetragonal ring with the C1 site occupied. The top sites are +indicated by the solid line, while the bottom sides are indicated by the dashed ones and prime labels. g) Top and side +views of TPDH-gr with C1 and C7’ sites occupied. The side view also shows the buckling height. h) Top and side +views of TPDH-gr with a fully hydrogenated tetragonal ring. +2 +Computational Methods +First-principles calculations were carried out within the Density Functional Theory (DFT) framework as implemented in +Quantum Espresso code [35]. Electron-ion interactions were dealt with Projected Augmented wave (PAW) and Ultra-soft +pseudopotentials for C and H atoms, respectively. They were obtained from the Standard Solid State Pseudopotentials +library (SSSP) [36, 37]. Exchange and correlation potential were used within the Generalized Gradient Approximation +(GGA) with the parameterization of Perdew, Burke, and Ernzerhof (GGA-PBE functional) [38]. Valence electrons were +treated with a set of plane waves basis set with a kinetic energy cutoff of 680 eV. The diagonalization of the density +matrix was performed with the Davidson iterative method with matrix overlap using the self-consistency threshold +of 10−6 eV. In the ionic relaxation calculations, the convergence thresholds were set to 10−3 eV and 10−2 eV/Å for +energy and forces, respectively. Brillouin zone (BZ) sampling was performed using a 12 × 12 × 1 (16 × 16 × 1) k-point +grid for SCF (NSCF) calculations following the scheme proposed by Monkhosrt and Pack [39]. For electronic structure +calculations, the k-points were chosen along the following path in the BZ: Γ(0, 0, 0) - M(0.5, 0.5, 0) - X(0.5, 0, 0) - +Γ(0, 0, 0) - Y (0, 0.5, 0) - M(0.5, 0.5, 0) - Γ(0, 0, 0). +We have also carried out fully atomistic molecular dynamics (MD) simulations using the large-scale atomic/molecular +massively parallel simulator (LAMMPS) code[40]. Atomic interactions were treated with the reactive force field +(ReaxFF) [41], with C-C interaction parameters developed by Chenoweth et al.[42]. All MD simulations were carried +out in the canonical (NVT) ensemble, with a time step of 0.25 fs, and using a Nosé-Hoover thermostat [43]. The +hydrogenation simulations were carried out considering a TPDH-gr membrane deposited on a graphene frame, as shown +in Fig. 1.c. The TPDH-graphene membrane is a 24x15 supercell, in which only its central part (16x11) is exposed to the +hydrogen atmosphere, resulting in a total number of 2112 available adsorption/reaction sites. The hydrogen atmosphere +was composed of 500 atoms in a volume of 60 000 Å3 on each side of the membrane, constrained to the exposed region +of the membrane. This methodology has been successfully applied to other systems, such as Me-graphane[16] and +graphone[44]. +3 + +a) +b) +) +d) +4.0 +4.0- +1111 +3.5 +3.5 +3.5 +3.5- +3.0 +3.0 +3.0 +3.0 +[eV/atom] +2.5 +2.5. +2.5 +2.5 +[eV/ato] +2.0 +2.0 +2.0 +1.5 - +1.5 +1.5 +1.5 +1.0 +1.0 +1.0 +1.0- +L +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0- +0.0 +C6' +C3 +C6 +C1 +C4 +C5 +C6' +C2 +C5 +C5' +C6. +C6' +C7 +C7 +C5' +C6 +Site +Site +Site +Site +C4 +g) +f) +h) +e) +C6 +-C6' +h = 1.185 A +h = 0.87 ARunning Title for Header +3 +Results and Discussion +3.1 +Ab initio Binding Energy and Hydrogenation Dynamics +TPDH-gr has a Pmmm (space group #47) symmetry; the 12 carbon atoms in its unit cell are arranged in an +orthorhombic lattice. The obtained optimized lattice parameters were: a = 4.94 Å, b = 6.97 Å with γ = 90o. There +are three different bond lengths (1.41, 1.50, and 1.44 Å .) involving the C atoms, as shown in Fig. 1.a. Except for the C +atoms bonded along the ⃗a direction in the tetragonal ring, the bond lengths are close to those sp2 in graphene (1.41 +Å)[45]. These results agree well with those reported by Batthacharya and Jana [32]. +The most favorable sites for H adsorption/reaction were investigated by evaluating the binding energy per adsorbed +atom, calculated as the energy difference between the hydrogenated structure and its parts: +Eb = − +�ET P DH+nH − (ET P DH + nEH) +n +� +where ET P DH+nH is the energy of TPDH-gr with n adsorbed H atoms, ET P DH is the energy of a TPDH-gr unit cell, +and EH the energy of an isolated H atom. The negative sign means that high energies indicate more favorable sites for +adsorption than others in the same structure. First, an H atom is adsorbed at each of the non-equivalent sites (Fig. 1.a). +The site corresponding to the highest energy is taken as the most favorable. Then, a second H is adsorbed at each of the +remaining sites, and the most favorable one is evaluated according to Eb. This process is repeated until the tetragonal +ring on the TPDH-gr is fully hydrogenated. We present the binding energies and obtained structures in Fig. 2. +The adsorption of the first H atom is more favorable on the C1 site, as seen in Fig. 2.a with Eb of 3.35 eV/atom. After +C1-Cx adsorption (with x = 2, 5, and 7), the bond length values increased to 1.51, 1.55, and 1.53 Å , respectively, +indicating a transition to sp3=like-bond in the C1 atom. It is worth mentioning that for sites located in the tetragonal +ring, the top and bottom configurations (Fig. 1.d) were considered. Adsorption of a single H atom at each of these sites +resulted in roughly the same results for Eb, as can be seen in Table 1S in Supplementary Material. +The adsorption of the second H atom (resulting in 16% hydrogen coverage) is more favorable at the C7′ site (Fig. 2.c), +with an Eb of +3.75 eV/atom. The resulting lattice distortions in the direction perpendicular to the structure plane +lead to a significant buckling of h = 0.87 Å , as seen in Fig. 2.f. The distortions of the structure and the fact that two +neighboring C atoms adsorb the pair of H atoms (but on opposite sides of the sheet) are in accordance with the results +reported by Boukhvalov and Katsnelson for the hydrogenation of graphene sheets [46]. +Interestingly, the adsorption of a third H atom gives the same Eb for both C5 and C6’ sites, as seen in Fig. 2.e. In this +case, the configuration in which the C1, C7’ and C5 sites are occupied was imposed, which will be justified later. The +resulting structure presents an overall increase in the Cx-C bond lengths (with x = 1, 7, and 5). The vertical distance +separating the C1 and C7 atoms is 1.02 Å versus 0.84 Å for the corresponding value between the C5 and C6 atoms. +The adsorption of a fourth H atom (33% hydrogen coverage) is more favorable at the C6’ site with Eb = 4.0 eV/ Å and +buckling of h = 1.185 Å(Fig. 2.g, h). It is clear that choosing the C5 or C6’ sites in the adsorption of the third H atom +leads basically to the same configuration (C1-C7’-C5-C6’). Therefore, choosing C5 or C6’ for the adsorption of the +third H atom is equivalent. +These calculations reveal a pattern for the hydrogenation of the tetragonal ring, which consists of two lines of H atoms +on opposite sides of the basal plane sheet, leading to the formation of well-delimited pentagonal ring strips along the +direction of the lattice vector a. DFT calculations confirm that this configuration is indeed more favorable. Molecular +Dynamics simulations, discussed below, produced similar results, +Reactive molecular dynamics simulations were carried out to study the dynamics and temperature effects on hydrogen +adsorption of bigger TPDH-graphene membranes (Fig. 1), which would be cost-prohibitive with DFT methods. +Representative MD snapshots of both sides of the TPDH-graphene membrane during the hydrogenation process (at +300K) are presented in Fig. 3 (a) - (c). +The H atoms are predominantly incorporated throughout the MD simulations on the C1 sites. Analyzing the hydrogena- +tion process, from Fig. 3 (a) to (c), we can see that the hydrogen-adsorbed C1 sites act as seeds to the hydrogenation of +their C1 neighbors, forming lines through the structure surface, which is an expected result, based on the DFT binding +energy ordering values. +In Fig. 4, we present the number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell, as a function +of the simulation time, for the different temperature values considered here. The hydrogenation occurs mainly at the C1 +sites for all temperatures. High rates of H incorporation indicate high reactivity for hydrogenation. At low temperatures +4 + +Running Title for Header +Figure 3: Representative MD snapshots at different simulation times: a) 4 ps, b) 7.5 ps, and c) 200 ps of the +hydrogenation of the TPDH-gr membrane. Results from simulations at 300k. +Figure 4: The number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell as a function of the +simulation time (at %) for 150, 300, 500, and 800 K. The color of the curves indicates the corresponding sites in the +unit cell (left, upper). +(150K), the C2 and C4 sites have approximately the same low adsorption rates, while the C3 sites exhibit insignificant +or no hydrogen incorporations. Increasing the temperature, C4, C2, and C3 sites become more reactive, while above +300K, the C1 site has a slight decrease in reactivity. +3.2 +Electronic Structure +In Fig. 5.a), we present pristine (non-Hydrogenated) TPDH-gr electronic band structure and the corresponding projected +density of states (pDOS) (obtained from DFT-GGA-PBE calculations). We can see that pristine TPDG-gr exhibits a +5 + +Top +Bottom +4 ps +7.5 ps +200 ps +a) +b) +c) +C3 +C2 +C1 +C480 +088482 +Hydrogen +bonded +200 +(at.%) +10 +150 +0 +800 K +100 +500 K +50 +Temperature +Time (ps) +300K +0 +150 KRunning Title for Header +Figure 5: Electronic band structures and the corresponding projected density of states (pDOS) for a) non-hydrogenated +TPDH-gr, b) TPDH-gr with the tetragonal ring partially hydrogenated (C1 and C7’ sites occupied), and c) tetragonal +ring fully hydrogenated. The total density of states is shown in black, while the blue and green curves represent the +projected DOS into orbitals s and p, respectively. +semimetallic behavior. The highest (lowest) valence (conduction) band is partially filled. These results are consistent +with previous works published in the literature [32]. +The effects of H adsorption in the tetragonal ring were investigated for the cases with a pair adsorbed in neighboring +atoms, in opposite sites of the sheet, and with all four sites of the ring occupied (Fig. 2.g,h respectively). +The adsorption of two hydrogen atoms in the C1 and C7’ sites results in the opening of the direct gaps by approximately +1 eV at k-points M and Γ, as shown in Fig. 5.b. Surprisingly, the valence and conduction bands overlap at the Fermi +level, giving rise to a Dirac cone-like, between the k-points Y and M. Near this point, the electronic dispersion is +unusually linear, and charge carriers behave like massless fermions, obeying the Dirac relativistic equation. It is expected +that unusual transport properties arise from this pattern in the band structure, as predicted and experimentally observed +for graphene [9]. The electronic band structure and the corresponding pDOS of the TPDH with full hydrogenation +of the tetragonal ring are shown in Fig. 5.c. We can see the appearance of narrow gaps (0.5 eV ) between k-points Γ +and M, and a very narrow direct gap at point Y . The Dirac cone-like is shifted near the Γ points with respect to the +half-hydrogenated structure. +6 + +a) +4 +Tot +3 +S +p +2 +M +e +1 +0 +E +1 +-1 +E +-3 +4 +M +X +Y +pDOS +b) +4 +Tot +3 +p +2 +M +e +1 +0 +E +1 +-1 +E +-2 +3 +4 +M +X +^ +M +< +pDOS +c) +4 +Tot +3 +S +p +2 +M +e +1 +f +0 +E +1 +-1 +E +-2 +4 +M +X +Y +M +pDOSRunning Title for Header +4 +Conclusions +This work investigated the effects of hydrogenation on the structural and electronic properties of tetra-penta-deca- +hexagonal-graphene (TPDH-gr) sheets. Molecular dynamics (MD) simulations revealed that H atoms are mainly +incorporated in the tetragonal ring (C1 sites) with up to 80% adsorption at 300 K (Fig. 4). The number of H atoms +incorporated on C2 and C4 sites varies according to the temperature. Hydrogenation produces a pattern where H lines +are formed on both sides of the sheet (Figs. 1.h and 3.c) generating well delimited pentagonal ring strips along ⃗a +direction. DFT calculations further corroborate that the complete hydrogenation of the tetragonal ring is energetically +favorable. +Electronic structure calculations for the partially hydrogenated structure show the formation of gaps and the emergence +of a Dirac cone-like between the points Γ and M. For the fully hydrogenated ring, narrow band gaps followed by wide +gaps are identified, and the Dirac cone-like is translated near the Γ point. This electronic profile strongly indicates +anisotropic transport properties, although these remain to be further explored in future works. +Conflicts of interest +There are no conflicts to declare. +Acknowledgements +The authors thank PRH.49 for funding and CCM-UFABC for the computational resources provided and CNPq +(#310045/2019-3) +References +[1] F. Diederich and M. Kivala, “All-Carbon Scaffolds by Rational Design,” Advanced Materials, vol. 22, pp. 803–812, +feb 2010. +[2] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, “C60: Buckminsterfullerene,” Nature, +vol. 318, no. 6042, pp. 162–163, 1985. +[3] S. Iijima, “Helical microtubules of graphitic carbon,” Nature, vol. 354, no. 6348, pp. 56–58, 1991. +[4] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. +Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science, vol. 306, pp. 666–669, oct 2004. +[5] F. Schedin, A. K. Geim, S. V. 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Katsnelson, “Chemical functionalization of graphene,” Journal of Physics: Condensed +Matter, vol. 21, p. 344205, jul 2009. +9 + diff --git a/DNE4T4oBgHgl3EQf6A5m/content/tmp_files/load_file.txt b/DNE4T4oBgHgl3EQf6A5m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..204b64c35d0368b5233773b5769c6c1b1cd7ae1d --- /dev/null +++ b/DNE4T4oBgHgl3EQf6A5m/content/tmp_files/load_file.txt @@ -0,0 +1,844 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf,len=843 +page_content='TETRA-PENTA-DECA-HEXAGONAL-GRAPHENE (TPDH-GRAPHENE) HYDROGENATION PATTERNS: DYNAMICS AND ELECTRONIC STRUCTURE Caique Campos, Matheus Medina, Pedro Alves da Silva Autreto Center for Natural and Human Sciences (CCNH) Federal University of ABC (UFABC) Santo André - SP, 09210-170, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='autreto@ufabc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='br Douglas Soares Galvao Physics Institute Gleb Wataghin (IFGW) State University of Campinas (UNICAMP) Campinas/SP, Brazil galvao@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='br ABSTRACT The advent of graphene has renewed the interest in other 2D carbon-based materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Bhattacharya and Jana have proposed a new carbon allotrope, composed of different polygonal carbon rings containing 4, 5, 6, and 10 atoms, named Tetra-Penta-Deca-Hexagonal-graphene (TPDH-graphene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' This unusual topology created material with interesting mechanical, electronic, and optical properties and several potential applications, including UV protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Like other 2D carbon structures, chemical functionalizations can be used to tune their TPDH-graphene properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In this work, we investigated the hydrogenation dynamics of TPDH-graphene and its effects on its electronic structure, combining DFT and fully atomistic reactive molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Our results show that H atoms are mainly incorporated on tetragonal ring sites (up to 80% at% at 300 K), leading to the appearance of well-delimited pentagonal carbon stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The electronic structure of the hydrogenated structures shows the formation of narrow bandgaps with the presence of Dirac cone-like structures, indicative of anisotropic transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1 Introduction The versatility in chemical bonding (different hybridizations) of carbon atoms allows the existence of a wide variety of different structures (allotropes) [1], such as fullerenes [2], nanotubes [3], and graphene [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Graphene is a 2D allotrope of sp2 carbon atoms tightly packed into a hexagonal honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It presents high carrier mobility (5000cm2/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='s )[4, 5], high thermal conductivity (5000WmK−1) [6], and Young modulus value of 1 TPa [7], one of the highest values ever measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It has unveiled new and unique physics phenomena, including the quantum Hall effect [8], the ambipolar electric field effect [4], and the massless charge carriers of Dirac fermions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' These remarkable properties have made graphene the subject of a large number of theoretical and experimental studies in different areas, such as catalysis [10], electronics [11], spintronics [12], twistronics [13], and gas sensors [14], to name just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' However, graphene is a null electronic gap material, even exhibiting extraordinary electronic properties, which limits its use in some applications [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Chemical functionalizations, such as hydrogenation, are one viable mechanism for altering graphene-like structures’ properties (including opening the gap [15–17] or changing the Fermi level [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Structural and electronic changes are introduced when the chemical species form covalent bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The partial hydrogenation of graphene introduces unsaturated sp3 carbon atoms that can be used to attach additional functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='05328v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 Running Title for Header Figure 1: (a) Schematics of the unit cell of tetra-penta-deca-hexagonal-graphene (TPDH) and the corresponding carbon-carbon bond-length values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The different colors indicate non-equivalent carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' (b) A 2 × 2 supercell illustrating the TPDHG rings and the pores of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The corresponding Unit cell vector values are indicated in the highlighted red rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' (c) The structural setup simulation used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' A TPDH membrane (indicated in blue) is deposited on a graphene frame (gray), and the TPDHG/graphene structure is immersed in a hydrogen atmosphere (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' See text for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Despite these limitations, the advent of graphene created a revolution in materials science and renewed the interest in 2D carbon allotropes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Among these structures, it is worth mentioning graphynes and biphenylene carbon networks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Graphynes are the generic name for families of 2D carbon porous structures containing hexagon rings connected by acetylenic groups and with sp and sp2 hybridized carbon atoms in the same lattice [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Graphdyines refer to the structural families where two acetylenic groups connect the hexagons [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' They can exhibit metallic and semiconducting behaviors [21] and have been exploited in different technological applications [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Biphenylene carbon networks (including biphenylene carbon and graphenylenes) are families of porous structures composed of mixed carbon rings (pentagons, hexagons, heptagons, octagons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=') [19, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Similarly to graphynes, they can be metallic, or semiconductors and have potential applications in catalysis [23], gas sensors [25], batteries [26], and energy storage applications [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Recently, new synthetic routes for graphynes [28, 29] and biphenylene carbon networks [30] have been reported increasing the interest in these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Bhattacharya and Jana [31] have proposed a new structure composed of two pentagons and a tetragonal ring called tetra-penta-octogonal graphene (TPO-graphene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It is metallic with a Dirac cone at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='7 eV above the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' More recently, they proposed another structure belonging to the tetra-pentagonal graphene family composed of sp2 carbon rings with 4, 5, 12, and 6 atoms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1) named tetra-penta-deca-hexagonal graphene (TPDH-graphene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It possesses thermal and dynamical stability and exhibits elastic anisotropy with Young’s modulus value larger than that of graphene in a specific direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Depending on the morphology, TPDH-graphene nanoribbons can exhibit metallic, or semiconductor behavior [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In this work, we have investigated the effects of hydrogenation on the structural and electronic properties of TPDH- graphene (TPDH-gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The hydrogenation of TPDH-gr sheets was investigated through reactive molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Structural optimization, energy, and electronic properties were further analyzed using ab initio (DFT) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In spite of graphene’s extraordinary properties, it is a null gap material, which Chemical functionalization is one viable mechanism to introduce specific modifications into graphene-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Structural and electronic changes are introduced when the chemical species being introduced form a covalent bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' For example, graphite oxides can form oxygen groups in graphene sheets dispersed in water and organic solvents [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Stankovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' prepared graphite oxides functionalized with isocyanates that were later exfoliated into graphene oxides dispersed in an aprotic polar solvent [34] in a stable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Partial hydrogenation of graphene sheets introduces unsaturated carbon atoms sp3 that neighbor unpaired with electrons that can be used to attach additional functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Chemical functionalization also allows one to change the electronic properties of the structure by opening a bandgap [15–17] or changing the Fermi level [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2 a) b c) C4 C3 C2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='41A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='50A C1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='43A A = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='97 b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='94 aRunning Title for Header Figure 2: Adsorption energies for TPDH-gr in a) the non-equivalent sites, b) with an H atom adsorbed in the C1 site, c) two H atoms adsorbed in the C1 and C7’ sites, and d) tree H atoms adsorbed in the C1, C7’ and C5 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' e) Non-equivalent sites in TPDH-gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' f) remaining sites in the tetragonal ring with the C1 site occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The top sites are indicated by the solid line, while the bottom sides are indicated by the dashed ones and prime labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' g) Top and side views of TPDH-gr with C1 and C7’ sites occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The side view also shows the buckling height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' h) Top and side views of TPDH-gr with a fully hydrogenated tetragonal ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2 Computational Methods First-principles calculations were carried out within the Density Functional Theory (DFT) framework as implemented in Quantum Espresso code [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Electron-ion interactions were dealt with Projected Augmented wave (PAW) and Ultra-soft pseudopotentials for C and H atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' They were obtained from the Standard Solid State Pseudopotentials library (SSSP) [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Exchange and correlation potential were used within the Generalized Gradient Approximation (GGA) with the parameterization of Perdew, Burke, and Ernzerhof (GGA-PBE functional) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Valence electrons were treated with a set of plane waves basis set with a kinetic energy cutoff of 680 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The diagonalization of the density matrix was performed with the Davidson iterative method with matrix overlap using the self-consistency threshold of 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In the ionic relaxation calculations, the convergence thresholds were set to 10−3 eV and 10−2 eV/Å for energy and forces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Brillouin zone (BZ) sampling was performed using a 12 × 12 × 1 (16 × 16 × 1) k-point grid for SCF (NSCF) calculations following the scheme proposed by Monkhosrt and Pack [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' For electronic structure calculations, the k-points were chosen along the following path in the BZ: Γ(0, 0, 0) - M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0) - X(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0, 0) - Γ(0, 0, 0) - Y (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0) - M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5, 0) - Γ(0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' We have also carried out fully atomistic molecular dynamics (MD) simulations using the large-scale atomic/molecular massively parallel simulator (LAMMPS) code[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Atomic interactions were treated with the reactive force field (ReaxFF) [41], with C-C interaction parameters developed by Chenoweth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' All MD simulations were carried out in the canonical (NVT) ensemble, with a time step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='25 fs, and using a Nosé-Hoover thermostat [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The hydrogenation simulations were carried out considering a TPDH-gr membrane deposited on a graphene frame, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The TPDH-graphene membrane is a 24x15 supercell, in which only its central part (16x11) is exposed to the hydrogen atmosphere, resulting in a total number of 2112 available adsorption/reaction sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The hydrogen atmosphere was composed of 500 atoms in a volume of 60 000 Å3 on each side of the membrane, constrained to the exposed region of the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' This methodology has been successfully applied to other systems, such as Me-graphane[16] and graphone[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 3 a) b) ) d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0- 1111 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 [eV/atom] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 [eV/ato] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0- L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content="0 C6' C3 C6 C1 C4 C5 C6' C2 C5 C5' C6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=" C6' C7 C7 C5' C6 Site Site Site Site C4 g) f) h) e) C6 C6' h = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='185 A h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='87 ARunning Title for Header 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='1 Ab initio Binding Energy and Hydrogenation Dynamics TPDH-gr has a Pmmm (space group #47) symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' the 12 carbon atoms in its unit cell are arranged in an orthorhombic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The obtained optimized lattice parameters were: a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='94 Å, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='97 Å with γ = 90o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' There are three different bond lengths (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='41, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='50, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='44 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=') involving the C atoms, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Except for the C atoms bonded along the ⃗a direction in the tetragonal ring, the bond lengths are close to those sp2 in graphene (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='41 Å)[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' These results agree well with those reported by Batthacharya and Jana [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The most favorable sites for H adsorption/reaction were investigated by evaluating the binding energy per adsorbed atom, calculated as the energy difference between the hydrogenated structure and its parts: Eb = − �ET P DH+nH − (ET P DH + nEH) n � where ET P DH+nH is the energy of TPDH-gr with n adsorbed H atoms, ET P DH is the energy of a TPDH-gr unit cell, and EH the energy of an isolated H atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The negative sign means that high energies indicate more favorable sites for adsorption than others in the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' First, an H atom is adsorbed at each of the non-equivalent sites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The site corresponding to the highest energy is taken as the most favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Then, a second H is adsorbed at each of the remaining sites, and the most favorable one is evaluated according to Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' This process is repeated until the tetragonal ring on the TPDH-gr is fully hydrogenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' We present the binding energies and obtained structures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The adsorption of the first H atom is more favorable on the C1 site, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='a with Eb of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='35 eV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' After C1-Cx adsorption (with x = 2, 5, and 7), the bond length values increased to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='51, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='55, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='53 Å , respectively, indicating a transition to sp3=like-bond in the C1 atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It is worth mentioning that for sites located in the tetragonal ring, the top and bottom configurations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='d) were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Adsorption of a single H atom at each of these sites resulted in roughly the same results for Eb, as can be seen in Table 1S in Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The adsorption of the second H atom (resulting in 16% hydrogen coverage) is more favorable at the C7′ site (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='c), with an Eb of +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='75 eV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The resulting lattice distortions in the direction perpendicular to the structure plane lead to a significant buckling of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='87 Å , as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The distortions of the structure and the fact that two neighboring C atoms adsorb the pair of H atoms (but on opposite sides of the sheet) are in accordance with the results reported by Boukhvalov and Katsnelson for the hydrogenation of graphene sheets [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Interestingly, the adsorption of a third H atom gives the same Eb for both C5 and C6’ sites, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In this case, the configuration in which the C1, C7’ and C5 sites are occupied was imposed, which will be justified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The resulting structure presents an overall increase in the Cx-C bond lengths (with x = 1, 7, and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The vertical distance separating the C1 and C7 atoms is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='02 Å versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='84 Å for the corresponding value between the C5 and C6 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The adsorption of a fourth H atom (33% hydrogen coverage) is more favorable at the C6’ site with Eb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='0 eV/ Å and buckling of h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='185 Å(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It is clear that choosing the C5 or C6’ sites in the adsorption of the third H atom leads basically to the same configuration (C1-C7’-C5-C6’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Therefore, choosing C5 or C6’ for the adsorption of the third H atom is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' These calculations reveal a pattern for the hydrogenation of the tetragonal ring, which consists of two lines of H atoms on opposite sides of the basal plane sheet, leading to the formation of well-delimited pentagonal ring strips along the direction of the lattice vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' DFT calculations confirm that this configuration is indeed more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Molecular Dynamics simulations, discussed below, produced similar results, Reactive molecular dynamics simulations were carried out to study the dynamics and temperature effects on hydrogen adsorption of bigger TPDH-graphene membranes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1), which would be cost-prohibitive with DFT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Representative MD snapshots of both sides of the TPDH-graphene membrane during the hydrogenation process (at 300K) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 3 (a) - (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The H atoms are predominantly incorporated throughout the MD simulations on the C1 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Analyzing the hydrogena- tion process, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 3 (a) to (c), we can see that the hydrogen-adsorbed C1 sites act as seeds to the hydrogenation of their C1 neighbors, forming lines through the structure surface, which is an expected result, based on the DFT binding energy ordering values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 4, we present the number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell, as a function of the simulation time, for the different temperature values considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The hydrogenation occurs mainly at the C1 sites for all temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' High rates of H incorporation indicate high reactivity for hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' At low temperatures 4 Running Title for Header Figure 3: Representative MD snapshots at different simulation times: a) 4 ps, b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 ps, and c) 200 ps of the hydrogenation of the TPDH-gr membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Results from simulations at 300k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Figure 4: The number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell as a function of the simulation time (at %) for 150, 300, 500, and 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The color of the curves indicates the corresponding sites in the unit cell (left, upper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' (150K), the C2 and C4 sites have approximately the same low adsorption rates, while the C3 sites exhibit insignificant or no hydrogen incorporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Increasing the temperature, C4, C2, and C3 sites become more reactive, while above 300K, the C1 site has a slight decrease in reactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='2 Electronic Structure In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='a), we present pristine (non-Hydrogenated) TPDH-gr electronic band structure and the corresponding projected density of states (pDOS) (obtained from DFT-GGA-PBE calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' We can see that pristine TPDG-gr exhibits a 5 Top Bottom 4 ps 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 ps 200 ps a) b) c) C3 C2 C1 C480 088482 Hydrogen bonded 200 (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='%) 10 150 0 800 K 100 500 K 50 Temperature Time (ps) 300K 0 150 KRunning Title for Header Figure 5: Electronic band structures and the corresponding projected density of states (pDOS) for a) non-hydrogenated TPDH-gr, b) TPDH-gr with the tetragonal ring partially hydrogenated (C1 and C7’ sites occupied), and c) tetragonal ring fully hydrogenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The total density of states is shown in black, while the blue and green curves represent the projected DOS into orbitals s and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' semimetallic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The highest (lowest) valence (conduction) band is partially filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' These results are consistent with previous works published in the literature [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The effects of H adsorption in the tetragonal ring were investigated for the cases with a pair adsorbed in neighboring atoms, in opposite sites of the sheet, and with all four sites of the ring occupied (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='g,h respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The adsorption of two hydrogen atoms in the C1 and C7’ sites results in the opening of the direct gaps by approximately 1 eV at k-points M and Γ, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Surprisingly, the valence and conduction bands overlap at the Fermi level, giving rise to a Dirac cone-like, between the k-points Y and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Near this point, the electronic dispersion is unusually linear, and charge carriers behave like massless fermions, obeying the Dirac relativistic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' It is expected that unusual transport properties arise from this pattern in the band structure, as predicted and experimentally observed for graphene [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The electronic band structure and the corresponding pDOS of the TPDH with full hydrogenation of the tetragonal ring are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' We can see the appearance of narrow gaps (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='5 eV ) between k-points Γ and M, and a very narrow direct gap at point Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The Dirac cone-like is shifted near the Γ points with respect to the half-hydrogenated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 6 a) 4 Tot 3 S p 2 M e 1 0 E 1 1 E 3 4 M X Y pDOS b) 4 Tot 3 p 2 M e 1 0 E 1 1 E 2 3 4 M X ^ M < pDOS c) 4 Tot 3 S p 2 M e 1 f 0 E 1 1 E 2 4 M X Y M pDOSRunning Title for Header 4 Conclusions This work investigated the effects of hydrogenation on the structural and electronic properties of tetra-penta-deca- hexagonal-graphene (TPDH-gr) sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Molecular dynamics (MD) simulations revealed that H atoms are mainly incorporated in the tetragonal ring (C1 sites) with up to 80% adsorption at 300 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' The number of H atoms incorporated on C2 and C4 sites varies according to the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Hydrogenation produces a pattern where H lines are formed on both sides of the sheet (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='h and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='c) generating well delimited pentagonal ring strips along ⃗a direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' DFT calculations further corroborate that the complete hydrogenation of the tetragonal ring is energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Electronic structure calculations for the partially hydrogenated structure show the formation of gaps and the emergence of a Dirac cone-like between the points Γ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' For the fully hydrogenated ring, narrow band gaps followed by wide gaps are identified, and the Dirac cone-like is translated near the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' This electronic profile strongly indicates anisotropic transport properties, although these remain to be further explored in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' Acknowledgements The authors thank PRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content='49 for funding and CCM-UFABC for the computational resources provided and CNPq (#310045/2019-3) References [1] F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE4T4oBgHgl3EQf6A5m/content/2301.05328v1.pdf'} diff --git a/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/2301.11242v1.pdf.txt b/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/2301.11242v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..134aa919211a01962d02073aabd5c601a3a42bd6 --- /dev/null +++ b/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/2301.11242v1.pdf.txt @@ -0,0 +1,1855 @@ +Regular Separability in Büchi VASS +Pascal Baumann � +Max Planck Institute for Software Systems (MPI-SWS), Germany +Roland Meyer � +TU Braunschweig, Germany +Georg Zetzsche � +Max Planck Institute for Software Systems (MPI-SWS), Germany +Abstract +We study the (ω-)regular separability problem for Büchi VASS languages: Given two Büchi VASS +with languages L1 and L2, check whether there is a regular language that fully contains L1 while +remaining disjoint from L2. We show that the problem is decidable in general and PSPACE-complete +in the 1-dimensional case, assuming succinct counter updates. The results rely on several arguments. +We characterize the set of all regular languages disjoint from L2. Based on this, we derive a (sound +and complete) notion of inseparability witnesses, non-regular subsets of L1. Finally, we show how to +symbolically represent inseparability witnesses and how to check their existence. +2012 ACM Subject Classification Theory of computation → Models of computation +Keywords and phrases Separability problem, Vector addition systems, Infinite words, Decidability +Funding Funded by the European Union (ERC, FINABIS, 101077902). Views and opinions expressed +are however those of the author(s) only and do not necessarily reflect those of the European Union +or the European Research Council Executive Agency. Neither the European Union nor the granting +authority can be held responsible for them. The second author was supported by the DFG project +EDS@SYN: Effective Denotational Semantics for Synthesis. +arXiv:2301.11242v1 [cs.FL] 26 Jan 2023 + +erc +EuropeanResearchCouncil +EstablishedbytheEuropeanCommission2 +Regular Separability in Büchi VASS +1 +Introduction +The separability problem asks, given languages L1 and L2, whether there exists a language +R that separates L1 and L2, meaning L1 ⊆ R and R ∩ L2 = ∅. Here, R is constrained to be +from a particular class S of admitted separators. Since safety verification of systems with +concurrent components is usually phrased as an intersection problem for finite-word languages, +and separators certify disjointness, deciding separability can be viewed as synthesizing safety +certificates. +Analogously, deciding separability for infinite-word languages is a way of +certifying liveness. If S is the class of (ω-)regular languages, we speak of regular separability. +Separability problems have been studied intensively over the last few years. If the input +languages are themselves regular and S is a subclass [42, 41, 40, 39, 43, 44, 35, 15], then +separability generalizes the classical subclass membership problem. Moreover, separability +for languages of infinite-state systems has received a significant amount of attention [17, 16, +14, 13, 10, 9, 12, 1, 51, 48, 11, 8]. Let us point out two prominent cases. +First, one of the main open problems in this line of research is whether regular separability +is decidable for (reachability) languages of vector addition systems with states (VASS): A +VASS consist of finitely many control states and a set of counters that can be incremented +and decremented, but not tested for zero. Moreover, each transition is labeled by a word +over the input alphabet. Here, a run is accepting if it reaches a final state with all counters +being zero. While there have been several decidability results for subclasses of the VASS +languages [17, 14, 13, 10, 9], the general case remains open. Second, a surprising result is +that if K and L are coverability languages of well-structured transition systems (WSTS), +then K and L are separable by a regular language if and only if they are disjoint [14]. As +VASS are one example of WSTS, this result also applies to their coverability languages. +Regular separability in Büchi VASS +In this paper, we study the regular separability +problem for Büchi VASS. These are VASS that accept languages of infinite words. A run +is accepting if it visits some final state infinitely often. Since no condition is placed on the +counter values, Büchi VASS languages are an infinite-word analogue of finite-word coverability +languages, where acceptance is defined by the reached state (not the counters). The regular +separability problem is to decide, given Büchi VASS V1 and V2, whether there exists an +ω-regular language R such that L(V1) ⊆ R and L(V2) ∩ R = ∅. +Our main results are that (i) regular separability for Büchi VASS is decidable, and +that (ii) for one-dimensional Büchi VASS (i.e. those with a single counter) the problem is +PSPACE-complete. Here, we assume that the counter updates are encoded in binary. +Given that Büchi VASS accept using final states and their transition systems are WSTS, +one may suspect that there is an analogue of the aforementioned result for WSTS: Namely, +that two languages of Büchi VASS are separable by an ω-regular language if and only if they +are disjoint. We show that this is not the case: There are Büchi VASS V1 and V2 such that +L(V1) and L(V2) are disjoint, but not separable by an ω-regular language. In fact, we show +an even larger disparity between these two problems for WSTS in the infinite-word case: We +exhibit a natural class of WSTS for which intersection is decidable but regular separability is +not. Thus, regular separability for Büchi VASS requires significantly new ideas and involves +several phenomena that do not occur for finite-word languages of VASS. +New phenomena and key ingredients +We first observe that we can assume one input +language to be fixed, namely an infinite-word version Dn of the Dyck language. Then, +following the basic separator approach from [17], we identify a small class B of ω-regular + +P. Baumann, R. Meyer, and G. Zetzsche +3 +languages such that L is separable from Dn if and only if L is included in a finite union of +sets from B. Here, a crucial insight is that a Büchi automaton can guarantee disjointness +from Dn without knowing exactly when the letter balance crosses zero. Note that a negative +letter balance is the exact condition for non-membership in Dn. In contrast, in the finite +word case, there are always separating automata that can tell when zero is crossed [17]. This +insight is also key to the example differentiating disjointness and separability in Büchi VASS, +and to the undecidability proof for certain WSTS despite decidable disjointness. +We then develop a decomposition of Büchi VASS languages into finitely many pieces, +which are induced by what we call profiles. Inspired by Büchi automata, the idea of a +profile is to fix the set of transitions that can and have to be taken infinitely often in a +run. Finding the right generalization to Büchi VASS, however, turned out to be non-trivial. +Our formulation refers to edges in the Karp-Miller graph, augmented by constraints that +guarantee the existence of an accepting run. The resulting decomposition has properties +similar to the decomposition of VASS languages into run ideals [33], which has been useful +for previous separability procedures [17, 12]. +We associate to each profile a system of linear inequalities and show that separability +holds if and only if each of these systems is feasible. While this yields decidability, checking +feasibility is not sufficient to obtain a PSPACE-upper bound in the one-dimensional case. +Instead, we use Farkas’ Lemma to obtain a dual system of inequalities so that separability +fails if and only if one dual system is satisfiable. A solution to a dual system yields a +pattern in the Karp-Miller graph, called inseparability flower, which witnesses inseparability. +Compared to prior witnesses for deciding properties of VASS languages (e.g. regularity [18], +language boundedness [7], and other properties [3]), inseparability flowers are quite unusual: +they contain a non-linear condition, requiring one vector to be a scalar multiple of another. +For one-dimensional Büchi VASS, the condition degenerates into a linear one. This allows +us to translate inseparability flowers into particular runs in a two-dimensional VASS subject +to additional linear constraints. Using methods from [5], this yields a PSPACE procedure. +Related work +It was already shown in 1976 that regular separability is undecidable for +context-free languages [47, 30]. Over the last decade, there has been intense interest in +deciding regular separability for subclasses of finite-word VASS reachability languages: The +problem is decidable for (i) reachability languages of one-dimensional VASS [13], (ii) cov- +erability languages of VASS [14], (iii) reachability languages of Parikh automata [9], and +(iv) commutative reachability languages of VASS [10]. Moreover, decidability still holds if one +input language is an arbitrary VASS language and the other is as in (i)-(iii) [17]. As discussed +above, for finite-word coverability languages of WSTS, regular separability is equivalent to +disjointness [14]. Moreover, the aforementioned undecidability for context-free languages has +been strengthened to visibly pushdown languages [32]. To our knowledge, for languages of +infinite words, separability has only been studied for regular input languages [38, 29]. +Our result makes use of Farkas’ Lemma to demonstrate the absence of what can be +understood as a linear ranking function (on letter balances). There are precursors to this. In +liveness verification [45], Farkas’ Lemma has been used to synthesize, in a complete way, linear +ranking functions proving the termination of while programs over integer variables. In the +context of separability for finite words, Farkas’ Lemma was used to distinguish separable from +non-separable instances [17], similar to our approach. The novelty here is the combination of +Farkas’ Lemma with the new notion of profiles needed to deal with infinite runs. +The languages of Büchi VASS have first been studied by Valk [49] and (in the determin- +istic case) Carstensen [6]. Some complexity results (such as EXPSPACE-complexity of the + +4 +Regular Separability in Büchi VASS +emptiness problem) were shown by Habermehl [28]. More recently, there have been several +papers on the topological complexity of Büchi VASS languages (and restrictions) [26, 20, 27]. +See the recent article by Finkel and Skrzypczak [27] for an overview. +2 +Preliminaries +Dyck Language +We use an infinite-word version of the Dyck language over n pairs of +matching letters ai, ¯ai. We denote the underlying alphabet by Σn := �n +i=1{ai, ¯ai}. The +Dyck language contains those infinite words where every occurrence of ¯ai has a matching +occurrence of ai to its left: Dn := {w ∈ Σω +n | ∀v ∈ prefix(w): ∀i ∈ [1, n]: ϕi(v) ≥ 0}. Here, +ϕi : Σ∗ +n → Z is the ith (letter) balance function that computes for a given word w the +difference |w|ai − |w|¯ai. We also use ϕ(w) for the vector (ϕ1(w), . . . , ϕn(w)) ∈ Zn. +Büchi VASS and Automata +A Büchi vector addition system with states (Büchi VASS) of +dimension d ∈ N over alphabet Σ is a tuple V = (Q, q0, T, F) consisting of a finite set of +states Q, an initial state q0 ∈ Q, a set of final states F ⊆ Q, and a finite set of transitions +T ⊆ Q×Σ∗ ×Zd ×Q. The size of the Büchi VASS is |V| := |Q|+1+|F|+� +(q,w,δ,q′)∈T |w|+ +�d +i=1 max{log |δ(i)|, 1}. If d = 0, we call V a Büchi automaton. +The semantics of the Büchi VASS is defined over configurations, which are elements of +Q × Nd. We call the second component in a configuration the counter valuation and refer to +the entry in dimension i as the value of counter i. The initial configuration is (q0, 0). We lift +the transitions of the Büchi VASS to a relation over configurations → ⊆ Q×Nd×Σ∗×Q×Nd +as follows: (q, m) +w +−→ (q′, m′) if there is (q, w, δ, q′) ∈ T so that m′ = m + δ. A run of the +Büchi VASS is an infinite sequence of transitions of the form (q0, 0) +w1 +−−→ (q1, m1) +w2 +−−→ · · · +Thus, the sequence starts in the initial configuration and makes sure the target of one +transition is the source of the next. The run is accepting if it visits final states infinitely +often, meaning there are infinitely many configurations (q, m) with q ∈ F. The run is said to +be labeled by the word w = w0w1 · · · in Σω. The language L(V) of the Büchi VASS consists +of all infinite words that label an accepting run. Note that we can always ensure that every +accepting run has an infinite-word label, by tracking in the state whether a non-ε-transition +has occurred since the last visit to a final state. An infinite-word language is (ω-)regular, if +it is the language of a Büchi automaton. As we only consider infinite-word languages, we +just call them languages. +Karp-Miller Graphs +We work with the Karp-Miller graph KM(V) associated with a Büchi +VASS V [31]. Since we are interested in infinite runs, we define the Karp-Miller graph as a +Büchi automaton. Its state set is a finite set of extended configurations, which are elements +of Q × (N ∪ {ω})d. The initial state is the initial configuration in the Büchi VASS. The final +states are those extended configurations (q, m) with q ∈ F. The transitions are labeled by +T, so instead of letters they carry full Büchi VASS transitions. An entry ω in an extended +configuration denotes the fact that a prefix of a run can be repeated to produce arbitrarily +high counter values. More precisely, the Karp-Miller graph is constructed as follows. From +an extended configuration (q, m) we have a transition labeled by (q1, a, δ, q2), if q = q1 and +m + δ remains non-negative. The latter addition is defined componentwise and assumes +ω + k := ω =: k + ω for all k ∈ Z. The result of taking the transition is the extended +configuration (q2, m2), where m2 is constructed from m + δ as follows. We raise to ω +all counters i for which there is an earlier configuration (q2, m1) with m1 ≤ m + δ and +m1(i) < [m + δ](i), earlier meaning on some path from (q0, 0) to (q, m). If this is the case, + +P. Baumann, R. Meyer, and G. Zetzsche +5 +q0 +q1 +q2 +e1|ε +0|ε +0|ε +0|¯a1 +−e1|a1 +e1|¯a1 +a1 +a2 +ε +a1¯a2¯a2 +a2¯a1¯a1 +x +Figure 1 Left: A Büchi VASS accepting a language S with S ∩ D1 = ∅ but S ̸ | D1. Here, e1 ∈ Z +is the one-dim. vector with entry 1. Right: A regular language that is not included in a finite union +of languages Pi,k and Si,k, but that is included in Sx,k for x = (1, 1), k = 1. The horizontal and +vertical dimensions denote the balance for a1 resp. a2. +the path from (q2, m1) to (q2, m + δ) can be repeated indefinitely to produce arbitrarily +high values for counter i. We refer to the repetition of such a path in a run as pumping. +The Karp-Miller graph over-approximates the language of the Büchi VASS in the following +sense. Every infinite sequence of transitions that leads to a run of the Büchi VASS is the +labeling of an infinite run in the Karp-Miller graph. Moreover, if the run of the Büchi VASS +is accepting, so is the run in the Karp-Miller graph. In the other direction, every finite +transition sequence in the Karp-Miller graph represents a transition sequence in the Büchi +VASS. The sequence in the Büchi VASS, however, may be longer to compensate negative +effects on ω-entries by pumping. +3 +Problem, Main Result, and Proof Outline +A language R is a regular separator for a pair of languages L1, L2, if R is regular, L1 ⊆ R, and +R ∩ L2 = ∅. We write L1 | L2 for the fact that a regular separator exists. We consider here +languages of Büchi VASS, and formulate the regular separability problem as follows. Given +Büchi VASS V1, V2, check whether L(V1) | L(V2) holds. Our main result is the following. +▶ Theorem 3.1. The regular separability problem for Büchi VASS is decidable. +It should be noted that our procedure is non-primitive recursive, as it explicitly constructs +the Karp-Miller graph of an input Büchi VASS, which can be of Ackermannian size [36, +Theorem 2]. In the case of VASS coverability languages (and even for more general WSTS), +it is known that regular separability is equivalent to disjointness [14]. +Thus, for finite +words, separability reduces to the much better understood problem of disjointness. For the +infinite-word languages considered here, the situation is different. +▶ Theorem 3.2. There are Büchi VASS languages L1, L2 with L1 ∩ L2 = ∅ and L1 ̸ | L2. +There are classes of WSTS where intersection is decidable but separability is not. +For the second statement, we introduce the class of weak Büchi reset VASS, which are VASS +with reset instructions, with the additional constraint that each run can only use resets a +finite number of times. Details can be found in Appendix F. +For the first statement of Theorem 3.2, we give an intuition and refer to Appendix A for +details. We choose L1 = L(V), where V is the Büchi VASS in Figure 1(left), and L2 = D1, +the Dyck language. To show L(V) ̸ | D1, suppose there is a Büchi automaton A with n +states such that L(V) ⊆ L(A) and L(A) ∩ D1 = ∅. Then A has to accept (an +1 ¯an+1 +1 +)ω ∈ L(V). +However, pumping yields that for some m > n the word (am +1 ¯an+1 +1 +)ω ∈ D1 also has to be + +6 +Regular Separability in Büchi VASS +accepted by A, contradiction. Moreover, to show L(V) ∩ D1 = ∅ we observe that in accepting +runs of V, almost every visit (meaning: all but finitely many) to the final state drops the +letter balance by 1. Therefore on any accepting run this balance eventually becomes negative, +yielding a word outside of D1. +In the remainder of the section, we outline the proof of Theorem 3.1. Assume we are +given L1 = L(V1) and L2 = L(V2) and this is a non-trivial instance of separability, meaning +L1, L2 are not regular and L1 ∩ L2 = ∅. For proving separability, we could enumerate +regular languages until we find a separator. The difficult part is disproving separability. +Inseparability of L1 and L2 is witnessed by a set of words W ⊆ L1 so that every regular +language R containing them already intersects L2, formally: W ⊆ R implies R ∩ L2 ̸= ∅. +Showing the existence of such a set W is difficult for two reasons. First, it is unclear which +sets of words ensure the universal quantification over all regular languages. Second, as we +have a non-trivial instance of separability, W (if it exists) will be a non-regular language. So +it is unclear how to represent it in a finite way and how to check its existence. +To address the first problem and understand the sets of words that disprove separability, +we use diagonalization. Call an (L2-)separator candidate a regular language that is disjoint +from L2. Let R1, R2, . . . be an enumeration of the separator candidates. If L1 is not separable +from L2, for every Ri there is a word wi ∈ L1 with wi /∈ Ri. We call such a set of words +W = {w1, w2, . . .} that escapes every separator candidate an inseparability witness. +▶ Observation 3.3. L1 ̸ | L2 if and only if there is an inseparability witness. +Our decision procedure will check the existence of an inseparability witness. We obtain the +procedure in four steps: the first is a simplification, the second is devoted to understanding +the separator candidates, the third is another simplification, and the last characterizes the +inseparability witnesses and checks their existence. +Step 1: Fixing L2 +We first reduce general regular separability to regular separability from +the Dyck language. The reduction is simple and works just as for finite words [17]. +▶ Lemma 3.4. Given Büchi VASS V1 and V2, we can compute a Büchi VASS V over Σn so +that L(V1) | L(V2) if and only if L(V) | Dn, where n is the dimension of V2. +Step 2: Understanding the Separator Candidates +To understand the regular languages +that are disjoint from Dn, we will define basic separators, sets Pi,k and Sx,k, on which we +elaborate in a moment. The following theorem says that finite unions of basic separators are +sufficient for regular separability. This is our first technical result and shown in Section 4. +▶ Theorem 3.5. If R ⊆ Σω +n is regular and R ∩ Dn = ∅, then R is included in a finite union +of basic separators. +For the definition of Pi,k, we note that the words outside Dn have, for some index i ∈ [1, n], +an earliest moment in time where the balance between ai and ¯ai falls below zero. To turn +this into a regular language, we impose an upper bound k ∈ N on the (positive) balance +between the letters ai and ¯ai that is maintained until the earliest moment is reached. This +yields the regular language +Pi,k := {w ∈ Σω +n | ∃v ∈ prefix(w): ϕi(v) < 0 ∧ ∀u ∈ prefix(v): ϕi(u) ≤ k}. +The family of languages Pi,k already captures the complement of Dn. The problem is +that we may need infinitely many such languages to cover the language R of interest. For + +P. Baumann, R. Meyer, and G. Zetzsche +7 +every bound k, a regular R with R∩D1 = ∅ may contain a word with a higher balance before +falling below zero, take for example R = a∗ +1¯aω +1 . The first insight is that if R can fall below +zero from arbitrarily high values, then the underlying Büchi automaton has to contain loops +with a negative balance. The R thus contains words uv with an unconstrained prefix and a +suffix that decomposes into v = v1v2 · · · so that every infix w = vℓ has a negative balance on +letter ai. The observation suggests the definition of a language that contains precisely the +words u.v. To make the language regular, we impose a bound k on the positive balance that +can be used during the infixes w. Call the resulting language Si,k. Unfortunately, taking the +Pi,k and the Si,k as basic separators is still not enough: Figure 1(right) exhibits a regular +language, disjoint from D1, that is not included in a finite union of Pi,k and Si,k, because it +contains infixes where the balance on each letter exceeds all bounds in each coordinate. +The second insight is that we can catch the remaining words with a version of Si,k that +weights coordinates with some x ∈ Nn. Let us give some intuition on this. The words from R +that we cannot catch with a Pi,k must come across, for each i that becomes negative, a loop +with positive balance on i (otherwise, the balance on those i would be bounded). But then, +the only way such words can avoid D1 is by ending up in a strongly connected component +where every loop (with a final state) makes progress towards crossing 0, i.e. is negative in +some coordinate. One can then conclude that even all Q≥0-linear combinations of loops +(a convex set) must avoid the positive orthant Qn +≥0 ⊂ Qn. By the Hyperplane Separation +Theorem (we use it in the form of Farkas’ Lemma), this is certified by a hyperplane that +separates all loop effects from Qn +≥0. This hyperplane is given by some orthogonal vector +x ∈ Nn, meaning that every loop balance must have negative scalar product with x. Hence, +we can catch these words by: +Sx,k := +� +u.v ∈ Σω +n +����� +a.) ∀f ∈ infix(v): ⟨x, ϕ(f)⟩ ≤ k, and +b.) v = v0.v1.v2 · · · ∧ ∀ℓ ∈ N: ⟨x, ϕ(vℓ)⟩ < 0 +� +. +Coming back to Figure 1(right), the weight vector x = (1, 1) guarantees that the weighted +balance decreases indefinitely and also the weighted balances of all infixes stay bounded. +In [17], a similar argument has been used to show sufficiency of basic separators. +Step 3: Pumpable Languages +With the basic separators at hand, the task is to understand +the sets of words witnessing inseparability. While studying this problem, we observed that +the argumentation for the Pi,k was always similar to the one for the Sx,k. This led us to +the question of whether we can get rid of the Pi,k in separators. The answer is positive, and +hinges on a new notion of pumpability for languages over Σn. +Call infinite words u and v equivalent, written u ∼ v, if v can be obtained from u by +removing and inserting finitely many letters: There are u0, v0 ∈ Σ∗ and w ∈ Σω such that +u = u0w and v = v0w. We say that a language L ⊆ Σω +n is pumpable if for every w ∈ L +and every k ∈ N, there exists a decomposition w = w0w1 and a word w′ +0 ∈ Σ∗ +n that is a +prefix of a word in Dn such that w′ +0.w1 ∈ L and the letter balance satisfies the following: +(a) ϕ(w′ +0) ≥ ϕ(w0) and (b) for the indices i ∈ [1, n] where ϕi becomes negative on some +prefix of w, we have ϕi(w′ +0) ≥ max{ϕi(w0), 0} + k. The consequence of this definition is that +a pumpable language leaves every language Pk := � +i∈[1,n] Pi,k. Indeed, for every word w ∈ L +and every k ∈ N, there is a word w′ ∈ L with w ∼ w′ where the letter balance exceeds k +before becoming negative, and thus w′ /∈ Pk. With the previous characterization of separator +candidates, what is left to separate L from Dn are the languages Sx,k. +▶ Lemma 3.6. If L ⊆ Σω +n is pumpable, then L | Dn if and only if L |limDn, where L |limDn +means L ⊆ � +x∈X Sx,k for some finite set X ⊆ Nn and some k ∈ N. + +8 +Regular Separability in Büchi VASS +q0 +q1 +q2 +(1, 0)|ε +0|ε +0|ε +(0, −1)|ε +(−1, 1)|ε +(1, −1)|ε +(q0, ω, 0) +(q1, ω, 0) +(q2, ω, 0) +(q1, 0, 0) +(q0, 0, 0) +(q2, 0, 0) +(q2, ω, ω) +(q1, ω, ω) +e1|ε +0|ε +0|ε +e1|ε +0|ε +0|ε +−e1|a1 +0|ε +0|¯a1 +−e1|a1 +e1|¯a1 +Figure 2 Left: The Büchi VASS ¯V constructed from the Büchi VASS V found in Figure 1(left). +Note how the added second counter tracks the letter balance of the now removed transition labels, +incrementing on letter a1 and decrementing on letter ¯a1. Right: The Büchi VASS Vpump corresponding +to V as given by Theorem 3.7. Here we did not mark the final states to reduce visual clutter; every +state that includes q1 is considered final. Similarly, the two labels above the loop in the top right +correspond to two distinct transitions. Note that Vpump essentially looks like KM(¯V), just with +different transition labels. +In our context, pumpability is interesting because we can turn every Büchi VASS language +into a pumpable language without affecting separability. +▶ Theorem 3.7. Let V be a d-dim. Büchi VASS over Σn. We can compute a d-dim. Büchi +VASS Vpump that satisfies the following: +1. L(Vpump) is pumpable, +2. there is a k ∈ N so that L(Vpump) ⊆ L(V) ⊆ L(Vpump) ∪ Pk, and +3. L(V) | Dn if and only if L(Vpump) | Dn. +The construction of Vpump employs the Karp-Miller graph in an original way, namely to +track the unboundedness of letter balances. Let ¯V be the (d + n)-dimensional Büchi VASS +obtained from V by tracking the effect of the letters from Σn in n additional counters. For +¯V, we construct the Karp-Miller graph. The relationship between the languages of KM(¯V) +and V is as follows. For all words where every letter balance stays non-negative, their runs +in V can be mimicked in KM(¯V). For all other words, where the balance eventually becomes +negative, this only holds if the corresponding counter in ¯V has been raised to ω beforehand. +Essentially, the new Büchi VASS Vpump restricts V to those runs that have counterparts in +KM(¯V). This is achieved with a simple product construction of V and KM(¯V). The thing +to note is that every word from L(V) that does not make it into L(Vpump) belongs to Pk, +where k is the maximum concrete number in KM(¯V): A run in V that cannot be mimicked +in KM(¯V) will at some point have a negative letter balance, before reaching ω in KM(¯V) in +that component; thus all counter values had been at most k until that point. +An example on how to construct ¯V and Vpump can be found in Figure 2, where both were +constructed for the Büchi VASS found in Figure 1(left). +In the proof of Theorem 3.5, we make use of Theorem 3.7 (recall that a regular language +is the language of a 0-dimensional Büchi VASS). This may look like cyclic reasoning, but it +is not: We will show Theorem 3.7(1)+(2) directly, using the arguments above. With this, we +prove Theorem 3.5, which in turn is used to derive Lemma 3.6 and Theorem 3.7(3). +Step 4: Non-Separability Witnesses and Decidability +Because of pumpability, it remains +to decide whether a Büchi VASS language L(V) is included in a finite union � +x∈X Sx,k for + +P. Baumann, R. Meyer, and G. Zetzsche +9 +some k. Part of the difficulty is that we have no bound on the cardinality of X. To circumvent +this, we decompose L(V) into a finite union � +π Lπ(V), where π is a profile, meaning a set of +edges in KM(V) seen infinitely often during a run of V. We then show that each Lπ(V) is +either (i) included in a single separator Sx,k or (ii) escapes every finite union � +x∈X Sx,k. +Here, it is key to show an even stronger fact: In case (i), not only Lπ(V) is included in +some Sx,k, but the entire set of runs in KM(V) that eventually remain in π. The advantage +of strengthening is that finiteness of KM(V) allows us to express inclusion in Sx,k, for some k, +as a finite system of linear inequalities over x: We say that (1) the balance of every primitive +cycle, weighted by x, is at most zero and (2) the balance, weighted by x, of some cycle +containing all edges from π is negative. Here, (1) and (2) correspond to Conditions a.) and +b.) of Sx,k. If they are met, then the runs of KM(V) along π are included in Sx,k for some k. +We then prove that if the system is not feasible, then V has runs that escape every +finite union � +x∈X Sx,k. To this end, we employ Farkas’ Lemma: It tells us that if there is +no solution, then the dual system has a solution. The solution of the dual system can be +interpreted as an executable linear combination of primitive cycles with non-negative balances. +We show that these cycles can be arranged in a pattern in KM(V) we call inseparability +flower. Such an inseparability flower then yields a sequence of runs ρ1, ρ2, . . . in KM(V) such +that ρk escapes Sx,k for every vector x. Finally, pumpability allows us to lift these runs of +KM(V) to runs of V and thus conclude inseparability. +This equips us with two possible decision procedures: We can either check solvability of +each system of inequalities, or detect inseparability flowers in KM(V). +4 +Basic Separators +We prove Theorem 3.5, that any regular language R over Σn with R ∩ Dn = ∅ is contained +in a finite union of languages Pi,k and Sx,k. Note that a single value of k is sufficient, since +we have Pi,k ⊆ Pi,k+1 and Sx,k ⊆ Sx,k+1 for each i, x, k. The proof decomposes the Büchi +automaton for R in a way that allows us to forget about connectedness issues and reason +over cycles (and their letter balances) using techniques from linear algebra. We make use of +the following basic fact from linear programming [46, Corollary 7.1f]. +▶ Theorem 4.1 (Farkas’ Lemma (variant), [46]). Let A ∈ Qm×n be a matrix and let b ∈ Qm +be a vector. Then the system Ax ≤ b has a solution x ∈ Qn +≥0 if and only if y⊤b ≥ 0 for +each vector y ∈ Qm +≥0 with y⊤A ≥ 0. +Decomposing with profiles +We decompose R = L(A) into a (not necessarily disjoint) union +of several languages, each linked to a so-called profile. We will later see that for pumpable R, +every such profile language already has to be contained in a single Sx,k. +▶ Definition 4.2. Let A be a Büchi automaton. A profile of A is a set π of transitions of A +for which there exists a cycle σπ in A such that (a) σπ contains exactly the transitions in π, +and (b) σπ starts (and ends) in a final state qπ. +We denote by Π(A) the finite set of profiles of A. Moreover, we associate to every accepting +run ρ of A its profile Π(ρ), which contains exactly the transitions appearing infinitely often +in ρ. This definition is sound, as the infinitely occurring transitions of an accepting run must +form a cycle due to repetition, which visits a final state due to acceptance. +Given a profile π of A, we define Lπ(A) ⊆ L(A) to be the language of all words that +have an accepting run ρ of A with Π(ρ) = π. Note that this language is still regular: From +A one can construct a Büchi automaton that guesses a point after which only transitions + +10 +Regular Separability in Büchi VASS +from π can occur, and once this point is reached it keeps a list of already used transitions +from π in each state. Then only once all transitions of π have been used the state becomes +final and the list is set back to empty. +This now allows us to view R as the union of the languages Lπ(A) with π ∈ Π(A). We +show that each language Lπ(A) is either contained in Sx,k for some x, k, or there is a cycle +that, assuming the pumpability from the previous section, makes Lπ(A) intersect Dn. +▶ Lemma 4.3. Let A be a Büchi automaton over Σn and let π be one of its profiles. Then +one of the following conditions holds: +(i) There is a number k ∈ N and a vector x ∈ Nn such that Lπ(A) ⊆ Sx,k, or +(ii) there is a cycle σ′ in A over w′ with ϕ(w′) ≥ 0, and σ′ contains all transitions from π. +Assume Lπ(A) ̸= ∅, otherwise Condition (i) trivially holds. We build a system Aπx ≤ b of +linear inequalities as follows. It contains one inequality ⟨x, ϕ(v)⟩ ≤ 0 for each word v read by +a primitive cycle of transitions in π. By primitive cycle we mean a cycle that does not repeat +a state. Moreover, the system contains the inequality ⟨x, ϕ(vπ)⟩ ≤ −1 for the cycle σπ over +vπ that justifies the profile π. Let us quickly remark that the solution space of the system +Aπx ≤ b is independent of the precise choice of the justifying cycle σπ: To see this, we claim +that Aπx ≤ b holds if and only if all primitive cyles in π have an x-weighted balance at most +zero, and at least one primitive cycle in π has a strictly negative x-weighted balance. For +the “if” direction, note that a sufficiently long repetition of σπ will contain each primitive +cycle as a (possibly non-contiguous) subsequence. This means, the repetition, and thus σπ, +must have a strictly negative x-weighted balance. For the converse, we observe that σπ can +be decomposed into primitive cycles. Thus, if σπ has strictly negative x-weighted balance, +then so must at least one of its constituent primitive cycles. +Applying Farkas’ Lemma to Aπx ≤ b either yields a solution x ∈ Qn +≥0 or a vector +y ∈ Qm +≥0 with y⊤Aπ ≥ 0 and y⊤b < 0. In both cases we assume wlog. that the given vector +has entries in N, as we can always multiply with the lcm of the denominators. +Suppose we have a solution x. We claim that then Lπ(A) ⊆ Sx,k, where k = |Qπ| · h and +h is the maximal length of a transition label of A. This is because x weights primitive cycles +non-positively, and k is chosen such that for any infix v of a word in Lπ(A), if |v| > k, then +v’s associated transition sequence has to contain a primitive cycle. Thus, infixes at almost +all start positions of a word in Lπ(A) must have x-weighted balance ≤ k. +If we obtain a vector y = (y1, . . . , ym), then we can view it as a selection of rows in the +matrix Aπ, where the jth row is being selected yj many times. Since each row corresponds +to a cycle, this is also a selection of cycles. Then by y⊤b < 0 we selected σπ, where we +can insert the other selected cycles. By y⊤Aπ ≥ 0 this forms a cycle σ′ as required, with +non-negative letter balance for all letter pairs. A detailed proof can be found in Appendix C. +Here, we used a system of linear inequalities Aπx ≤ b, which was solely dependent on A +and π. We reasoned that if this system has a solution, then Condition (i) has to hold. This +is a fact that we want to refer to in a later proof, and therefore we formalize it here. +▶ Corollary 4.4. If A is a Büchi automaton with a profile π for which there is an x ∈ Nn +with Aπx ≤ b, then Lπ(A) ⊆ Sx,k for some k ∈ N. +With Theorem 3.7 and Lemma 4.3, we can now show Theorem 3.5. Suppose R = L(A) for +some Büchi automaton A. First, applying Theorem 3.7 with d = 0 yields a Büchi automaton +Apump such that L(A) ⊆ L(Apump) ∪ Pℓ for some ℓ ∈ N and L(Apump) ∩ Dn = ∅. Therefore, +it suffices to show that L(Apump) is included in a finite union of languages Sx,k. Suppose not. +Then the set L(Apump) decomposes into the sets Lπ(Apump) for π ∈ Π(Apump). By Lemma 4.3, +we know that for some π, Condition (ii) must hold: Otherwise, each Lπ(Apump) would be + +P. Baumann, R. Meyer, and G. Zetzsche +11 +included in some Sx,k. But if (ii) holds for π, then there is a cycle σ′ in Apump that contains +π (and thus visits a final state) and reads a word v with ϕ(v) ≥ 0. Now for some finite prefix +u, the word uvω belongs to L(Apump). Since ϕ(v) ≥ 0, there is some lower bound B ∈ Z such +that for each i ∈ [1, n] and every prefix p of uvω, we have ϕi(p) ≥ B. Finally, since L(Apump) is +pumpable, we can exchange a prefix in w = uvω to obtain another word w′ ∈ L(Apump) where +every prefix p has ϕ(p) ≥ 0. Hence w′ ∈ Dn and thus L(Apump) ∩ Dn ̸= ∅, a contradiction. +5 +Deciding Regular Separability +We now present the algorithm to decide, given a Büchi VASS V whether L(V) | Dn. We +first employ Theorem 3.7, because for pumpable languages we only have to deal with one +type of basic separators. The next step is to generalize the notion of profiles from Büchi +automata to Büchi VASS. Recall that for a sequence χ of transitions in V, δ(χ) denotes its +effect on the counters of V. If χ is a transition sequence in KM(V), then χ is labeled with +a transition sequence of V, so we define δ(χ) accordingly. Since we consider Büchi VASS +with input alphabet Σn, we write ϕ(χ) for the image of the input word under ϕ. Again, this +notation is used for transition sequences in KM(V). We also write ∆(χ) = (δ(χ), ϕ(χ)). +▶ Definition 5.1. Let V be a Büchi VASS. A profile for V is a set π of edges in KM(V) for +which there exists a cycle σ in KM(V) such that (i) σ contains exactly the edges in π, (ii) σ +starts (and ends) in a final state, and (iii) δ(σ) ≥ 0. +Clearly, every Büchi VASS has a finite set of profiles, which we denote by Π(V). Moreover, +Π(V) can be constructed effectively: Given a set of edges, a simple reduction to checking +unboundedness of a counter can be used to check if it is a profile. Furthermore, to every run +ρ of V, we can associate a profile: The run ρ must have a corresponding run in KM(V), which +has a finite set Π(ρ) of edges that are used infinitely often. Thus, ρ decomposes as ρ0ρ1 such +that ρ1 only contains edges from π. Then, ρ1 decomposes into σ1σ2 · · · such that each σi +uses every edge from Π(ρ) at least once and starts (and ends) in a final state. Since ≤ is a +well-quasi ordering on Nn, there are r < s such that δ(σr · · · σs) ≥ 0. Thus, σ = σr · · · σs is +our desired transition sequence showing that Π(ρ) is a profile. For each π ∈ Π(V), we denote +by Lπ(V) the set of all words accepted by runs ρ of V for which Π(ρ) = π. Then clearly: +▶ Lemma 5.2. L(V) = � +π∈Π(V) Lπ(V). +A system of inequalities for each profile +Our next step is to associate with each profile +π ∈ Π(V) a system of linear inequalities. We need some terminology. A π-cycle is a cycle σ +in KM(V) that only contains edges in π. If in addition, σ visits each state of KM(V) at most +once, except for the initial state, which is visited twice, then σ is a primitive π-cycle. Clearly, +a primitive π-cycle has length ≤ |π|. Moreover, from every π-cycle σ, one can successively cut +out primitive π-cycles until it is empty. Therefore, if τ1, . . . , τm are the primitive π-cycles of +KM(V), then there are numbers r1, . . . , rm ∈ N such that ∆(σ) = r1 ·∆(τ1)+· · ·+rm ·∆(τm). +We call σ a complete π-cycle if this holds for some r1, . . . , rm ≥ 1. Observe that if π is a profile, +then this is always witnessed by a complete π-cycle: Take any cycle σ witnessing that π is a +profile. Then σ|π| contains each primitive π-cycle as a subsequence. Hence, the cycle σm·|π| +is complete: We can carry out the cutting in each factor σ|π| so as to cut some τi at least +once. Moreover, σm·|π| still witnesses that π is a profile, since δ(σm·|π|) = m · |π| · δ(σ) ≥ 0. +Let us now construct the system of inequalities associated with π. Let σ be a complete +π-cycle witnessing that π is a profile and let τ1, . . . , τm be the primitive π-cycles. +Let +Aπ ∈ Z(m+1)×n be the matrix with rows ϕ(τ1), . . . , ϕ(τm), ϕ(σ), and let b ∈ Zm+1 be the + +12 +Regular Separability in Büchi VASS +column vector (0, . . . , 0, −1). Then clearly, Aπx ≤ b is equivalent to ⟨x, ϕ(σ)⟩ < 0 and +⟨x, ϕ(τ)⟩ ≤ 0 for each primitive π-cycle τ. +Inseparability flowers +An inseparability flower is a structure in the Karp-Miller graph +KM(V) as depicted to the right. It +consists of a final state q and three +cycles α, β, γ that all start in q and +that meet the given conditions. +q +α +β +γ +δ(αβγ) ≥ 0 +ϕ(αβ) ≥ 0 +ϕ(αβγ) ∈ Q · ϕ(α) +Let us give some intuition on why such a flower is the relevant structure to look for. True +to its name, an inseparability flower guarantees the existence of an inseparability witness, i.e. +a family of words accepted by the pumpable Büchi VASS V that escape every basic separator +Sx,k. Such a family of words therefore needs an accepting run for each member, and the three +conditions of the flower provide such runs: The first condition ensures that the three cycles +actually correspond to a transition sequence enabled in V. The second condition guarantees +that for every x ∈ Nn, the x-weighted letter balance of α or of β is positive; unless they are +both zero, in which case the third condition ensures that αβγ has x-weighted balance zero. +This allows us to construct, for each k, a run that escapes Sx,k for all x: By sufficiently +repeating each cycle α, β, and γ, we obtain a run that for each x ∈ Nn, will either (i) have +infixes with x-weighted balance > k, or (ii) attain some x-weighted balance infinitely often. +Each of these properties rules out membership in Sx,k. Proposition 5.5 proves this formally. +▶ Theorem 5.3. Let V be a Büchi VASS such that L(V) is pumpable. Then the following are +equivalent: +(i) L(V) ̸ | Dn. (ii) There is a profile π ∈ Π(V) such that the system Aπx ≤ b +has no solution x ∈ Nn. (iii) There exists an inseparability flower in KM(V). +The decision procedure +Before we prove Theorem 5.3, let us see how to use it to decide +separability. Given Büchi VASS V1 and V2, we can compute V so that L(V1) | L(V2) if and +only if L(V) | Dn, by Lemma 3.4. Then Theorem 3.7 tells us that L(Vpump) is pumpable +and we have L(V) | Dn if and only if L(Vpump) | Dn. Finally, by Theorem 5.3, we can check +whether L(Vpump) | Dn by checking the systems Aπx ≤ b for satisfiability: If there is a +solution for every π ∈ Π(Vpump), then we have separability; otherwise, we have inseparability. +Since the systems Aπx ≤ b are constructed directly from KM(Vpump), we need to explicitly +construct the latter. Therefore our procedure may take Ackermann time, because Karp-Miller +graphs can be Ackermann large [36, Theorem 2]. +▶ Example 5.4. Consider the instance of regular separability where our two inputs are the +Büchi VASS V found in Figure 1(left), and another Büchi VASS accepting the language D1. +Since we are already in the case of wanting to decide L(V) | D1, we can skip the first step of +applying Lemma 3.4. The second step is to apply Theorem 3.7 and construct Vpump, which +we have already done for this case in Figure 2(right). +Now we have to construct KM(Vpump), which can be found in Figure 3. There are two +relevant parts of KM(Vpump), where we can find cycles involving a final state: (1) the part on +the right, where the state tuples contain ω twice and the counter value is 0, and (2) the part +at the top with triple ωs. In the following we will only write down the states, as the counter +values and the other contents of the state tuples will be clear from context. +For part (1), the Büchi VASS Vpump has only a single profile π1 containing only the two +edges between q1 and q2. Since each π1-cycle σ only consists of repetitions of the primitive +cycle q1 +0|ε +−−→ q2 +0|¯a1 +−−−→ q1, we have ϕ(σ) < 0. Therefore the system Aπ1x ≤ b trivially has a +solution x = 1. + +P. Baumann, R. Meyer, and G. Zetzsche +13 +((q0, ω, 0), 1) +((q1, ω, 0), 1) +((q2, ω, 0), 1) +((q2, ω, ω), 0) +((q1, ω, ω), 0) +((q1, 0, 0), 0) +((q0, 0, 0), 0) +((q2, 0, 0), 0) +((q1, ω, 0), ω) +((q0, ω, 0), ω) +((q2, ω, 0), ω) +((q2, ω, ω), ω) +((q1, ω, ω), ω) +e1|ε +0|ε +0|ε +e1|ε +0|ε +0|ε +−e1|a1 +0|¯a1 +0|ε +e1|¯a1 +e1|ε +0|ε +0|ε +−e1|a1 +0|ε +0|¯a1 +−e1|a1 +e1|¯a1 +Figure 3 The Karp-Miller graph KM(Vpump) of the Büchi VASS Vpump from Figure 2(left). Here +we did not mark the final states to reduce visual clutter; every state that includes q1 is considered +final. For similar reasons, we also only labelled the edges of the graph with letters and counter +effects. The proper edge labels would be full transitions of Vpump, including source and target state. +Regarding part (2), Vpump has exactly two more profiles: profile π2 containing only the +two edges between q1 and q2, and profile π3, which additionally contains the two loop edges +on q2. The cycles of π2 look almost exactly like the cycles of π1 with only the counter values +of the nodes in the graph being different. Thus, the system Aπ2x ≤ b is the exact same +system as Aπ1x ≤ b and also trivially has a solution x = 1. +For π3, we have as primitive cycles both the loop edges on q2 as well as the primitive +cycle of π2. To obtain a complete π3-cycle, we simply insert both loops into the π2-cycle +at q2 forming the cycle σ = q1 +0|ε +−−→ q2 +−e1|a1 +−−−−→ q2 +e1|¯a1 +−−−→ q2 +0|¯a1 +−−−→ q1. Since σ contains all +primitive cycles exactly once without overlap, it is automatically complete. We also have +δ(σ) = 0, meaning σ is a cycle witnessing π3 as a profile. Thus these cycles lead to the +following system of inequalities Aπ3x ≤ b: +1 · x1 ≤ 0 +loop 1 +−1 · x1 ≤ 0 +loop 2 +−1 · x1 ≤ 0 +π2-cycle +−1 · x1 ≤ −1 +complete π3-cycle +Clearly this system has no solution; the first and last inequality are contradictory. Therefore +we conclude regular inseparability for L(V) and D1. +While not part of the decision procedure, for an inseparable instance of the problem +as we have here, we can also find an inseparability flower in KM(Vpump). +In this case +we have α = q1 +0|ε +−−→ q2 +0|¯a1 +−−−→ q1, β = q1 +0|ε +−−→ q2 +−e1|a1 +−−−−→ q2 +−e1|a1 +−−−−→ q2 +0|¯a1 +−−−→ q1, and +γ = q1 +0|ε +−−→ q2 +e1|¯a1 +−−−→ q2 +e1|¯a1 +−−−→ q2 +0|¯a1 +−−−→ q1. This selection of cycles meets all the requirements +of a flower: δ(αβγ) = 0, ϕ(αβ) = 0, and ϕ(αβγ) = −3 = 3 · ϕ(α). +Inseparability flowers disprove separability +The remainder of this section is devoted to +proving Theorem 5.3. The implication “(i)⇒(ii)” follows by applying Corollary 4.4 to KM(V), +viewed as a Büchi automaton; see Lemma D.1. For “(iii)⇒(i)”, we employ Lemma 3.6: + +14 +Regular Separability in Büchi VASS +▶ Proposition 5.5. If L(V) is pumpable and KM(V) has an insep. flower, then L(V) ̸ | Dn. +Proof. Suppose there is an inseparability flower α, β, γ in KM(V) and also L(V) | Dn. By +Lemma 3.6, there is a k ∈ N and a finite set X ⊆ Nn such that L(V) ⊆ � +x∈X Sx,k. We claim +that for every x ∈ Nn, at least one of the following holds: +⟨x, ϕ(α)⟩ > 0, +⟨x, ϕ(β)⟩ > 0, +or ⟨x, ϕ(αβγ)⟩ = 0. +(1) +Indeed, if ⟨x, ϕ(α)⟩ ≤ 0 and ⟨x, ϕ(β)⟩ ≤ 0, then ϕ(αβ) ≥ 0 implies that ⟨x, ϕ(α)⟩ = +⟨x, ϕ(β)⟩ = 0. Since ϕ(αβγ) = N · ϕ(α) for some N ∈ Q, we have ϕ(αβγ) = 0. This +proves the claim. Because of (1), the sequence αk+1βk+1γk+1 either has an infix χ with +⟨x, ϕ(χ)⟩ > k or we have ⟨x, ϕ(αk+1βk+1γk+1)⟩ = 0. Since δ(αk+1βk+1γk+1) ≥ 0, there is +a run ρ such that ραk+1βk+1γk+1 is a run in V. Hence, ρ(αk+1βk+1γk+1)ω is a run in V +whose word cannot belong to Sx,k for any x ∈ Nn, contradicting L(V) ⊆ � +x∈X Sx,k. +◀ +Constructing inseparability flowers +It remains to show the implication “(ii)⇒(iii)”. Suppose +there is a profile π ∈ Π(V) whose associated system of inequalities Aπx ≤ b is unsatisfiable. +By Farkas’ Lemma, there exists a y ∈ Nm+1 such that y⊤Aπ ≥ 0 and y⊤b < 0. From this +vector y, we now construct an inseparability flower in KM(V). +Let σ be the complete π-cycle in KM(V) that was chosen to construct Aπ. Let τ1, . . . , τm +be the primitive π-cycles. Since σ is complete, there is a vector r = (r1, . . . , rm) ∈ Nm so that +r1, . . . , rm ≥ 1 and ∆(σ) = r1·∆(τ1)+· · ·+rm·∆(τm). Moreover, since σ contains every edge +of π, we can wlog. write σ = σ0 · · · σm such that between σi−1 and σi, σ arrives in the initial +state of τi. The decomposition allows us to insert further repetitions of the primitive cycles. +For z = (z1, . . . , zm) ∈ Nm with z ≥ r, we define σz as σ0τ z1−r1 +1 +σ1 · · · τ zm−rm +m +σm. Then +∆(σz) = z1 ·∆(τ1)+· · ·+zm ·∆(τm). In particular, for s, t ≥ r, we have ∆(σsσt) = ∆(σs+t). +Recall that every transition in a Karp-Miller graph is labeled by a VASS transition, and +so every transition sequence χ in KM(V) is labeled by a transition sequence in V, which we +denote by trans(χ). We now define the transition sequences α, β, and γ as trans(σz) for +suitable vectors z. For α, we take trans(σ), the transitions labeling the complete π-cycle. +Observe that σ = σr. We proceed to define β = trans(σs) and γ = trans(σt). The choice of +the vectors s and t has to meet the requirements on an inseparability flower: ϕ(αβ) ≥ 0, +δ(αβγ) ≥ 0, and ϕ(αβγ) ∈ Q · ϕ(α). +Step I: Building β. +We will define s so that ϕ(αβ) = ϕ(σrσs) = ϕ(σr+s) ≥ 0. The +remaining two requirements (i.e. δ(αβγ) ≥ 0 and ϕ(αβγ) ∈ Q · ϕ(α)) will be ensured with +an appropriate choice of t in Step II. Let us now describe how to pick s. Recall that y is the +vector from the application of Farkas’ Lemma. It can be understood as assigning a repetition +count yi to every primitive cycle τi in the profile and a repetition count ym+1 to the complete +π-cycle σ. Since y⊤Aπ ≥ 0, and since our goal is to make ϕ(αβ) non-negative, we will use y +to construct a vector ˆy = (ˆy1, . . . , ˆym) ∈ Nm so that ϕ(σ ˆy) = y⊤Aπ. The right definition is +ˆyi := yi + ym+1 · ri for i ∈ [1, m], because +y⊤Aπ = +m +� +i=1 +yi · ϕ(τi) + ym+1 · ϕ(σ) = +m +� +i=1 +(yi + ym+1ri)ϕ(τi) = ϕ(σ ˆy). +We now choose M ∈ N such that s = M · ˆy − r ≥ r. This is possible since all entries in ˆy +are positive, due to ym+1 > 0 by y⊤b < 0, and ri > 0 for all i by definition. Then we have +ϕ(αβ) = ϕ(σrσs) = ϕ(σr+s) = ϕ(σM·ˆy) = M · ϕ(σ ˆy) ≥ 0. + +P. Baumann, R. Meyer, and G. Zetzsche +15 +Step II: Building γ. +It remains to define t so that γ = trans(σt) satisfies δ(αβγ) = +δ(σr+s+t) ≥ 0 and ϕ(αβγ) ∈ Q · ϕ(α). The idea is to choose t so that r + s + t is a positive +multiple of r. Such a choice is possible, because r has positive entries everywhere: We pick +N ∈ N such that t := N · r − s − r ≥ r. Then indeed δ(αβγ) = δ(σr+s+t) = δ(σN·r) = +N · δ(σr) = N · δ(σ) ≥ 0 and ϕ(αβγ) = ϕ(σr+s+t) = ϕ(σN·r) = N · ϕ(σr) = N · ϕ(α). +6 +One-dimensional Büchi VASS +Our second contribution is the precise complexity of separability for the 1-dimensional case. +▶ Theorem 6.1. Regular separability for 1-dimensional Büchi VASS with binary encoded +updates is PSPACE-complete. +For the lower bound, we use a simple reduction from the disjointness problem L1 ∩L2 +?= ∅ +for finite-word languages of 1-dim. VASS [24]. However, we also show that separability is +PSPACE-hard even if the input languages are promised to be disjoint. See Appendix E.1. +For the upper bound, we rely on the results in Section 5, but need a modification. There, +to simplify the exposition, we first make the input language pumpable, which may incur an +Ackermannian blowup. A closer look at the results, however, reveals that we can also check +separability directly on the Karp-Miller graph of ¯V as defined in Section 3. +▶ Proposition 6.2. Let V be a Büchi VASS with L(V) ⊆ Σω +n. Then L(V) ̸ | Dn if and only +if KM(¯V) has an inseparability flower. +Proposition 6.2 allows us to phrase inseparability as the existence of a run in ¯V that +satisfies certain constraints. Recall that if V is 1-dimensional and over Σ1, then ¯V has two +counters the second of which tracks the letter balance. +▶ Corollary 6.3. Let V be a 1-dimensional Büchi VASS with L(V) ⊆ Σω +1 and L(V) ∩ D1 = ∅. +Then L(V) ̸ | D1 if and only if there exist states p, q, r with r final, and a run in ¯V as follows: +(q0, 0, 0) +∗−→ +σ1 +� +�� +� +(p, x1, y1) +∗−→ (p, x2, y2) +∗−→ +σ2 +� +�� +� +(q, x3, y3) +∗−→ (q, x4, y4) +∗−→ +α +� +�� +� +(r, x5, y5) +∗−→ (r, x6, y6) +∗−→ +γ +� +�� +� +(r, x7, y7) +� +�� +� +β +∗−→ (r, x8, y8) +(1) y3 < y4 and also (a) x3 ≤ x4 +or (b) x1 < x2 and y1 ≤ y2 +(2) y5 ≤ y7 +(3) x5 ≤ x8 +(4) if y5 = y6, then y5 = y8. +Observe that an inseparability flower in KM(¯V) must carry ω in the second coordinate, +meaning the letter balance is unbounded. Otherwise, it would yield an accepting run of ¯V, +which cannot exist because L(V) ∩ D1 = ∅. If the flower has ω in the second coordinate, we +can construct a finite run as above. The cycles σ1 and σ2 plus Condition 1 ensure that indeed +the second coordinate becomes ω. Condition 2 is ϕ(αβ) ≥ 0. Condition 3 says δ(αβγ) ≥ 0. +Finally, to express ϕ(αβγ) ∈ Q · ϕ(α), note that for integers a ∈ Q · b iff b = 0 implies a = 0. +Condition 4 expresses that y6 − y5 = 0 implies y8 − y5 = 0. +In order to apply Corollary 6.3 for deciding L(V1) | L(V2) for 1-dim. Büchi VASS V1, V2 +with binary counter updates, we would like to follow the approach for the general case and +use Lemma 3.4 to first construct V so that L(V1) | L(V2) if and only if L(V) | D1. From V, +we would then construct the 2-dimensional Büchi VASS ¯V that tracks the letter balance, and +on ¯V we would then check the conditions of Corollary 6.3. The problem is that, under binary +updates, the intermediary V may become exponentially large. We use the fact that also ¯V +has binary counters available. This allows us to directly construct a compact variant of ¯V: + +16 +Regular Separability in Büchi VASS +▶ Lemma 6.4. Given 1-dim. Büchi VASS V1, V2 with binary updates, there is a a 1-dim. +Büchi VASS V with L(V1) ∩ L(V2) = ∅ iff L(V) ∩ D1 = ∅, L(V1) | L(V2) iff L(V) | D1, and +we can construct in time polynomial in |V1| + |V2| the 2-dim. Büchi VASS ¯V (binary updates). +Detecting constrained runs in 2-VASS +It remains to check for the existence of runs in +¯V as described in Corollary 6.3, and to check whether L(V1) ∩ L(V2) = ∅. Both of these +problems reduce to what we call the constrained runs problem for 2-VASS. Recall that +Presburger arithmetic is the first-order theory of (N, +, <, 0, 1). We will use the existential +fragment to express conditions on counter values of VASS like the ones from Corollary 6.3. +The constrained runs problem is the following: +Given A 2-dim. VASS V (with updates encoded in binary), a number m ∈ N, states q1, . . . , qm +in V, a quantifier-free Presburger formula ψ(x1, y1, . . . , xm, ym), and s, t ∈ [1, m], s ≤ t. +Question Does there exist a run (q0, 0, 0) +∗−→ (q1, x1, y1) +∗−→ · · · +∗−→ (qm, xm, ym) that visits a +final state between (qs, xs, ys) and (qt, xt, yt) and satisfies ψ(x1, y1, . . . , xm, ym)? +Lemma 6.4 and Corollary 6.3 imply that if L(V1) ∩ L(V2) = ∅, then L(V1) | L(V2) reduces +to the constrained runs problem on ¯V. Moreover, checking L(V1) ∩ L(V2) = ∅ reduces via a +product construction to checking emptiness of a 2-VASS. Such a 2-VASS has an accepting +run iff (q0, 0, 0) +∗−→ (q, x, y) +∗−→ (q, x′, y′) with (x, y) ≤ (x′, y′) and q final. Hence, this problem +also reduces to the constrained runs problem for 2-VASS. We thus need to show: +▶ Proposition 6.5. The constrained runs problem for 2-VASS is solvable in PSPACE. +For Proposition 6.5, we show that if there is a constrained run, then there is one with at +most exponential counter values along the way. For this, we use methods from [5]. +Complexity in higher dimension +We leave open two natural questions: (i) What is the +complexity of regular separability for Büchi d-VASS, for each d ≥ 2? (ii) What is the +complexity of regular separability for Büchi VASS (where the dimension is part of the input)? +Given that the regular separability and the disjointness problem usually (but not al- +ways [32, 48]) coincide regarding decidability, we expect the complexity of regular separability +to be PSPACE in every fixed dimension d and EXPSPACE in general. The lower bounds +follow from Theorem 6.1 for fixed d and from [14] (because disjointness is EXPSPACE- +complete [22, 34]). 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And +moreover, if ϕ1(v) > 0, then this decreases the counter. Furthermore, in order to visit q1, +the balance has to drop once. Therefore, between any two (not necessarily successive) visits +to the final state q1, one of the following holds: (i) the counter strictly decreases or (ii) the +input word v satisfies ϕ1(v) < 0. Since q1 is visited infinitely often, we can decompose +w = uv1v2 . . . such that after reading vℓ, we are in (q1, xℓ) and we have x1 ≤ x2 ≤ · · · . Then +“(i)” cannot happen for any vℓ and thus we have ϕ1(vℓ) < 0 for every ℓ. Hence, the claim is +proven. +It remains to show L(V) ̸ | D1. +Towards a contradiction, suppose there is a Büchi +automaton A with n states such that L(V) ⊆ L(A) and L(A) ∩ D1 = ∅. Note that V accepts +(an +1 ¯an+1 +1 +)ω: We drive up the counter to n in q0 and then read each an +1 ¯an+1 +1 +in a loop from q1 +to q1. However, a run of A must cycle on some non-empty infix of an +1 and thus, for some +m > n, also accept w = (am +1 ¯an+1 +1 +)ω. Since w ∈ D1, that is a contradiction. +◀ +The second part of Theorem 3.2 is proven in Appendix F. + +P. Baumann, R. Meyer, and G. Zetzsche +21 +A.2 +Proof of Lemma 3.4 +▶ Lemma 3.4. Given Büchi VASS V1 and V2, we can compute a Büchi VASS V over Σn so +that L(V1) | L(V2) if and only if L(V) | Dn, where n is the dimension of V2. +For the proof, we need the concept of rational transductions of infinite words. +Rational transductions +A finite state Büchi transducer is a tuple T = (Q, Σ, Γ, E, q0, Qf) +consists of a finite set of states Q, an input alphabet A, an initial state q0 ∈ Q, a set of +final states Qf ⊆ Q, and a transition relation E ⊆ Q × (Σ ∪ {ε}) × (Γ ∪ {ε}) × Q. For a +transition (q, a, b, q′) ∈ E, we also write q +(a,b) +−−−→ q′. The transducer T recognizes the binary +relation T(T ) ⊆ Σω × Γω containing precisely those pairs (u, v) ∈ Σω × Γω, for which there +is a transition sequence +q0 +(a1,b1) +−−−−→ q1 +(a2,b2) +−−−−→ . . . +such that u = a1a2 · · · , v = b1b2 · · · , and for infinitely many i ∈ N, we have qi ∈ Qf. We say +that a relation T ⊆ Σω × Γω is rational if there is a finite-state Büchi transducer T with +T = T(T ). For a language L ⊆ Γω and a relation T ⊆ Σω × Γω, we define +TL = {u ∈ Σω | ∃v ∈ L: (u, v) ∈ T}. +Moreover, for relations T ⊆ Σω × Γω and S ⊆ Θω × Σω, we define +S ◦ T = {(u, w) ∈ Θω × Γω | ∃v ∈ Σω : (u, v) ∈ S, (v, w) ∈ T}. +Using a simple product construction, we observe that for rational transductions S and T, the +relation S ◦T is (effectively) rational as well. By simply exchanging the two input coordinates, +one can also show that if T ⊆ Σω × Γω is rational, then so is +T −1 = {(u, v) ∈ Γω × Σω | (v, u) ∈ T}. +The following is also entirely straightforward. +▶ Lemma A.2. A language L ⊆ Σω is a Büchi VASS language if and only if there exists a +rational transduction T and a number n ∈ N such that L = TDn. Moreover, the translation +can be performed in exponential time. +Here, the automaton underlying an n-dim. Büchi VASS is translated into a transducer with +input in Dn and vice-versa. More precisely, for h ∈ N an operation of +h on the ith counter +is translated into the string (ai)h, whereas −h is translated into (¯ai)h. The 0-vector is +hereby translated into a1¯a1 instead of ε, to ensure that every infinite run of the Büchi VASS +actually corresponds to an infinite word in Dn. The only reason why this construction is +not feasible in polynomial time, is because we assume that counter operations of Büchi +VASS are encoded in binary. In particular, the string (ai)h mentioned above takes h steps +to write down, whereas the size of the Büchi VASS is only dependent on log h. However, +the construction only takes polynomial time, if counter updates are encoded in unary, or if +strings such as (ai)h are subjected to some exponential compression. +We also need the following lemma. The proof is exactly the same as the corresponding +proof in [17]. The only difference is that we have infinite instead of finite words. +▶ Lemma A.3. Let T ⊆ Σω × Γω be rational and L ⊆ Σω and K ⊆ Γω. Then L | TK if and +only if T −1L | K. + +22 +Regular Separability in Büchi VASS +Proof. Suppose L ⊆ R and R ∩ TK = ∅ for some regular R. Then clearly T −1L ⊆ T −1R +and T −1R ∩ K = ∅. Therefore, the regular set T −1R witnesses T −1L | K. Conversely, +if T −1L | K, then K | T −1L and hence, by the first direction, (T −1)−1K | L. +Since +(T −1)−1 = T, this reads TK | L and thus L | TK. +◀ +We are now ready to prove Lemma 3.4. +Proof of Lemma 3.4. Given Büchi VASS V1 and V2, where V2 is n-dimensional, Lemma A.2 +allows us to compute in exponential time a rational transduction T such that L(V2) = TDn. +We apply Lemma A.2 again to construct a Büchi VASS V for T −1L(V1). Then we have +L(V1) | L(V2) ⇐⇒ L(V1) | TDn ⇐⇒ T −1L(V1) | Dn ⇐⇒ L(V) | Dn, +where the second equivalence is due to Lemma A.3. +◀ +B +Proof Details for Pumpability +Let us formally define the constructions of ¯V and Vpump. +▶ Definition B.1. Let V = (Q, q0, T, F) be a d-dimensional Büchi VASS over Σn. Then +¯V = (Q, q0, ¯T, F) is the (d+n)-dimensional Büchi VASS over Σn with transitions constructed +as follows: (q, ε, (δ, ϕ(w)), q′) ∈ ¯T if and only if (q, w, δ, q′) ∈ T. +Furthermore, Vpump = (Qpump, qpump,0, Tpump, Fpump) is the d-dimensional Büchi VASS +over Σn constructed as follows: +Qpump = Q × (N ∪ {ω})d+n, i.e. the states of KM(¯V), +qpump,0 = (q0, 0), +Fpump = F × (N ∪ {ω})d+n, and +(qpump, w, δ, q′ +pump) ∈ Tpump if and only if there is a transition (qpump, t, q′ +pump) in KM(¯V) +labelled by t = (q, ε, (δ, ϕ(w)), q′) ∈ ¯T. +We split the first two parts of Theorem 3.7 into Lemmas B.2 and B.3, which we prove +separately. The third part later follows from Lemma 3.6, which, in turn, follows from +Theorem 3.5. +▶ Lemma B.2. L(Vpump) is pumpable. +Proof. Consider some w ∈ L(Vpump) and some k ∈ N. Let ρ be an accepting run of V over w. +By construction of Lpump, there exists a corresponding run ¯ρ in KM(¯V). Let Ω ⊆ [d+1, d+n] +be the set of coordinates where the states of KM(¯V) carry ω eventually during ¯ρ. Then at +some point, ¯ρ visits an extended configuration (q, ¯m) ∈ Q × Nd+n +ω +where all coordinates +from Ω in ¯m are ω. Decompose ¯ρ = ¯ρ0¯ρ1 so that ¯ρ0 reaches (q, ¯m). Let ρ = ρ0ρ1 and +w = w0w1 be the corresponding decompositions of ρ and w. Then ρ0 reaches a configuration +(q, m) ∈ Q × Nd in V . +Let ℓ = maxi∈[1,n]{0, −ϕi(w0)} + k. By the construction of Karp-Miller graphs, there +exists a run ¯ρ′ +0 in ¯V that reaches a configuration (q, ¯m′) ∈ Nd+n such that ¯m′(i) ≥ m(i) for +i ∈ [1, d + n] \ Ω, and ¯m′(i) > m(i) + ℓ for i ∈ Ω. Then ¯ρ′ +0 corresponds to a run ρ′ +0 in V. It +reaches a configuration (q, m′) with m′ ≥ m and thus ρ′ +0ρ1 is a run of V. It reads a word +w′ +0w1 ∈ Lpump, where w′ +0 is the prefix read by ρ′ +0. Since w′ +0 was also read by ¯ρ′ +0 in ¯V, it is a +prefix of some word in Dn, as mandated by the additional counters of ¯V. +We claim that now ϕ(w′ +0) ≥ ϕ(w0) and for every i ∈ [1, n] where w ever becomes +negative, we have ϕi(w′ +0) ≥ max{ϕi(w0), 0} + k. The first condition follows from the fact + +P. Baumann, R. Meyer, and G. Zetzsche +23 +that ϕi(w′ +0) = m′(d + i) ≥ m(d + i) = ϕi(w0). For the second condition, note that if w +ever becomes negative in coordinate i, then ¯ρ must necessarily visit a configuration where in +coordinate i, there is an ω. In particular, we have d + i ∈ Ω and thus ϕi(w′ +0) = m′(d + i) ≥ +m(d + i) + ℓ = ϕi(w0) + maxi∈[1,n]{0, −ϕi(w0)} + k ≥ max{ϕi(w0), 0} + k. +◀ +▶ Lemma B.3. There exists a k ∈ N such that L(Vpump) ⊆ L(V) ⊆ L(Vpump) ∪ Pk. +Proof. The inclusion L(Vpump) ⊆ L(V) is obvious from the construction. For the second +inclusion, define k ∈ N to be the largest number occurring in the states of KM(¯V). We +claim that then L(V) ⊆ Pk ∪ L(Vpump). Let w ∈ L be accepted by a run ρ in V and suppose +w /∈ Pi,k for some i ∈ [1, n]. If u is a prefix of w, then we say that i ∈ [1, n] is crossing at u if +ϕi(u) < 0 and ϕi(v) ≥ 0 for every prefix v of u. Observe that whenever i is crossing at u, +then ϕi(v) > k for some prefix v of u: Otherwise, w would belong to Pi,k. This implies that +ρ has a corresponding run in KM(¯V): Whenever a counter in [d + 1, d + n] drops below zero, +it must have been higher than k before and thus been set to ω. Therefore, w is also accepted +by KM(¯V) and thus w ∈ L(Vpump). +◀ +C +Proof Details for Basic Separators +Proof of Lemma 4.3. First of all, if Lπ(A) is empty, then Condition (i) trivially holds. +Thus, in the following we assume that Lπ(A) ̸= ∅ and in particular that the final state qπ +associated with the profile π is reachable from A’s initial state. +We want to set up a system of linear inequalities that has a solution x if and only if +there is a k such that Lπ(A) ⊆ Sx,k. Therefore let us talk about some requirements that are +necessary for the above inclusion to hold. These requirements will be on cycles of transitions +in π, and we make sure that they can be expressed as linear inequalities. +The cycle σπ that contains exactly the transitions in π has to be over a word vπ ∈ Σ∗ +n with +⟨x, ϕ(vπ)⟩ ≤ −1. Otherwise, we can just repeat σπ infinitely often and prepend any prefix +leading to qπ from A’s initial state, yielding a word that violates requirement b.) of Sx,k. +Any primitive cycle σ in A of transitions in π has to be over a word v with ⟨x, ϕ(v)⟩ ≤ 0. +Otherwise we repeat σπ infinitely often from some arbitrary prefix reaching qπ like before, +and then perform k + 1 insertions of the cycle σ into each copy of σπ. This yields a word +that violates requirement a.) of Sx,k, and we can do this for any k. +We can now use these requirements on cycles to construct a linear system of inequalities +Aπx ≤ b for x. For the cycle σπ corresponding to word vπ ∈ Σ∗ +n, we add the inequality +x1ϕ1(vπ) + x2ϕ2(vπ) + . . . + xnϕn(vπ) ≤ −1, +and for each primitive cycle σ of transitions in π over a word v ∈ Σ∗ +n, we add the inequality +x1ϕ1(v) + x2ϕ2(v) + . . . + xnϕn(v) ≤ 0. +Let us argue that the precise choice of the justifying cycle σπ does not affect the +satisfiability of the system Aπx ≤ b. To this end we argue that x ∈ Nn is a valid solution to +the system if and only if (1) all primitive cycles have an x-weighted balance at most zero, +and (2) at least one primitive cycle has a strictly negative x-weighted balance. Constraint +(1) is clearly equivalent to the inequalities added for each primitive cycle. +For constraint (2), assume that the inequality ⟨x, ϕ(vπ)⟩ ≤ −1 holds. Now observe that +any valid choice of σπ is a cycle and therefore can be constructed by inserting primitive cycles +into each other a finite number of times. If all primitive cycles had non-negative x-weighted +balance, then the x-weighted balance for σπ could not be negative. + +24 +Regular Separability in Büchi VASS +For the other implication direction, assume that constraint (2) holds, and let the primitive +cycle with negative x-weighted balance be σ′. Since any valid choice of σπ contains each +transition in π, its |π|-fold repetition σ| +ππ contains each primitive cycle as a (possibly +non-contiguous) subsequence. Now, if we delete σ′ from σ| +ππ, the remaining (possibly not +connected) transition sequences still combine to form a collection of cycles, since σ′ is a cycle. +Thus, the summed-up x-weighted balance of this collection is the sum of x-weighted balances +of primitive cycles, and can therefore be at most zero by Condition (1). Then adding σ′ +back in gives us that ⟨x, ϕ(v| +ππ|)⟩ is negative, and therefore ⟨x, ϕ(vπ)⟩ is as well. Since the +letter balance can only have integer-values, and weighting by x ∈ N does not change this, it +follows that ⟨x, ϕ(vπ)⟩ ≤ −1. +The characterization of solutions x via constraints (1) and (2) is clearly independent +of σπ, meaning the precise choice of the latter does not affect satisfiability of the system +Aπx ≤ b. Furthermore, the restriction of x ∈ Nd is not a meaningful one, as we can always +compute a solution in Qd from one in Nd, as we explain below. +Applying Farkas’ Lemma (Theorem 4.1) to this system of equations, we either obtain a +vector x ∈ Qn +≥0 as a suitable solution, or we obtain a vector y ∈ Qm +≥0 with y⊤Aπ ≥ 0⊤ and +y⊤b < 0, where m is the number of rows of Aπ. +In the first case, we can multiply the entries of x by their denominators’ least common +multiple, say ℓ, to yield a suitable vector ℓ · x = x′ ∈ Nn. Furthermore we set k = |Qπ| · h, +where |Qπ| ⊇ {qπ} is the set of all states of A adjacent to transitions in π, and h is the +length of the longest word appearing as a transition label of A. With this we can show that +Lπ(A) ⊆ Sx′,k: Each word w ∈ Lπ(A) decomposes into uv with v = v0v1v2 · · · such that +u leads to qπ from A’s initial state and each vj corresponds to some cycle σj on qπ, that +contains each transition of π at least once. Then we have ⟨x′, ϕ(vj)⟩ = ℓ·⟨x, ϕ(vj)⟩ < ℓ·0 = 0 +as required by Sx,k: each cycle σj can be obtained by starting with σπ, which contributes at +most −1 to this value, and inserting finitely many primitive cycles, which all add at most 0. +Moreover, we need to show ⟨x′, ϕ(f)⟩ ≤ k for every infix f of v. Towards a contradiction +assume there is at least one infix f of v, for which this does not hold. +Since f fulfils +⟨x′, ϕ(f)⟩ > |Qπ| · h, and h is the maximum length of a transition label, the transition +sequence corresponding to f has to be longer than |Qπ|. Thus this sequence repeats a state +and therefore has to contain a primitive cycle. However, all such primitive cycles add at +most 0 to the value ⟨x′, ϕ(f)⟩, meaning one could delete the word corresponding to this cycle +from f and still fulfil the aforementioned requirement. One can repeatedly remove primitive +cycles until one obtains a word f ′ of length |f ′| ≤ |Qπ| · h with ⟨x′, ϕ(f ′)⟩ > |Qπ| · h. This is +a contradiction, therefore infixes such as f cannot exist. +In the other case we also multiply y with the least common multiple of its entries, say ℓ, +to yield ℓ · y = y′ ∈ Nm. Furthermore, each row of the matrix Aπ essentially contains the +ϕ-values of its corresponding cycle. The requirement y′⊤Aπ = ℓy⊤Aπ ≥ ℓ · 0⊤ = 0⊤ can +then be seen as a selection of cycles, whose combined ϕ-values are all 0 or above. Moreover, +the requirement y′⊤b = ℓy⊤b < ℓ · 0 = 0 ensures that σπ is selected at least once, because +all other entries of b are 0, meaning we would have y⊤b = 0 if σπ was not selected. This +means we can combine all the selected cycles into one large cycle σ′ via matching states, +which is possible because σπ visits all states in Qπ. Since the combined ϕ-values of all the +cycles selected by y are 0 or above, we have that σ′ corresponds to a word w′ with ϕ(w′) ≥ 0. +Finally, σ′ also contains all transitions of π as required, because it contains the cycle σπ. +◀ +Regarding Theorem 3.5 +We mentioned in 4 that a single value of k is sufficient for a +finite union of basic separators Pi,k and Sx,k. This is because we have Pi,k ⊆ Pi,k+1 and + +P. Baumann, R. Meyer, and G. Zetzsche +25 +Sx,k ⊆ Sx,k+1 for each i ∈ [1, n], x ∈ Nn, k ∈ N. Therefore it suffices to show the following: +Let A be a Büchi automaton with L(A) = R ⊆ Σω +n and R ∩ Dn = ∅. Then there is a +finite set X ⊆ Nn and a number k ∈ N such that R ⊆ � +i∈[1,n] Pi,k ∪ � +x∈X Sx,k. +Here R is a separator candidate in the sense of the original phrasing of the theorem, +because it is ω-regular and disjoint from Dn. +Proof of Theorem 3.5. We begin by invoking Theorem 3.7 on A to obtain a Büchi auto- +maton Apump, whose language is pumpable, and a number ℓ such that L(Apump) ⊆ L(A) ⊆ +L(Apump) ∪ Pℓ. +Using the theorem this way is feasible, because Büchi automata can +be seen as 0-dimensional Büchi VASS. Since L(A) ∩ Dn = ∅ and L(Apump) ⊆ L(A) we +have L(Apump) ∩ Dn = ∅. +It now suffices to show that the basic separators theorem +holds for L(Apump): If there are X, k such that L(Apump) ⊆ � +i∈[1,n] Pi,k ∪ � +x∈X Sx,k then +L(A) ⊆ L(Apump) ∪ Pℓ ⊆ � +i∈[1,n] Pi,o ∪ � +x∈X Sx,o, where o = max(k, ℓ). +Now consider the decomposition L(Apump) = � +π∈Π(Apump) Lπ(Apump). If we can show that +each language Lπ(Apump) is contained in a finite union of basic separators, then we are done. +In the following let us fix a profile π of Apump. +We now invoke Lemma 4.3 on Apump and π. If Condition (i) holds, then this already yields +x, k such that Lπ(Apump) ⊆ Sx,k, and we need not concern ourselves with the languages Pi,k. +In the other case, Condition (ii) yields a cycle c′ in Apump that contains all transitions +in π and is over a word w′ with ϕ(w′) ≥ 0. Since Condition (i) did not hold, we know +that Lπ(Apump) is not empty, which means that all states adjacent to transitions of π are +reachable from A’s initial state, including the final state qπ associated with π. Let u′ be a +word that reaches qπ from Apump’s initial state. Then ˜w = u′(w′)ω ∈ L(Apump). +Let m be the lowest value of ϕi for any index i and prefix of ˜w, formally m = +mini∈[1,n],v∈prefix( ˜ +w) ϕi(v). Since ϕ(w′) ≥ 0 we know that m ∈ Z is well-defined. Moreover, +since L(Apump) is pumpable, there is a decomposition ˜w = u0w1 and a word v0 ∈ Σ∗ +n such +that v0w1 ∈ L(Apump), ϕ(v0) ≥ ϕ(u0), and ϕi(v0) ≥ ϕi(u0) + |m| for all indices i where there +is a v ∈ prefix( ˜w) with ϕi ≥ 0. Then swapping u0 for v0 in ˜w can only increase the ϕi-values +of its prefixes, and in fact all such values that fell below 0 are now raised above 0 by choice +of |m|. This means that v0w1 ∈ L(Apump) ∩ Dn, which is a contradiction. +◀ +D +Proof Details for Decidability +▶ Lemma 3.6. If L ⊆ Σω +n is pumpable, then L | Dn if and only if L |limDn, where L |limDn +means L ⊆ � +x∈X Sx,k for some finite set X ⊆ Nn and some k ∈ N. +Proof. The “if” direction is trivial. Conversely, let L | Dn. By Theorem 3.5, we have L ⊆ +� +i∈[1,n] Pi,k ∪ � +x∈X Sx,k for some finite X ⊆ Nn and k ∈ N. We claim that L ⊆ � +x∈X Sx,k, +which yields L |limDn. Indeed, given u ∈ L, pumpability yields a word u′ ∈ L such that +u′ ∼ u and u′ /∈ Pi,k for any i ∈ [1, n]. Since u′ ∈ L ⊆ Pk ∪� +x∈X Sx,k, we conclude u′ ∈ Sx,k. +Finally, observe that membership in Sx,k is not affected by changing a finite prefix of a word. +Therefore, we also have u ∈ Sx,k. +◀ +▶ Lemma D.1. Let π ∈ Π(V). If Aπx ≤ b for x ∈ Nn, then Lπ(V) ⊆ Sx,k for some k ∈ N. +Proof. We regard KM(V) as a Büchi automaton. Then, π is in particular a profile for KM(V). +Moreover, the cycle witnessing that π is a profile is also an admissible cycle for π in KM(V) +as a Büchi automaton. Thus, Corollary 4.4 implies Lπ(V) ⊆ Lπ(KM(V)) ⊆ Sx,k for some +k ∈ N. +◀ + +26 +Regular Separability in Büchi VASS +E +Proof Details for One-dimensional Büchi VASS +E.1 +Theorem 6.1: PSPACE-hardness +We begin with the straightforward reduction from intersection emptiness of finite-word +languages of 1-dim. VASS. Suppose L1, L2 ⊆ Σ∗ are finite-word languages of 1-dim. VASS with +succinct counter updates, and acceptance by final state. Checking whether the intersection +L1 ∩ L2 is empty is PSPACE-complete [24]. We construct Büchi VASS for L1#ω and L2#ω, +where # is a fresh letter. Since L1 and L2 are coverability languages of finitely-branching +WSTS, we know from [14, Theorem 7] that L1 ∩ L2 = ∅ if and only if L1 | L2. Furthermore, +with a fresh letter #, it is easy to observe that L1 | L2 if and only if L1#ω | L2#ω. +Hardness for disjoint languages +In the PSPACE-hardness proof above, one can notice that +the languages L1#ω and L2#ω are regularly separable if and only if they are disjoint. In +order to further highlight the disparity between the finite-word case of WSTS languages +(where disjointness and separability coincide [14]) and the infinite-word case, we want to +present a proof that PSPACE-hardness already holds if the input languages are promised to +be disjoint: Note that with this promise, separability in the finite-word case becomes trivial. +Here, we reduce directly from configuration reachability in bounded one-counter automata, +which was shown to be PSPACE-hard in [23]. +A bounded one-counter automaton B = (VB, b) consists of a 1-dim. VASS VB equipped +with a bound b ∈ N on its counter values. This means transitions of B are enabled if and +only if they meet the firing restrictions of a VASS and also lead to a configuration (q, m) +with m ≤ b. Here, counter values and the bound b are encoded in binary. In particular, the +size of B is that of the underlying VASS plus log b, and the size of a configuration (q, m) is +log m. +Now we want to construct two 1-dim. Büchi VASS V1 and V2, whose languages are always +disjoint, but are ω-regular separable if and only if (q, m) is not reachable from (q0, 0) in +B = (VB, b). Let T be the set of transitions of B. We use Σ = T ∪ {#} ∪ Σ1 as the alphabet +for V1 and V2. Let VD1 be the 1-dim. Büchi VASS accepting D1, i.e. VD1 consists of a single +state, both initial and final, with two loops e1|a1 and −e1|¯a1. Furthermore let VS be the +1-dim. Büchi VASS from Figure 1(left) accepting the language S with S ∩D1 = ∅ but S ̸ | D1, +which we talked about in Section 3 (see the proof of the first statement in Theorem 3.2). +We start constructing V1 by using a copy of VB with all states being non-final and every +transition t ∈ T labelled with t itself. Then we add a copy of VD1 with its only state still +being final. To connect the two copies, we add the transition −m|# from state q of VB to +the initial state of VD1. +For V2 we also start with a copy of VB with all non-final states and transitions labeled +with themselves, but we also invert every transition effect, changing it from z ∈ Z to −z. +Then we add a new initial state q′ +0 with the same outgoing transitions as the initial state +of VB, except we change their original effects z to b − z. These new transitions of q′ +0 are +labelled with their original copies from T. Additionally, we add a copy of VS with q2 still +being a final state. The two copies are then connected with a transition m − b|# from q to +the initial state of VS. If (q, m) = (q0, 0), we also add a transition 0|# from q′ +0 to the initial +state of VS. +Now let R1 be the set of all transition sequences over T that cover (q, m) in VB and do +not necessarily respect the bound b. Formally, ρ ∈ R1 if ρ leads from (q0, 0) to (q, m′) in VB +for some m′ ∈ N with m′ ≥ m. Moreover let R2 be the set of all transition sequences over +T that reach q with a counter value below m, when respecting the upper bound b, but not + +P. Baumann, R. Meyer, and G. Zetzsche +27 +necessarily the lower bound 0 of VASS counters. Formally, ρ ∈ R2 if ρ leads from (q0, 0) to +(q, z′) in B′ for some z′ ∈ Z with z′ ≤ m, where B′ = (V′, b) and V′ is just V interpreted as +a Z-VASS. We now want to argue that there are languages L1, L2 such that L(V1) = R1#L1 +and L(V2) = R2#L2. +L(V1) = R1#L1 is easy to see, since V1 simulates VB faithfully, and can only read # if a +configuration (q, m) or greater is reached. For L(V2) = R2#L2 observe that before reading +#, V2 essentially simulates VB with inverted counter values, starting with b instead of 0. +Since V can go above b, this essentially simulates going below 0 in B. The # can also only +be read if in q the counter is valued at least m − b, which corresponds to at most m before +inversion. Let us now show that L(V1) ∩ L(V2) = ∅, and furthermore L(V1) | L(V2) if and +only if (q0, 0) → (q, m) in B. +The configuration (q, m) not being reachable in B is equivalent to R1 and R2 being +disjoint. In this case L(V1) and L(V2) also have to be disjoint, since the prefixes before the +’#’ of their words cannot coincide. They are however ω-regular separable: With Q being the +states of B, an exponential size Büchi automaton A with states Q × {0, . . . , b} can simulate +B. To make A accept all words with prefixes in R1, we add a final state with loops on all +input letters, that is reachable by every transition that would make the counter value go +above b. Now L(V1) ⊆ L(A) is clear. Since transition sequences that do not respect the +bound b cannot be prefixes of elements of R2, L(A) ∩ L(V2) = ∅ immediately follows. Thus, +L(A) is an ω regular separator for L(V1) and L(V2), which also means that they are disjoint. +If (q, m) is reachable in B, we have a finite transition sequence ρ ∈ R1 ∩ R2. Reading +ρ then leads to (q, m) in V1, respectively to (q, b − m) in V2. Therefore if # is read right +after, the counter value of either Büchi VASS would be 0. This implies that ρ#D1 ⊆ L(V1) +and ρ#S ⊆ L(V1), as the second component of either VASS would be simulated faithfully +after this prefix. A regular separator A for L(N1) and L(N2) would therefore have to accept +all words in ρ#D1 but no words in ρ#S. By adding a new initial state qinit to A and +adding all outgoing transitions of states reachable via ρ# in the original A to qinit, we obtain +an ω-regular separator for D1 and S. This is a contradiction, since we established earlier +that these languages are not ω-regular separable as shown in the proof of the first half of +Theorem 3.2. +It remains to show that L(V1) ∩ L(V2) = ∅ in the case where (q, m) is reachable in B. +For transition sequences ρ over T, we know that ρ ∈ R1 ∩ R2 if and only if ρ reaches (q, m) +in B. Therefore two words w1 ∈ L(V1) and w2 ∈ L(V2) can only agree on a prefix ρ#, if ρ +has this property. However, in this case w1 = ρ#w′ +1 for some w′ +1 ∈ D1 and w2 = ρ#w′ +2 for +some w′ +2 ∈ S. This yields w1 ̸= w2 since D1 and S are disjoint. +E.2 +Proof of Proposition 6.2 +▶ Proposition 6.2. Let V be a Büchi VASS with L(V) ⊆ Σω +n. Then L(V) ̸ | Dn if and only +if KM(¯V) has an inseparability flower. +Proof. We first invoke Theorem 3.7 to obtain Vpump with L(Vpump) ̸ | Dn if and only if +L(V) ̸ | Dn. Recall that Vpump was constructed as the product of V and KM(¯V), which means +that every cycle of Vpump is also a cycle of KM(¯V). +For the if direction, we get that KM(Vpump) contains an inseparability flower by The- +orem 5.3. Its three cycles then correspond to three cycles of Vpump, which then also appear +in KM(¯V), where they still fulfill the requirements of an inseparability flower. +For the only if direction, observe that KM(Vpump) is essentially the product construction of +KM(V) and KM(¯V). Furthermore, any transition sequence permitted by ¯V is also permitted + +28 +Regular Separability in Büchi VASS +by V, as the former only added restrictions in the form of more counters, but did not remove +any. Thus, each cycle of KM(¯V) (including the ones that make up its inseparability flower) +also appears as a cycle in KM(Vpump). This implies that KM(Vpump) also has an inseparability +flower, and by Theorem 5.3 it follows that L(Vpump) ̸ | Dn. +◀ +E.3 +Proof of Lemma 6.4 +▶ Lemma 6.4. Given 1-dim. Büchi VASS V1, V2 with binary updates, there is a a 1-dim. +Büchi VASS V with L(V1) ∩ L(V2) = ∅ iff L(V) ∩ D1 = ∅, L(V1) | L(V2) iff L(V) | D1, and +we can construct in time polynomial in |V1| + |V2| the 2-dim. Büchi VASS ¯V (binary updates). +Proof. Recall that ¯V is constructed from a Büchi VASS V over alphabet Σn by adding n +additional counters and for each transition t replacing its label w ∈ Σ∗ +n with ε and instead +adding to t an effect of ϕ(w) on the n additional counters. A precise definition can be found +in Appendix B. To now proof Lemma 6.4 we have to argue that we can modify Lemma 3.4 +to directly construct ¯V instead of V, and that the modified version is feasible in polynomial +time. +If we analyze the proof of Lemma 3.4 in Appendix A then we obtain exponential time +complexity for this construction. The bottleneck here is Lemma A.2. However, we already +mentioned in the proof of Lemma A.2, that its associated complexity shrinks from exponential +to polynomial time, if we can somehow compress the exponentially long transition labels +that we end up with. An adequate compression for this is replacing an exponentially long +string w ∈ Σ∗ +n by its effect on the letter balance δ(w), which is exactly what we do when +going from V to ¯V. Since we encode δ(w) in binary, this is an exponential compression, and +therefore the time complexity of constructing ¯V directly is only polynomial, as required. +Note that for two 1-dimensional Büchi VASS as input, we have n = 1. But our proof +shows that constructing ¯V in polynomial time would still be feasible for Büchi VASS of +arbitrary dimension n. +◀ +E.4 +Proof of Proposition 6.5 +▶ Proposition 6.5. The constrained runs problem for 2-VASS is solvable in PSPACE. +Proof. We show that if there is a constrained run, then there is one where all counters have +at most exponential values along the way. For this, we rely on a result from [5] about linear +path schemes. +A linear path scheme (LPS) for a 2-dimensional VASS V is a regular expression of the +form S = σ0λ1σ1 · · · λnσn. Its alphabet is the set T of transition of V, and each infix λi +corresponds to a cycle of transitions in V. +Each LPS S induces a reachability relation →S over configurations of V, where (q, x, y) →S +(q′, x′, y′) if and only if there are numbers x1, . . . , xn ∈ N such that σ0λx1 +1 σ1 · · · λxn +m σn is a run +of V from (q, x, y) to (q′, x′, y′). In [5, Theorem 3.1], it is shown that for any two states q, q′ +in a 2-VASS V, there exists a set S of LPSs, each of which is of polynomial length, such that +for x, y, x′, y′ ∈ N, (q′, x′, y′) is reachable from (q, x, y) if and only if (q, x, y) →S (q′, x′, y′) +for some S from S. +In [5], this yields a PSPACE algorithm for configuration reachability in 2-dimensional VASS: +If there is run reaching a certain configuration, then there is one of the form σ0λx1 +1 σ1 · · · λxn +n σn +for some LPS σ0λ1σ1 · · · λnσn of polynomial length. Now the fact that σ0λx1 +1 σ1 · · · λxn +n σn is +a run between two given configurations can be expressed using a set of linear inequalities + +P. Baumann, R. Meyer, and G. Zetzsche +29 +over x1, . . . , xn. Since each solvable polynomial-sized set of linear inequalities has a solution +with at most exponential entries, this yields a run where all counters are at most exponential. +We only need to extend this argument from [5] slightly: First, we want to guess a system +of linear inequalities, whose solutions would satisfy the Presburger formula ψ. To this end, +we view ψ as a propositional formula by treating each atomic formula as a proposition. +For Presburger arithmetic, an atomic formula is either an equality t1 = t2 or an inequality +t1 < t2, where t1, t2 are additive terms over variables and/or the constants 0, 1. With this +propositional view of ψ, we can guess an assignment to its propositions, and verify that +its a satisfying assignment, feasible in polynomial space. If this assignment sets an atomic +formula of the form t1 = t2 to false, this means that t1 < t2 or t2 < t1 has to hold. Similarly +if t1 < t2 is set to false then t1 = t2 or t2 < t1 has to hold. In both cases, we simply guess +one of the two atomic formulas that have to hold instead. With these guesses together with +the unchanged atomic formulas that were set to true, we obtain a system of equalities and +inequalities, whose solutions would satisfy ψ. Formally, this system is comprised of matrices +A ∈ Zℓ×m, C ∈ Zk×m and vectors b ∈ Zℓ, d ∈ Zk with entries encoded in binary, such that +x ∈ Nm is a solution if and only if Ax < b and Cx = d. In fact unary encodings would +suffice for our definition of Presburger, since an entry of e.g. 3 would have come from a term +of the form y + y + y for a variable y, meaning all entries are polynomial in the size of ψ. +However, we do not require unary encodings and can also work with binary ones. +Now we only need to check that there is a constrained run (q0, 0, 0) +∗−→ (q1, x1, y1) +∗−→ +· · · +∗−→ (qm, xm, ym), whose counter values indeed satisfy these equalities and inequalities. +This is the case if Az < b and Cz = d, where z = (x1, y1, . . . , xm, ym). If such a constrained +run exists, then for each i ∈ [1, m], there is an LPS for the part (qi−1, xi−1, yi−1) +∗−→ (qi, xi, yi) +such that said run conforms to each of these LPSs. By imposing (a) the linear inequalities +of [5], which make sure that all counters stay non-negative, and (b) our linear inequalities +Az < b and equalities Cx = d, we obtain a new (poynomial-size) system of linear inequalities +over the exponents in the LPSs. +By [50] systems like these have minimal solutions with at most exponential entries, +yielding an overall run with at most exponential counter values. Binary encoding then means +that these solutions only take up polynomial space. More specifically, this implies that we +can simply guess configurations (q1, x1, y1) to (qm, xm, ym) of the constrained run in PSPACE, +and then check that equalities and inequalities of our system hold for them, i.e. that they +are actual solutions to the system. This concludes the description of our decision procedure. +As a final remark, note that [50] assumes inequalities of the form t1 ≤ t2 rather than +t1 < t2. However, since we seek solutions in Nm, we can simply express t1 < t2 as t1 + 1 ≤ t2 +to circumvent this issue. +◀ +F +Regular Separability vs. Intersection +In this section we prove the second part of Theorem 3.2. To this end we present a class of +WSTS such that, for their ω-languages, intersection is decidable whereas regular separability +is not. A d-dimensional reset Büchi VASS over alphabet Σ is a tuple V = (Q, q0, T, F). The +only difference to Büchi VASS is in the finite set of transitions which, besides adding a vector, +may reset a counter, T ⊆ Q×(Zd ∪{r1, . . . , rd})×Σ∗ ×Q. The configurations are defined like +for Büchi VASS, but the transition relation has to be adapted. We have (q, m) +w +−→ (q′, m′) if +there is a transition (q, x, w, q′) such that either (i) x ∈ Zd and m′ = m + x or (ii) x = ri for +some i ∈ [1, d] and m′(j) = m(j) for j ∈ [1, d] \ {i} and m′(i) = 0. +Acceptance is defined +as before, and so is the language (of infinite words) L(V). + +30 +Regular Separability in Büchi VASS +q0 +q +q1 +q2 +q3 +ε +0|a1 +0|¯a1 +ε +0|ε +0|¯a1 +−e1 + ed+1|a1 +e1 − ed+1|¯a1 +Vε +Figure 4 Weak Büchi reset VASS V′ in the proof of Theorem F.2. +For general Büchi reset VASS, emptiness and intersection are undecidable [37, Theorem 10]. +We consider a slight restriction of the model that makes the problems decidable. A Büchi +reset VASS is weak if there is no path from a final state to a reset transition. In particular, an +accepting run can only perform finitely many resets. Note that the usual product construction +of V1 and V2 to yield a Büchi reset VASS for L(V1) ∩ L(V2) preserves weakness. +▶ Theorem F.1. For weak Büchi reset VASS, emptiness (hence intersection) is decidable. +Here, emptiness can be decided using standard techniques. We order the configurations +Q×Nd in the usual way: We have (q, m) ≤ (q′, m′) if q = q′ and m ≤ m′. First, one observes +that for any Büchi VASS V, the set U(V) ⊆ Q × Nd of all configurations (q, m) from which +an infinite accepting run can start, is upward closed. Moreover, using a saturation procedure, +we can effectively compute the finitely many minimal elements (q1, m1), . . . , (qℓ, mℓ) of U(V). +The details can be found in Lemma F.3 at the end of this section. Then, for a weak Büchi +reset VASS V, we do the following. We construct the Büchi VASS V′, which is obtained from +V by deleting all reset transitions. Now L(V) is non-empty if and only if V, as a reset VASS, +can cover any of the configurations (q1, m1), . . . , (qℓ, mℓ) of U(V′). Whether the latter is the +case can be decided because coverability is decidable in reset VASS [19, 25, 2]. +▶ Theorem F.2. For weak B. reset VASS over Σ1, regular separability from D1 is undecidable. +We reduce from the place boundedness problem for reset VASS. A reset VASS is a Büchi +reset VASS without input words and without final states. For k ∈ N, we say that a reset +VASS V is k-place bounded if for every reachable configurations (q, m), we have m(1) ≤ k. +Moreover, we call V place bounded if V is k-place bounded for some k ∈ N. The place +boundedness problem then asks whether a given reset VASS is place bounded. The place +boundedness problem (more generally, the boundedness problem) for reset VASS is known +to be undecidablei [19, Theorem 8] (for a simpler proof, see [37, Theorem 18]). +Given a d-dim. reset VASS V, we build a (d + 1)-dim. weak Büchi reset VASS V′ with +L(V′) = {w ∈ S1,k | k ∈ N, V can reach some (q, m) ∈ Q × Nd with m(1) ≥ k}. +(2) +Before we describe V′, observe that L(V′) | D1 iff V is place bounded. If V is k-place bounded, +then L(V′) ⊆ S1,k and thus L(V′) | D1. On the other hand, if V is not place bounded, then +L(V′) = � +k∈N S1,k. As for the Büchi VASS in Figure 1 (left), one can show L(V′) ̸ | D1. +The construction is depicted in Figure 4. The dashed box contains Vε, which is obtained +from V by changing every transition (p, u, q) into (p, (u, 0), ε, q). In the figure, q stands +for arbitrary states of Vε, meaning for every state q in Vε, we have a transition (q, 0, ε, q1). +Observe that in the states q1, q2, q3, V′ behaves exactly like the Büchi VASS in Figure 1(left), +except that the additional counter ensures that for each infix the balance on letter ai is +bounded by k from configurations (q1, (k, u)). Thus the accepted language from (q1, (k, u)) +is exactly S1,k. This shows that V′ accepts the language (2). +▶ Lemma F.3. Let V be a Büchi VASS. We can compute the set U(V) of minimal configura- +tions from which there is an infinite accepting run. + +P. Baumann, R. Meyer, and G. Zetzsche +31 +Proof. It is decidable whether a given Büchi VASS has an accepting run [21, 28]. We +strengthen this result to checking whether a given Büchi VASS V has an accepting run +starting in a downward-closed set of configurations. The downward-closed set is given as a +finite union I of ideals, each represented by a generalized configuration (q, m) ∈ Q×(N∪{ω})d. +The algorithm is as follows. We construct an instrumented Büchi VASS VI from V and I +that starts in a gadget for I from which it moves to V. This gadget selects one of the ideals, +say (q, m), and increments each counter c to at most m(c). Note that m(c) may be ω, in +which case we may put an arbitrary value to this counter. After this initial phase, VI moves +to state q of V. The states in the gadget are not accepting, so VI will eventually move to V +to obtain an infinite accepting run. To be precise, we have in V an accepting run from a +configuration in I if and only if VI has an accepting run. +With this, we can saturate a set of markings S, initially S = ∅. We repeatedly ask for +an accepting run starting in a downward-closed set of configurations represented by a set +of ideals I. Initially, we just ask for any run, I = Q × {ωd}. If such a run does not exist, +we return S. If such a run exists, we can reconstruct a configuration (q, m) ∈ I, m ∈ Nd, +with which VI moved from the gadget for I to V. This can be done with an enumeration. +We add (q, m) to S and refine the downward-closed set represented by I by subtracting the +upward-closure of the new S. The subtraction can be computed effectively and yields a new +set of ideals with which we repeat the check of an accepting run. The process terminates: the +set S represents an upward-closed set of configurations, and every infinite sequence of such +sets becomes stationary due to the wqo. In the moment when the set becomes stationary, we +will no longer find an accepting run and return. +◀ + diff --git a/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/load_file.txt b/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..806886d6f39b360c595bccb1dbf01b642d58d7e7 --- /dev/null +++ b/G9FIT4oBgHgl3EQfXCt6/content/tmp_files/load_file.txt @@ -0,0 +1,1565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf,len=1564 +page_content='Regular Separability in Büchi VASS Pascal Baumann � Max Planck Institute for Software Systems (MPI-SWS), Germany Roland Meyer � TU Braunschweig, Germany Georg Zetzsche � Max Planck Institute for Software Systems (MPI-SWS), Germany Abstract We study the (ω-)regular separability problem for Büchi VASS languages: Given two Büchi VASS with languages L1 and L2, check whether there is a regular language that fully contains L1 while remaining disjoint from L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We show that the problem is decidable in general and PSPACE-complete in the 1-dimensional case, assuming succinct counter updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The results rely on several arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We characterize the set of all regular languages disjoint from L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Based on this, we derive a (sound and complete) notion of inseparability witnesses, non-regular subsets of L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, we show how to symbolically represent inseparability witnesses and how to check their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Models of computation Keywords and phrases Separability problem, Vector addition systems, Infinite words, Decidability Funding Funded by the European Union (ERC, FINABIS, 101077902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Neither the European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The second author was supported by the DFG project EDS@SYN: Effective Denotational Semantics for Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='11242v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='FL] 26 Jan 2023 erc EuropeanResearchCouncil EstablishedbytheEuropeanCommission2 Regular Separability in Büchi VASS 1 Introduction The separability problem asks, given languages L1 and L2, whether there exists a language R that separates L1 and L2, meaning L1 ⊆ R and R ∩ L2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, R is constrained to be from a particular class S of admitted separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since safety verification of systems with concurrent components is usually phrased as an intersection problem for finite-word languages, and separators certify disjointness, deciding separability can be viewed as synthesizing safety certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Analogously, deciding separability for infinite-word languages is a way of certifying liveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If S is the class of (ω-)regular languages, we speak of regular separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Separability problems have been studied intensively over the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If the input languages are themselves regular and S is a subclass [42, 41, 40, 39, 43, 44, 35, 15], then separability generalizes the classical subclass membership problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, separability for languages of infinite-state systems has received a significant amount of attention [17, 16, 14, 13, 10, 9, 12, 1, 51, 48, 11, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us point out two prominent cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' First, one of the main open problems in this line of research is whether regular separability is decidable for (reachability) languages of vector addition systems with states (VASS): A VASS consist of finitely many control states and a set of counters that can be incremented and decremented, but not tested for zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, each transition is labeled by a word over the input alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, a run is accepting if it reaches a final state with all counters being zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' While there have been several decidability results for subclasses of the VASS languages [17, 14, 13, 10, 9], the general case remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Second, a surprising result is that if K and L are coverability languages of well-structured transition systems (WSTS), then K and L are separable by a regular language if and only if they are disjoint [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' As VASS are one example of WSTS, this result also applies to their coverability languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Regular separability in Büchi VASS In this paper, we study the regular separability problem for Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' These are VASS that accept languages of infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A run is accepting if it visits some final state infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since no condition is placed on the counter values, Büchi VASS languages are an infinite-word analogue of finite-word coverability languages, where acceptance is defined by the reached state (not the counters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The regular separability problem is to decide, given Büchi VASS V1 and V2, whether there exists an ω-regular language R such that L(V1) ⊆ R and L(V2) ∩ R = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Our main results are that (i) regular separability for Büchi VASS is decidable, and that (ii) for one-dimensional Büchi VASS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' those with a single counter) the problem is PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, we assume that the counter updates are encoded in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given that Büchi VASS accept using final states and their transition systems are WSTS, one may suspect that there is an analogue of the aforementioned result for WSTS: Namely, that two languages of Büchi VASS are separable by an ω-regular language if and only if they are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We show that this is not the case: There are Büchi VASS V1 and V2 such that L(V1) and L(V2) are disjoint, but not separable by an ω-regular language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In fact, we show an even larger disparity between these two problems for WSTS in the infinite-word case: We exhibit a natural class of WSTS for which intersection is decidable but regular separability is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, regular separability for Büchi VASS requires significantly new ideas and involves several phenomena that do not occur for finite-word languages of VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' New phenomena and key ingredients We first observe that we can assume one input language to be fixed, namely an infinite-word version Dn of the Dyck language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then, following the basic separator approach from [17], we identify a small class B of ω-regular P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 3 languages such that L is separable from Dn if and only if L is included in a finite union of sets from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, a crucial insight is that a Büchi automaton can guarantee disjointness from Dn without knowing exactly when the letter balance crosses zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that a negative letter balance is the exact condition for non-membership in Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In contrast, in the finite word case, there are always separating automata that can tell when zero is crossed [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This insight is also key to the example differentiating disjointness and separability in Büchi VASS, and to the undecidability proof for certain WSTS despite decidable disjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We then develop a decomposition of Büchi VASS languages into finitely many pieces, which are induced by what we call profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Inspired by Büchi automata, the idea of a profile is to fix the set of transitions that can and have to be taken infinitely often in a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finding the right generalization to Büchi VASS, however, turned out to be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Our formulation refers to edges in the Karp-Miller graph, augmented by constraints that guarantee the existence of an accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The resulting decomposition has properties similar to the decomposition of VASS languages into run ideals [33], which has been useful for previous separability procedures [17, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We associate to each profile a system of linear inequalities and show that separability holds if and only if each of these systems is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' While this yields decidability, checking feasibility is not sufficient to obtain a PSPACE-upper bound in the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Instead, we use Farkas’ Lemma to obtain a dual system of inequalities so that separability fails if and only if one dual system is satisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A solution to a dual system yields a pattern in the Karp-Miller graph, called inseparability flower, which witnesses inseparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Compared to prior witnesses for deciding properties of VASS languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' regularity [18], language boundedness [7], and other properties [3]), inseparability flowers are quite unusual: they contain a non-linear condition, requiring one vector to be a scalar multiple of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For one-dimensional Büchi VASS, the condition degenerates into a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This allows us to translate inseparability flowers into particular runs in a two-dimensional VASS subject to additional linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Using methods from [5], this yields a PSPACE procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Related work It was already shown in 1976 that regular separability is undecidable for context-free languages [47, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Over the last decade, there has been intense interest in deciding regular separability for subclasses of finite-word VASS reachability languages: The problem is decidable for (i) reachability languages of one-dimensional VASS [13], (ii) cov- erability languages of VASS [14], (iii) reachability languages of Parikh automata [9], and (iv) commutative reachability languages of VASS [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, decidability still holds if one input language is an arbitrary VASS language and the other is as in (i)-(iii) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' As discussed above, for finite-word coverability languages of WSTS, regular separability is equivalent to disjointness [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, the aforementioned undecidability for context-free languages has been strengthened to visibly pushdown languages [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To our knowledge, for languages of infinite words, separability has only been studied for regular input languages [38, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Our result makes use of Farkas’ Lemma to demonstrate the absence of what can be understood as a linear ranking function (on letter balances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are precursors to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In liveness verification [45], Farkas’ Lemma has been used to synthesize, in a complete way, linear ranking functions proving the termination of while programs over integer variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the context of separability for finite words, Farkas’ Lemma was used to distinguish separable from non-separable instances [17], similar to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The novelty here is the combination of Farkas’ Lemma with the new notion of profiles needed to deal with infinite runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The languages of Büchi VASS have first been studied by Valk [49] and (in the determin- istic case) Carstensen [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Some complexity results (such as EXPSPACE-complexity of the 4 Regular Separability in Büchi VASS emptiness problem) were shown by Habermehl [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' More recently, there have been several papers on the topological complexity of Büchi VASS languages (and restrictions) [26, 20, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' See the recent article by Finkel and Skrzypczak [27] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 2 Preliminaries Dyck Language We use an infinite-word version of the Dyck language over n pairs of matching letters ai, ¯ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We denote the underlying alphabet by Σn := �n i=1{ai, ¯ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The Dyck language contains those infinite words where every occurrence of ¯ai has a matching occurrence of ai to its left: Dn := {w ∈ Σω n | ∀v ∈ prefix(w): ∀i ∈ [1, n]: ϕi(v) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, ϕi : Σ∗ n → Z is the ith (letter) balance function that computes for a given word w the difference |w|ai − |w|¯ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We also use ϕ(w) for the vector (ϕ1(w), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , ϕn(w)) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS and Automata A Büchi vector addition system with states (Büchi VASS) of dimension d ∈ N over alphabet Σ is a tuple V = (Q, q0, T, F) consisting of a finite set of states Q, an initial state q0 ∈ Q, a set of final states F ⊆ Q, and a finite set of transitions T ⊆ Q×Σ∗ ×Zd ×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The size of the Büchi VASS is |V| := |Q|+1+|F|+� (q,w,δ,q′)∈T |w|+ �d i=1 max{log |δ(i)|, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If d = 0, we call V a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The semantics of the Büchi VASS is defined over configurations, which are elements of Q × Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We call the second component in a configuration the counter valuation and refer to the entry in dimension i as the value of counter i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The initial configuration is (q0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We lift the transitions of the Büchi VASS to a relation over configurations → ⊆ Q×Nd×Σ∗×Q×Nd as follows: (q, m) w −→ (q′, m′) if there is (q, w, δ, q′) ∈ T so that m′ = m + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A run of the Büchi VASS is an infinite sequence of transitions of the form (q0, 0) w1 −−→ (q1, m1) w2 −−→ · · · Thus, the sequence starts in the initial configuration and makes sure the target of one transition is the source of the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The run is accepting if it visits final states infinitely often, meaning there are infinitely many configurations (q, m) with q ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The run is said to be labeled by the word w = w0w1 · · · in Σω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The language L(V) of the Büchi VASS consists of all infinite words that label an accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that we can always ensure that every accepting run has an infinite-word label, by tracking in the state whether a non-ε-transition has occurred since the last visit to a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' An infinite-word language is (ω-)regular, if it is the language of a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' As we only consider infinite-word languages, we just call them languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Karp-Miller Graphs We work with the Karp-Miller graph KM(V) associated with a Büchi VASS V [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since we are interested in infinite runs, we define the Karp-Miller graph as a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Its state set is a finite set of extended configurations, which are elements of Q × (N ∪ {ω})d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The initial state is the initial configuration in the Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The final states are those extended configurations (q, m) with q ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The transitions are labeled by T, so instead of letters they carry full Büchi VASS transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' An entry ω in an extended configuration denotes the fact that a prefix of a run can be repeated to produce arbitrarily high counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' More precisely, the Karp-Miller graph is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' From an extended configuration (q, m) we have a transition labeled by (q1, a, δ, q2), if q = q1 and m + δ remains non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The latter addition is defined componentwise and assumes ω + k := ω =: k + ω for all k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The result of taking the transition is the extended configuration (q2, m2), where m2 is constructed from m + δ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We raise to ω all counters i for which there is an earlier configuration (q2, m1) with m1 ≤ m + δ and m1(i) < [m + δ](i), earlier meaning on some path from (q0, 0) to (q, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If this is the case, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 5 q0 q1 q2 e1|ε 0|ε 0|ε 0|¯a1 −e1|a1 e1|¯a1 a1 a2 ε a1¯a2¯a2 a2¯a1¯a1 x Figure 1 Left: A Büchi VASS accepting a language S with S ∩ D1 = ∅ but S ̸ | D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, e1 ∈ Z is the one-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' vector with entry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Right: A regular language that is not included in a finite union of languages Pi,k and Si,k, but that is included in Sx,k for x = (1, 1), k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The horizontal and vertical dimensions denote the balance for a1 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' the path from (q2, m1) to (q2, m + δ) can be repeated indefinitely to produce arbitrarily high values for counter i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We refer to the repetition of such a path in a run as pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The Karp-Miller graph over-approximates the language of the Büchi VASS in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Every infinite sequence of transitions that leads to a run of the Büchi VASS is the labeling of an infinite run in the Karp-Miller graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, if the run of the Büchi VASS is accepting, so is the run in the Karp-Miller graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the other direction, every finite transition sequence in the Karp-Miller graph represents a transition sequence in the Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The sequence in the Büchi VASS, however, may be longer to compensate negative effects on ω-entries by pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 3 Problem, Main Result, and Proof Outline A language R is a regular separator for a pair of languages L1, L2, if R is regular, L1 ⊆ R, and R ∩ L2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We write L1 | L2 for the fact that a regular separator exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We consider here languages of Büchi VASS, and formulate the regular separability problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given Büchi VASS V1, V2, check whether L(V1) | L(V2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Our main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The regular separability problem for Büchi VASS is decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It should be noted that our procedure is non-primitive recursive, as it explicitly constructs the Karp-Miller graph of an input Büchi VASS, which can be of Ackermannian size [36, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the case of VASS coverability languages (and even for more general WSTS), it is known that regular separability is equivalent to disjointness [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, for finite words, separability reduces to the much better understood problem of disjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the infinite-word languages considered here, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are Büchi VASS languages L1, L2 with L1 ∩ L2 = ∅ and L1 ̸ | L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are classes of WSTS where intersection is decidable but separability is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the second statement, we introduce the class of weak Büchi reset VASS, which are VASS with reset instructions, with the additional constraint that each run can only use resets a finite number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Details can be found in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the first statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2, we give an intuition and refer to Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We choose L1 = L(V), where V is the Büchi VASS in Figure 1(left), and L2 = D1, the Dyck language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To show L(V) ̸ | D1, suppose there is a Büchi automaton A with n states such that L(V) ⊆ L(A) and L(A) ∩ D1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then A has to accept (an 1 ¯an+1 1 )ω ∈ L(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, pumping yields that for some m > n the word (am 1 ¯an+1 1 )ω ∈ D1 also has to be 6 Regular Separability in Büchi VASS accepted by A, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, to show L(V) ∩ D1 = ∅ we observe that in accepting runs of V, almost every visit (meaning: all but finitely many) to the final state drops the letter balance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore on any accepting run this balance eventually becomes negative, yielding a word outside of D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the remainder of the section, we outline the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Assume we are given L1 = L(V1) and L2 = L(V2) and this is a non-trivial instance of separability, meaning L1, L2 are not regular and L1 ∩ L2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For proving separability, we could enumerate regular languages until we find a separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The difficult part is disproving separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Inseparability of L1 and L2 is witnessed by a set of words W ⊆ L1 so that every regular language R containing them already intersects L2, formally: W ⊆ R implies R ∩ L2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Showing the existence of such a set W is difficult for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' First, it is unclear which sets of words ensure the universal quantification over all regular languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Second, as we have a non-trivial instance of separability, W (if it exists) will be a non-regular language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' So it is unclear how to represent it in a finite way and how to check its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To address the first problem and understand the sets of words that disprove separability, we use diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Call an (L2-)separator candidate a regular language that is disjoint from L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' be an enumeration of the separator candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If L1 is not separable from L2, for every Ri there is a word wi ∈ L1 with wi /∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We call such a set of words W = {w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='} that escapes every separator candidate an inseparability witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L1 ̸ | L2 if and only if there is an inseparability witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Our decision procedure will check the existence of an inseparability witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We obtain the procedure in four steps: the first is a simplification, the second is devoted to understanding the separator candidates, the third is another simplification, and the last characterizes the inseparability witnesses and checks their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Step 1: Fixing L2 We first reduce general regular separability to regular separability from the Dyck language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The reduction is simple and works just as for finite words [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given Büchi VASS V1 and V2, we can compute a Büchi VASS V over Σn so that L(V1) | L(V2) if and only if L(V) | Dn, where n is the dimension of V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Step 2: Understanding the Separator Candidates To understand the regular languages that are disjoint from Dn, we will define basic separators, sets Pi,k and Sx,k, on which we elaborate in a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The following theorem says that finite unions of basic separators are sufficient for regular separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is our first technical result and shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If R ⊆ Σω n is regular and R ∩ Dn = ∅, then R is included in a finite union of basic separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the definition of Pi,k, we note that the words outside Dn have, for some index i ∈ [1, n], an earliest moment in time where the balance between ai and ¯ai falls below zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To turn this into a regular language, we impose an upper bound k ∈ N on the (positive) balance between the letters ai and ¯ai that is maintained until the earliest moment is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This yields the regular language Pi,k := {w ∈ Σω n | ∃v ∈ prefix(w): ϕi(v) < 0 ∧ ∀u ∈ prefix(v): ϕi(u) ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The family of languages Pi,k already captures the complement of Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The problem is that we may need infinitely many such languages to cover the language R of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 7 every bound k, a regular R with R∩D1 = ∅ may contain a word with a higher balance before falling below zero, take for example R = a∗ 1¯aω 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The first insight is that if R can fall below zero from arbitrarily high values, then the underlying Büchi automaton has to contain loops with a negative balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The R thus contains words uv with an unconstrained prefix and a suffix that decomposes into v = v1v2 · · · so that every infix w = vℓ has a negative balance on letter ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The observation suggests the definition of a language that contains precisely the words u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To make the language regular, we impose a bound k on the positive balance that can be used during the infixes w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Call the resulting language Si,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Unfortunately, taking the Pi,k and the Si,k as basic separators is still not enough: Figure 1(right) exhibits a regular language, disjoint from D1, that is not included in a finite union of Pi,k and Si,k, because it contains infixes where the balance on each letter exceeds all bounds in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The second insight is that we can catch the remaining words with a version of Si,k that weights coordinates with some x ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us give some intuition on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The words from R that we cannot catch with a Pi,k must come across, for each i that becomes negative, a loop with positive balance on i (otherwise, the balance on those i would be bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' But then, the only way such words can avoid D1 is by ending up in a strongly connected component where every loop (with a final state) makes progress towards crossing 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' is negative in some coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' One can then conclude that even all Q≥0-linear combinations of loops (a convex set) must avoid the positive orthant Qn ≥0 ⊂ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By the Hyperplane Separation Theorem (we use it in the form of Farkas’ Lemma), this is certified by a hyperplane that separates all loop effects from Qn ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This hyperplane is given by some orthogonal vector x ∈ Nn, meaning that every loop balance must have negative scalar product with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence, we can catch these words by: Sx,k := � u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='v ∈ Σω n ����� a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') ∀f ∈ infix(v): ⟨x, ϕ(f)⟩ ≤ k, and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') v = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='v2 · · · ∧ ∀ℓ ∈ N: ⟨x, ϕ(vℓ)⟩ < 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Coming back to Figure 1(right), the weight vector x = (1, 1) guarantees that the weighted balance decreases indefinitely and also the weighted balances of all infixes stay bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In [17], a similar argument has been used to show sufficiency of basic separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Step 3: Pumpable Languages With the basic separators at hand, the task is to understand the sets of words witnessing inseparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' While studying this problem, we observed that the argumentation for the Pi,k was always similar to the one for the Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This led us to the question of whether we can get rid of the Pi,k in separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The answer is positive, and hinges on a new notion of pumpability for languages over Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Call infinite words u and v equivalent, written u ∼ v, if v can be obtained from u by removing and inserting finitely many letters: There are u0, v0 ∈ Σ∗ and w ∈ Σω such that u = u0w and v = v0w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We say that a language L ⊆ Σω n is pumpable if for every w ∈ L and every k ∈ N, there exists a decomposition w = w0w1 and a word w′ 0 ∈ Σ∗ n that is a prefix of a word in Dn such that w′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='w1 ∈ L and the letter balance satisfies the following: (a) ϕ(w′ 0) ≥ ϕ(w0) and (b) for the indices i ∈ [1, n] where ϕi becomes negative on some prefix of w, we have ϕi(w′ 0) ≥ max{ϕi(w0), 0} + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The consequence of this definition is that a pumpable language leaves every language Pk := � i∈[1,n] Pi,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Indeed, for every word w ∈ L and every k ∈ N, there is a word w′ ∈ L with w ∼ w′ where the letter balance exceeds k before becoming negative, and thus w′ /∈ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With the previous characterization of separator candidates, what is left to separate L from Dn are the languages Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If L ⊆ Σω n is pumpable, then L | Dn if and only if L |limDn, where L |limDn means L ⊆ � x∈X Sx,k for some finite set X ⊆ Nn and some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 8 Regular Separability in Büchi VASS q0 q1 q2 (1, 0)|ε 0|ε 0|ε (0, −1)|ε (−1, 1)|ε (1, −1)|ε (q0, ω, 0) (q1, ω, 0) (q2, ω, 0) (q1, 0, 0) (q0, 0, 0) (q2, 0, 0) (q2, ω, ω) (q1, ω, ω) e1|ε 0|ε 0|ε e1|ε 0|ε 0|ε −e1|a1 0|ε 0|¯a1 −e1|a1 e1|¯a1 Figure 2 Left: The Büchi VASS ¯V constructed from the Büchi VASS V found in Figure 1(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note how the added second counter tracks the letter balance of the now removed transition labels, incrementing on letter a1 and decrementing on letter ¯a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Right: The Büchi VASS Vpump corresponding to V as given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here we did not mark the final states to reduce visual clutter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' every state that includes q1 is considered final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Similarly, the two labels above the loop in the top right correspond to two distinct transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that Vpump essentially looks like KM(¯V), just with different transition labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In our context, pumpability is interesting because we can turn every Büchi VASS language into a pumpable language without affecting separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a d-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS over Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We can compute a d-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS Vpump that satisfies the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L(Vpump) is pumpable, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' there is a k ∈ N so that L(Vpump) ⊆ L(V) ⊆ L(Vpump) ∪ Pk, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L(V) | Dn if and only if L(Vpump) | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The construction of Vpump employs the Karp-Miller graph in an original way, namely to track the unboundedness of letter balances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let ¯V be the (d + n)-dimensional Büchi VASS obtained from V by tracking the effect of the letters from Σn in n additional counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For ¯V, we construct the Karp-Miller graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The relationship between the languages of KM(¯V) and V is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For all words where every letter balance stays non-negative, their runs in V can be mimicked in KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For all other words, where the balance eventually becomes negative, this only holds if the corresponding counter in ¯V has been raised to ω beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Essentially, the new Büchi VASS Vpump restricts V to those runs that have counterparts in KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is achieved with a simple product construction of V and KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The thing to note is that every word from L(V) that does not make it into L(Vpump) belongs to Pk, where k is the maximum concrete number in KM(¯V): A run in V that cannot be mimicked in KM(¯V) will at some point have a negative letter balance, before reaching ω in KM(¯V) in that component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' thus all counter values had been at most k until that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' An example on how to construct ¯V and Vpump can be found in Figure 2, where both were constructed for the Büchi VASS found in Figure 1(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5, we make use of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 (recall that a regular language is the language of a 0-dimensional Büchi VASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This may look like cyclic reasoning, but it is not: We will show Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7(1)+(2) directly, using the arguments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With this, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5, which in turn is used to derive Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Step 4: Non-Separability Witnesses and Decidability Because of pumpability, it remains to decide whether a Büchi VASS language L(V) is included in a finite union � x∈X Sx,k for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 9 some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Part of the difficulty is that we have no bound on the cardinality of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To circumvent this, we decompose L(V) into a finite union � π Lπ(V), where π is a profile, meaning a set of edges in KM(V) seen infinitely often during a run of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We then show that each Lπ(V) is either (i) included in a single separator Sx,k or (ii) escapes every finite union � x∈X Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, it is key to show an even stronger fact: In case (i), not only Lπ(V) is included in some Sx,k, but the entire set of runs in KM(V) that eventually remain in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The advantage of strengthening is that finiteness of KM(V) allows us to express inclusion in Sx,k, for some k, as a finite system of linear inequalities over x: We say that (1) the balance of every primitive cycle, weighted by x, is at most zero and (2) the balance, weighted by x, of some cycle containing all edges from π is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, (1) and (2) correspond to Conditions a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') of Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If they are met, then the runs of KM(V) along π are included in Sx,k for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We then prove that if the system is not feasible, then V has runs that escape every finite union � x∈X Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To this end, we employ Farkas’ Lemma: It tells us that if there is no solution, then the dual system has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The solution of the dual system can be interpreted as an executable linear combination of primitive cycles with non-negative balances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We show that these cycles can be arranged in a pattern in KM(V) we call inseparability flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Such an inseparability flower then yields a sequence of runs ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' in KM(V) such that ρk escapes Sx,k for every vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, pumpability allows us to lift these runs of KM(V) to runs of V and thus conclude inseparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This equips us with two possible decision procedures: We can either check solvability of each system of inequalities, or detect inseparability flowers in KM(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 4 Basic Separators We prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5, that any regular language R over Σn with R ∩ Dn = ∅ is contained in a finite union of languages Pi,k and Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that a single value of k is sufficient, since we have Pi,k ⊆ Pi,k+1 and Sx,k ⊆ Sx,k+1 for each i, x, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The proof decomposes the Büchi automaton for R in a way that allows us to forget about connectedness issues and reason over cycles (and their letter balances) using techniques from linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We make use of the following basic fact from linear programming [46, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1 (Farkas’ Lemma (variant), [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let A ∈ Qm×n be a matrix and let b ∈ Qm be a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then the system Ax ≤ b has a solution x ∈ Qn ≥0 if and only if y⊤b ≥ 0 for each vector y ∈ Qm ≥0 with y⊤A ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Decomposing with profiles We decompose R = L(A) into a (not necessarily disjoint) union of several languages, each linked to a so-called profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We will later see that for pumpable R, every such profile language already has to be contained in a single Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let A be a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A profile of A is a set π of transitions of A for which there exists a cycle σπ in A such that (a) σπ contains exactly the transitions in π, and (b) σπ starts (and ends) in a final state qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We denote by Π(A) the finite set of profiles of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, we associate to every accepting run ρ of A its profile Π(ρ), which contains exactly the transitions appearing infinitely often in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This definition is sound, as the infinitely occurring transitions of an accepting run must form a cycle due to repetition, which visits a final state due to acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given a profile π of A, we define Lπ(A) ⊆ L(A) to be the language of all words that have an accepting run ρ of A with Π(ρ) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that this language is still regular: From A one can construct a Büchi automaton that guesses a point after which only transitions 10 Regular Separability in Büchi VASS from π can occur, and once this point is reached it keeps a list of already used transitions from π in each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then only once all transitions of π have been used the state becomes final and the list is set back to empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This now allows us to view R as the union of the languages Lπ(A) with π ∈ Π(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We show that each language Lπ(A) is either contained in Sx,k for some x, k, or there is a cycle that, assuming the pumpability from the previous section, makes Lπ(A) intersect Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let A be a Büchi automaton over Σn and let π be one of its profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then one of the following conditions holds: (i) There is a number k ∈ N and a vector x ∈ Nn such that Lπ(A) ⊆ Sx,k, or (ii) there is a cycle σ′ in A over w′ with ϕ(w′) ≥ 0, and σ′ contains all transitions from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Assume Lπ(A) ̸= ∅, otherwise Condition (i) trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We build a system Aπx ≤ b of linear inequalities as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It contains one inequality ⟨x, ϕ(v)⟩ ≤ 0 for each word v read by a primitive cycle of transitions in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By primitive cycle we mean a cycle that does not repeat a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, the system contains the inequality ⟨x, ϕ(vπ)⟩ ≤ −1 for the cycle σπ over vπ that justifies the profile π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us quickly remark that the solution space of the system Aπx ≤ b is independent of the precise choice of the justifying cycle σπ: To see this, we claim that Aπx ≤ b holds if and only if all primitive cyles in π have an x-weighted balance at most zero, and at least one primitive cycle in π has a strictly negative x-weighted balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the “if” direction, note that a sufficiently long repetition of σπ will contain each primitive cycle as a (possibly non-contiguous) subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This means, the repetition, and thus σπ, must have a strictly negative x-weighted balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the converse, we observe that σπ can be decomposed into primitive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, if σπ has strictly negative x-weighted balance, then so must at least one of its constituent primitive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Applying Farkas’ Lemma to Aπx ≤ b either yields a solution x ∈ Qn ≥0 or a vector y ∈ Qm ≥0 with y⊤Aπ ≥ 0 and y⊤b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In both cases we assume wlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' that the given vector has entries in N, as we can always multiply with the lcm of the denominators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose we have a solution x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that then Lπ(A) ⊆ Sx,k, where k = |Qπ| · h and h is the maximal length of a transition label of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is because x weights primitive cycles non-positively, and k is chosen such that for any infix v of a word in Lπ(A), if |v| > k, then v’s associated transition sequence has to contain a primitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, infixes at almost all start positions of a word in Lπ(A) must have x-weighted balance ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If we obtain a vector y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , ym), then we can view it as a selection of rows in the matrix Aπ, where the jth row is being selected yj many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since each row corresponds to a cycle, this is also a selection of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then by y⊤b < 0 we selected σπ, where we can insert the other selected cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By y⊤Aπ ≥ 0 this forms a cycle σ′ as required, with non-negative letter balance for all letter pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A detailed proof can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, we used a system of linear inequalities Aπx ≤ b, which was solely dependent on A and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We reasoned that if this system has a solution, then Condition (i) has to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is a fact that we want to refer to in a later proof, and therefore we formalize it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If A is a Büchi automaton with a profile π for which there is an x ∈ Nn with Aπx ≤ b, then Lπ(A) ⊆ Sx,k for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, we can now show Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose R = L(A) for some Büchi automaton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' First, applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 with d = 0 yields a Büchi automaton Apump such that L(A) ⊆ L(Apump) ∪ Pℓ for some ℓ ∈ N and L(Apump) ∩ Dn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, it suffices to show that L(Apump) is included in a finite union of languages Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then the set L(Apump) decomposes into the sets Lπ(Apump) for π ∈ Π(Apump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, we know that for some π, Condition (ii) must hold: Otherwise, each Lπ(Apump) would be P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 11 included in some Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' But if (ii) holds for π, then there is a cycle σ′ in Apump that contains π (and thus visits a final state) and reads a word v with ϕ(v) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now for some finite prefix u, the word uvω belongs to L(Apump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since ϕ(v) ≥ 0, there is some lower bound B ∈ Z such that for each i ∈ [1, n] and every prefix p of uvω, we have ϕi(p) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, since L(Apump) is pumpable, we can exchange a prefix in w = uvω to obtain another word w′ ∈ L(Apump) where every prefix p has ϕ(p) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence w′ ∈ Dn and thus L(Apump) ∩ Dn ̸= ∅, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 5 Deciding Regular Separability We now present the algorithm to decide, given a Büchi VASS V whether L(V) | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We first employ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7, because for pumpable languages we only have to deal with one type of basic separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The next step is to generalize the notion of profiles from Büchi automata to Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that for a sequence χ of transitions in V, δ(χ) denotes its effect on the counters of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If χ is a transition sequence in KM(V), then χ is labeled with a transition sequence of V, so we define δ(χ) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since we consider Büchi VASS with input alphabet Σn, we write ϕ(χ) for the image of the input word under ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Again, this notation is used for transition sequences in KM(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We also write ∆(χ) = (δ(χ), ϕ(χ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A profile for V is a set π of edges in KM(V) for which there exists a cycle σ in KM(V) such that (i) σ contains exactly the edges in π, (ii) σ starts (and ends) in a final state, and (iii) δ(σ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Clearly, every Büchi VASS has a finite set of profiles, which we denote by Π(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, Π(V) can be constructed effectively: Given a set of edges, a simple reduction to checking unboundedness of a counter can be used to check if it is a profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, to every run ρ of V, we can associate a profile: The run ρ must have a corresponding run in KM(V), which has a finite set Π(ρ) of edges that are used infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, ρ decomposes as ρ0ρ1 such that ρ1 only contains edges from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then, ρ1 decomposes into σ1σ2 · · · such that each σi uses every edge from Π(ρ) at least once and starts (and ends) in a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since ≤ is a well-quasi ordering on Nn, there are r < s such that δ(σr · · · σs) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, σ = σr · · · σs is our desired transition sequence showing that Π(ρ) is a profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For each π ∈ Π(V), we denote by Lπ(V) the set of all words accepted by runs ρ of V for which Π(ρ) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then clearly: ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L(V) = � π∈Π(V) Lπ(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A system of inequalities for each profile Our next step is to associate with each profile π ∈ Π(V) a system of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We need some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A π-cycle is a cycle σ in KM(V) that only contains edges in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If in addition, σ visits each state of KM(V) at most once, except for the initial state, which is visited twice, then σ is a primitive π-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Clearly, a primitive π-cycle has length ≤ |π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, from every π-cycle σ, one can successively cut out primitive π-cycles until it is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, if τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , τm are the primitive π-cycles of KM(V), then there are numbers r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , rm ∈ N such that ∆(σ) = r1 ·∆(τ1)+· · ·+rm ·∆(τm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We call σ a complete π-cycle if this holds for some r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , rm ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Observe that if π is a profile, then this is always witnessed by a complete π-cycle: Take any cycle σ witnessing that π is a profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then σ|π| contains each primitive π-cycle as a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence, the cycle σm·|π| is complete: We can carry out the cutting in each factor σ|π| so as to cut some τi at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, σm·|π| still witnesses that π is a profile, since δ(σm·|π|) = m · |π| · δ(σ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us now construct the system of inequalities associated with π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let σ be a complete π-cycle witnessing that π is a profile and let τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , τm be the primitive π-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let Aπ ∈ Z(m+1)×n be the matrix with rows ϕ(τ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , ϕ(τm), ϕ(σ), and let b ∈ Zm+1 be the 12 Regular Separability in Büchi VASS column vector (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , 0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then clearly, Aπx ≤ b is equivalent to ⟨x, ϕ(σ)⟩ < 0 and ⟨x, ϕ(τ)⟩ ≤ 0 for each primitive π-cycle τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Inseparability flowers An inseparability flower is a structure in the Karp-Miller graph KM(V) as depicted to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It consists of a final state q and three cycles α, β, γ that all start in q and that meet the given conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' q α β γ δ(αβγ) ≥ 0 ϕ(αβ) ≥ 0 ϕ(αβγ) ∈ Q · ϕ(α) Let us give some intuition on why such a flower is the relevant structure to look for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' True to its name, an inseparability flower guarantees the existence of an inseparability witness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' a family of words accepted by the pumpable Büchi VASS V that escape every basic separator Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Such a family of words therefore needs an accepting run for each member, and the three conditions of the flower provide such runs: The first condition ensures that the three cycles actually correspond to a transition sequence enabled in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The second condition guarantees that for every x ∈ Nn, the x-weighted letter balance of α or of β is positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' unless they are both zero, in which case the third condition ensures that αβγ has x-weighted balance zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This allows us to construct, for each k, a run that escapes Sx,k for all x: By sufficiently repeating each cycle α, β, and γ, we obtain a run that for each x ∈ Nn, will either (i) have infixes with x-weighted balance > k, or (ii) attain some x-weighted balance infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Each of these properties rules out membership in Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5 proves this formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a Büchi VASS such that L(V) is pumpable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then the following are equivalent: (i) L(V) ̸ | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' (ii) There is a profile π ∈ Π(V) such that the system Aπx ≤ b has no solution x ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' (iii) There exists an inseparability flower in KM(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The decision procedure Before we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, let us see how to use it to decide separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given Büchi VASS V1 and V2, we can compute V so that L(V1) | L(V2) if and only if L(V) | Dn, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 tells us that L(Vpump) is pumpable and we have L(V) | Dn if and only if L(Vpump) | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, we can check whether L(Vpump) | Dn by checking the systems Aπx ≤ b for satisfiability: If there is a solution for every π ∈ Π(Vpump), then we have separability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' otherwise, we have inseparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since the systems Aπx ≤ b are constructed directly from KM(Vpump), we need to explicitly construct the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore our procedure may take Ackermann time, because Karp-Miller graphs can be Ackermann large [36, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Consider the instance of regular separability where our two inputs are the Büchi VASS V found in Figure 1(left), and another Büchi VASS accepting the language D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since we are already in the case of wanting to decide L(V) | D1, we can skip the first step of applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The second step is to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 and construct Vpump, which we have already done for this case in Figure 2(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now we have to construct KM(Vpump), which can be found in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are two relevant parts of KM(Vpump), where we can find cycles involving a final state: (1) the part on the right, where the state tuples contain ω twice and the counter value is 0, and (2) the part at the top with triple ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the following we will only write down the states, as the counter values and the other contents of the state tuples will be clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For part (1), the Büchi VASS Vpump has only a single profile π1 containing only the two edges between q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since each π1-cycle σ only consists of repetitions of the primitive cycle q1 0|ε −−→ q2 0|¯a1 −−−→ q1, we have ϕ(σ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore the system Aπ1x ≤ b trivially has a solution x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 13 ((q0, ω, 0), 1) ((q1, ω, 0), 1) ((q2, ω, 0), 1) ((q2, ω, ω), 0) ((q1, ω, ω), 0) ((q1, 0, 0), 0) ((q0, 0, 0), 0) ((q2, 0, 0), 0) ((q1, ω, 0), ω) ((q0, ω, 0), ω) ((q2, ω, 0), ω) ((q2, ω, ω), ω) ((q1, ω, ω), ω) e1|ε 0|ε 0|ε e1|ε 0|ε 0|ε −e1|a1 0|¯a1 0|ε e1|¯a1 e1|ε 0|ε 0|ε −e1|a1 0|ε 0|¯a1 −e1|a1 e1|¯a1 Figure 3 The Karp-Miller graph KM(Vpump) of the Büchi VASS Vpump from Figure 2(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here we did not mark the final states to reduce visual clutter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' every state that includes q1 is considered final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For similar reasons, we also only labelled the edges of the graph with letters and counter effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The proper edge labels would be full transitions of Vpump, including source and target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Regarding part (2), Vpump has exactly two more profiles: profile π2 containing only the two edges between q1 and q2, and profile π3, which additionally contains the two loop edges on q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The cycles of π2 look almost exactly like the cycles of π1 with only the counter values of the nodes in the graph being different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, the system Aπ2x ≤ b is the exact same system as Aπ1x ≤ b and also trivially has a solution x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For π3, we have as primitive cycles both the loop edges on q2 as well as the primitive cycle of π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To obtain a complete π3-cycle, we simply insert both loops into the π2-cycle at q2 forming the cycle σ = q1 0|ε −−→ q2 −e1|a1 −−−−→ q2 e1|¯a1 −−−→ q2 0|¯a1 −−−→ q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since σ contains all primitive cycles exactly once without overlap, it is automatically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We also have δ(σ) = 0, meaning σ is a cycle witnessing π3 as a profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus these cycles lead to the following system of inequalities Aπ3x ≤ b: 1 · x1 ≤ 0 loop 1 −1 · x1 ≤ 0 loop 2 −1 · x1 ≤ 0 π2-cycle −1 · x1 ≤ −1 complete π3-cycle Clearly this system has no solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' the first and last inequality are contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore we conclude regular inseparability for L(V) and D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' While not part of the decision procedure, for an inseparable instance of the problem as we have here, we can also find an inseparability flower in KM(Vpump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In this case we have α = q1 0|ε −−→ q2 0|¯a1 −−−→ q1, β = q1 0|ε −−→ q2 −e1|a1 −−−−→ q2 −e1|a1 −−−−→ q2 0|¯a1 −−−→ q1, and γ = q1 0|ε −−→ q2 e1|¯a1 −−−→ q2 e1|¯a1 −−−→ q2 0|¯a1 −−−→ q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This selection of cycles meets all the requirements of a flower: δ(αβγ) = 0, ϕ(αβ) = 0, and ϕ(αβγ) = −3 = 3 · ϕ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Inseparability flowers disprove separability The remainder of this section is devoted to proving Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The implication “(i)⇒(ii)” follows by applying Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 to KM(V), viewed as a Büchi automaton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' see Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For “(iii)⇒(i)”, we employ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6: 14 Regular Separability in Büchi VASS ▶ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If L(V) is pumpable and KM(V) has an insep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' flower, then L(V) ̸ | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose there is an inseparability flower α, β, γ in KM(V) and also L(V) | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6, there is a k ∈ N and a finite set X ⊆ Nn such that L(V) ⊆ � x∈X Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that for every x ∈ Nn, at least one of the following holds: ⟨x, ϕ(α)⟩ > 0, ⟨x, ϕ(β)⟩ > 0, or ⟨x, ϕ(αβγ)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' (1) Indeed, if ⟨x, ϕ(α)⟩ ≤ 0 and ⟨x, ϕ(β)⟩ ≤ 0, then ϕ(αβ) ≥ 0 implies that ⟨x, ϕ(α)⟩ = ⟨x, ϕ(β)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since ϕ(αβγ) = N · ϕ(α) for some N ∈ Q, we have ϕ(αβγ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Because of (1), the sequence αk+1βk+1γk+1 either has an infix χ with ⟨x, ϕ(χ)⟩ > k or we have ⟨x, ϕ(αk+1βk+1γk+1)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since δ(αk+1βk+1γk+1) ≥ 0, there is a run ρ such that ραk+1βk+1γk+1 is a run in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence, ρ(αk+1βk+1γk+1)ω is a run in V whose word cannot belong to Sx,k for any x ∈ Nn, contradicting L(V) ⊆ � x∈X Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ Constructing inseparability flowers It remains to show the implication “(ii)⇒(iii)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose there is a profile π ∈ Π(V) whose associated system of inequalities Aπx ≤ b is unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By Farkas’ Lemma, there exists a y ∈ Nm+1 such that y⊤Aπ ≥ 0 and y⊤b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' From this vector y, we now construct an inseparability flower in KM(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let σ be the complete π-cycle in KM(V) that was chosen to construct Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , τm be the primitive π-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since σ is complete, there is a vector r = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , rm) ∈ Nm so that r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , rm ≥ 1 and ∆(σ) = r1·∆(τ1)+· · ·+rm·∆(τm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, since σ contains every edge of π, we can wlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' write σ = σ0 · · · σm such that between σi−1 and σi, σ arrives in the initial state of τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The decomposition allows us to insert further repetitions of the primitive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , zm) ∈ Nm with z ≥ r, we define σz as σ0τ z1−r1 1 σ1 · · · τ zm−rm m σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then ∆(σz) = z1 ·∆(τ1)+· · ·+zm ·∆(τm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In particular, for s, t ≥ r, we have ∆(σsσt) = ∆(σs+t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that every transition in a Karp-Miller graph is labeled by a VASS transition, and so every transition sequence χ in KM(V) is labeled by a transition sequence in V, which we denote by trans(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We now define the transition sequences α, β, and γ as trans(σz) for suitable vectors z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For α, we take trans(σ), the transitions labeling the complete π-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Observe that σ = σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We proceed to define β = trans(σs) and γ = trans(σt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The choice of the vectors s and t has to meet the requirements on an inseparability flower: ϕ(αβ) ≥ 0, δ(αβγ) ≥ 0, and ϕ(αβγ) ∈ Q · ϕ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Step I: Building β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We will define s so that ϕ(αβ) = ϕ(σrσs) = ϕ(σr+s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The remaining two requirements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' δ(αβγ) ≥ 0 and ϕ(αβγ) ∈ Q · ϕ(α)) will be ensured with an appropriate choice of t in Step II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us now describe how to pick s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that y is the vector from the application of Farkas’ Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It can be understood as assigning a repetition count yi to every primitive cycle τi in the profile and a repetition count ym+1 to the complete π-cycle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since y⊤Aπ ≥ 0, and since our goal is to make ϕ(αβ) non-negative, we will use y to construct a vector ˆy = (ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , ˆym) ∈ Nm so that ϕ(σ ˆy) = y⊤Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The right definition is ˆyi := yi + ym+1 · ri for i ∈ [1, m], because y⊤Aπ = m � i=1 yi · ϕ(τi) + ym+1 · ϕ(σ) = m � i=1 (yi + ym+1ri)ϕ(τi) = ϕ(σ ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We now choose M ∈ N such that s = M · ˆy − r ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is possible since all entries in ˆy are positive, due to ym+1 > 0 by y⊤b < 0, and ri > 0 for all i by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then we have ϕ(αβ) = ϕ(σrσs) = ϕ(σr+s) = ϕ(σM·ˆy) = M · ϕ(σ ˆy) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 15 Step II: Building γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It remains to define t so that γ = trans(σt) satisfies δ(αβγ) = δ(σr+s+t) ≥ 0 and ϕ(αβγ) ∈ Q · ϕ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The idea is to choose t so that r + s + t is a positive multiple of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Such a choice is possible, because r has positive entries everywhere: We pick N ∈ N such that t := N · r − s − r ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then indeed δ(αβγ) = δ(σr+s+t) = δ(σN·r) = N · δ(σr) = N · δ(σ) ≥ 0 and ϕ(αβγ) = ϕ(σr+s+t) = ϕ(σN·r) = N · ϕ(σr) = N · ϕ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 6 One-dimensional Büchi VASS Our second contribution is the precise complexity of separability for the 1-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Regular separability for 1-dimensional Büchi VASS with binary encoded updates is PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the lower bound, we use a simple reduction from the disjointness problem L1 ∩L2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='= ∅ for finite-word languages of 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VASS [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, we also show that separability is PSPACE-hard even if the input languages are promised to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the upper bound, we rely on the results in Section 5, but need a modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There, to simplify the exposition, we first make the input language pumpable, which may incur an Ackermannian blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A closer look at the results, however, reveals that we can also check separability directly on the Karp-Miller graph of ¯V as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a Büchi VASS with L(V) ⊆ Σω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then L(V) ̸ | Dn if and only if KM(¯V) has an inseparability flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 allows us to phrase inseparability as the existence of a run in ¯V that satisfies certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that if V is 1-dimensional and over Σ1, then ¯V has two counters the second of which tracks the letter balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a 1-dimensional Büchi VASS with L(V) ⊆ Σω 1 and L(V) ∩ D1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then L(V) ̸ | D1 if and only if there exist states p, q, r with r final, and a run in ¯V as follows: (q0, 0, 0) ∗−→ σ1 � �� � (p, x1, y1) ∗−→ (p, x2, y2) ∗−→ σ2 � �� � (q, x3, y3) ∗−→ (q, x4, y4) ∗−→ α � �� � (r, x5, y5) ∗−→ (r, x6, y6) ∗−→ γ � �� � (r, x7, y7) � �� � β ∗−→ (r, x8, y8) (1) y3 < y4 and also (a) x3 ≤ x4 or (b) x1 < x2 and y1 ≤ y2 (2) y5 ≤ y7 (3) x5 ≤ x8 (4) if y5 = y6, then y5 = y8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Observe that an inseparability flower in KM(¯V) must carry ω in the second coordinate, meaning the letter balance is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Otherwise, it would yield an accepting run of ¯V, which cannot exist because L(V) ∩ D1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If the flower has ω in the second coordinate, we can construct a finite run as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The cycles σ1 and σ2 plus Condition 1 ensure that indeed the second coordinate becomes ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Condition 2 is ϕ(αβ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Condition 3 says δ(αβγ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, to express ϕ(αβγ) ∈ Q · ϕ(α), note that for integers a ∈ Q · b iff b = 0 implies a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Condition 4 expresses that y6 − y5 = 0 implies y8 − y5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In order to apply Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 for deciding L(V1) | L(V2) for 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V1, V2 with binary counter updates, we would like to follow the approach for the general case and use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 to first construct V so that L(V1) | L(V2) if and only if L(V) | D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' From V, we would then construct the 2-dimensional Büchi VASS ¯V that tracks the letter balance, and on ¯V we would then check the conditions of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The problem is that, under binary updates, the intermediary V may become exponentially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We use the fact that also ¯V has binary counters available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This allows us to directly construct a compact variant of ¯V: 16 Regular Separability in Büchi VASS ▶ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V1, V2 with binary updates, there is a a 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V with L(V1) ∩ L(V2) = ∅ iff L(V) ∩ D1 = ∅, L(V1) | L(V2) iff L(V) | D1, and we can construct in time polynomial in |V1| + |V2| the 2-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS ¯V (binary updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Detecting constrained runs in 2-VASS It remains to check for the existence of runs in ¯V as described in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, and to check whether L(V1) ∩ L(V2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Both of these problems reduce to what we call the constrained runs problem for 2-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that Presburger arithmetic is the first-order theory of (N, +, <, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We will use the existential fragment to express conditions on counter values of VASS like the ones from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The constrained runs problem is the following: Given A 2-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VASS V (with updates encoded in binary), a number m ∈ N, states q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , qm in V, a quantifier-free Presburger formula ψ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , xm, ym), and s, t ∈ [1, m], s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Question Does there exist a run (q0, 0, 0) ∗−→ (q1, x1, y1) ∗−→ · · · ∗−→ (qm, xm, ym) that visits a final state between (qs, xs, ys) and (qt, xt, yt) and satisfies ψ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , xm, ym)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 imply that if L(V1) ∩ L(V2) = ∅, then L(V1) | L(V2) reduces to the constrained runs problem on ¯V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, checking L(V1) ∩ L(V2) = ∅ reduces via a product construction to checking emptiness of a 2-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Such a 2-VASS has an accepting run iff (q0, 0, 0) ∗−→ (q, x, y) ∗−→ (q, x′, y′) with (x, y) ≤ (x′, y′) and q final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence, this problem also reduces to the constrained runs problem for 2-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We thus need to show: ▶ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The constrained runs problem for 2-VASS is solvable in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5, we show that if there is a constrained run, then there is one with at most exponential counter values along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For this, we use methods from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Complexity in higher dimension We leave open two natural questions: (i) What is the complexity of regular separability for Büchi d-VASS, for each d ≥ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' (ii) What is the complexity of regular separability for Büchi VASS (where the dimension is part of the input)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given that the regular separability and the disjointness problem usually (but not al- ways [32, 48]) coincide regarding decidability, we expect the complexity of regular separability to be PSPACE in every fixed dimension d and EXPSPACE in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The lower bounds follow from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1 for fixed d and from [14] (because disjointness is EXPSPACE- complete [22, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, it is not clear how to show the upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The clearest obstacle is that inseparability flowers involve a non-linear condition: The requirement ϕ(αβγ) ∈ Q · ϕ(α) is not expressible in Presburger arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are several generic results providing EXPSPACE upper bounds for detecting particular types of runs in VASS [18, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, the numerical properties directly expressible there are confined to Presburger arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The only reason we could obtain the PSPACE upper bound for d = 1 is that the non-linear condition degenerates into a linear condition in dimension one: It is equivalent to “ϕ(αβγ) = 0 or ϕ(α) ̸= 0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' References 1 Parosh Aziz Abdulla, Mohamed Faouzi Atig, Vrunda Dave, and Shankara Narayanan Krishna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' On the Separability Problem of String Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In Igor Konnov and Laura Kovács, editors, 31st 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1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1007/BFb0017144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 7 Pierre Chambart, Alain Finkel, and Sylvain Schmitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Forward analysis and model checking for trace bounded WSTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Comput.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=', 381(1-3):274– 279, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 9 Lorenzo Clemente, Wojciech Czerwinski, Slawomir Lasota, and Charles Paperman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Regular Separability of Parikh Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 117:1–117:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Separability of Reachability Sets of Vector Addition Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, volume 66 of LIPIcs, pages 24:1–24:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4230/LIPIcs.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Noncanonical extensions of bottom-up parsing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' SIAM Journal on Computing, 5(2), 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 48 Ramanathan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thinniyam and Georg Zetzsche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Regular Separability and Intersection Emptiness are Independent Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' of the 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019), volume 150 of LIPIcs, Dagstuhl, Germany, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Schloss Dagstuhl–Leibniz-Zentrum für Informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 49 Rüdiger Valk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Infinite behaviour of petri nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Theoretical computer science, 25(3):311–341, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 50 Joachim von zur Gathen and Malte Sieveking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A bound on solutions of linear integer equalities and inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proceedings of the American Mathematical Society, 72(1):155–158, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 51 Georg Zetzsche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Separability by piecewise testable languages and downward closures beyond subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In Anuj Dawar and Erich Grädel, editors, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' of the Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2018), pages 929–938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ACM, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1145/3209108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3209201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A Proof Details for Overview A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1 Proof of Part 1 of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 Here we proof the first part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2, which is the following: ▶ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There are Büchi VASS languages L1, L2 with L1 ∩ L2 = ∅ and L1 ̸ | L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We choose L1 = L(V), where V is the Büchi VASS in Figure 1(left), and L2 = D1, the Dyck language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that each w ∈ L(V) can be written as w = uv1v2 · · · with ϕ1(vℓ) < 0 for every ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This clearly implies L(V) ∩ D1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose w ∈ L(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that on q2, reading a1 decrements the counter and reading ¯a1 increments the counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, from a configuration (q2, x), a word v read in q2 can have balance ϕ1(v) at most x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' And moreover, if ϕ1(v) > 0, then this decreases the counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, in order to visit q1, the balance has to drop once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, between any two (not necessarily successive) visits to the final state q1, one of the following holds: (i) the counter strictly decreases or (ii) the input word v satisfies ϕ1(v) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since q1 is visited infinitely often, we can decompose w = uv1v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' such that after reading vℓ, we are in (q1, xℓ) and we have x1 ≤ x2 ≤ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then “(i)” cannot happen for any vℓ and thus we have ϕ1(vℓ) < 0 for every ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hence, the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It remains to show L(V) ̸ | D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Towards a contradiction, suppose there is a Büchi automaton A with n states such that L(V) ⊆ L(A) and L(A) ∩ D1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that V accepts (an 1 ¯an+1 1 )ω: We drive up the counter to n in q0 and then read each an 1 ¯an+1 1 in a loop from q1 to q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, a run of A must cycle on some non-empty infix of an 1 and thus, for some m > n, also accept w = (am 1 ¯an+1 1 )ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since w ∈ D1, that is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ The second part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 is proven in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given Büchi VASS V1 and V2, we can compute a Büchi VASS V over Σn so that L(V1) | L(V2) if and only if L(V) | Dn, where n is the dimension of V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the proof, we need the concept of rational transductions of infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Rational transductions A finite state Büchi transducer is a tuple T = (Q, Σ, Γ, E, q0, Qf) consists of a finite set of states Q, an input alphabet A, an initial state q0 ∈ Q, a set of final states Qf ⊆ Q, and a transition relation E ⊆ Q × (Σ ∪ {ε}) × (Γ ∪ {ε}) × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For a transition (q, a, b, q′) ∈ E, we also write q (a,b) −−−→ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The transducer T recognizes the binary relation T(T ) ⊆ Σω × Γω containing precisely those pairs (u, v) ∈ Σω × Γω, for which there is a transition sequence q0 (a1,b1) −−−−→ q1 (a2,b2) −−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' such that u = a1a2 · · · , v = b1b2 · · · , and for infinitely many i ∈ N, we have qi ∈ Qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We say that a relation T ⊆ Σω × Γω is rational if there is a finite-state Büchi transducer T with T = T(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For a language L ⊆ Γω and a relation T ⊆ Σω × Γω, we define TL = {u ∈ Σω | ∃v ∈ L: (u, v) ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, for relations T ⊆ Σω × Γω and S ⊆ Θω × Σω, we define S ◦ T = {(u, w) ∈ Θω × Γω | ∃v ∈ Σω : (u, v) ∈ S, (v, w) ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Using a simple product construction, we observe that for rational transductions S and T, the relation S ◦T is (effectively) rational as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By simply exchanging the two input coordinates, one can also show that if T ⊆ Σω × Γω is rational, then so is T −1 = {(u, v) ∈ Γω × Σω | (v, u) ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The following is also entirely straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A language L ⊆ Σω is a Büchi VASS language if and only if there exists a rational transduction T and a number n ∈ N such that L = TDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, the translation can be performed in exponential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, the automaton underlying an n-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS is translated into a transducer with input in Dn and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' More precisely, for h ∈ N an operation of +h on the ith counter is translated into the string (ai)h, whereas −h is translated into (¯ai)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The 0-vector is hereby translated into a1¯a1 instead of ε, to ensure that every infinite run of the Büchi VASS actually corresponds to an infinite word in Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The only reason why this construction is not feasible in polynomial time, is because we assume that counter operations of Büchi VASS are encoded in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In particular, the string (ai)h mentioned above takes h steps to write down, whereas the size of the Büchi VASS is only dependent on log h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, the construction only takes polynomial time, if counter updates are encoded in unary, or if strings such as (ai)h are subjected to some exponential compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We also need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The proof is exactly the same as the corresponding proof in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The only difference is that we have infinite instead of finite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let T ⊆ Σω × Γω be rational and L ⊆ Σω and K ⊆ Γω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then L | TK if and only if T −1L | K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 22 Regular Separability in Büchi VASS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose L ⊆ R and R ∩ TK = ∅ for some regular R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then clearly T −1L ⊆ T −1R and T −1R ∩ K = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, the regular set T −1R witnesses T −1L | K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Conversely, if T −1L | K, then K | T −1L and hence, by the first direction, (T −1)−1K | L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since (T −1)−1 = T, this reads TK | L and thus L | TK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ We are now ready to prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given Büchi VASS V1 and V2, where V2 is n-dimensional, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 allows us to compute in exponential time a rational transduction T such that L(V2) = TDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 again to construct a Büchi VASS V for T −1L(V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then we have L(V1) | L(V2) ⇐⇒ L(V1) | TDn ⇐⇒ T −1L(V1) | Dn ⇐⇒ L(V) | Dn, where the second equivalence is due to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ B Proof Details for Pumpability Let us formally define the constructions of ¯V and Vpump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V = (Q, q0, T, F) be a d-dimensional Büchi VASS over Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then ¯V = (Q, q0, ¯T, F) is the (d+n)-dimensional Büchi VASS over Σn with transitions constructed as follows: (q, ε, (δ, ϕ(w)), q′) ∈ ¯T if and only if (q, w, δ, q′) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, Vpump = (Qpump, qpump,0, Tpump, Fpump) is the d-dimensional Büchi VASS over Σn constructed as follows: Qpump = Q × (N ∪ {ω})d+n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' the states of KM(¯V), qpump,0 = (q0, 0), Fpump = F × (N ∪ {ω})d+n, and (qpump, w, δ, q′ pump) ∈ Tpump if and only if there is a transition (qpump, t, q′ pump) in KM(¯V) labelled by t = (q, ε, (δ, ϕ(w)), q′) ∈ ¯T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We split the first two parts of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 into Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3, which we prove separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The third part later follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6, which, in turn, follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L(Vpump) is pumpable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Consider some w ∈ L(Vpump) and some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let ρ be an accepting run of V over w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By construction of Lpump, there exists a corresponding run ¯ρ in KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let Ω ⊆ [d+1, d+n] be the set of coordinates where the states of KM(¯V) carry ω eventually during ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then at some point, ¯ρ visits an extended configuration (q, ¯m) ∈ Q × Nd+n ω where all coordinates from Ω in ¯m are ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Decompose ¯ρ = ¯ρ0¯ρ1 so that ¯ρ0 reaches (q, ¯m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let ρ = ρ0ρ1 and w = w0w1 be the corresponding decompositions of ρ and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then ρ0 reaches a configuration (q, m) ∈ Q × Nd in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let ℓ = maxi∈[1,n]{0, −ϕi(w0)} + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By the construction of Karp-Miller graphs, there exists a run ¯ρ′ 0 in ¯V that reaches a configuration (q, ¯m′) ∈ Nd+n such that ¯m′(i) ≥ m(i) for i ∈ [1, d + n] \\ Ω, and ¯m′(i) > m(i) + ℓ for i ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then ¯ρ′ 0 corresponds to a run ρ′ 0 in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It reaches a configuration (q, m′) with m′ ≥ m and thus ρ′ 0ρ1 is a run of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It reads a word w′ 0w1 ∈ Lpump, where w′ 0 is the prefix read by ρ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since w′ 0 was also read by ¯ρ′ 0 in ¯V, it is a prefix of some word in Dn, as mandated by the additional counters of ¯V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that now ϕ(w′ 0) ≥ ϕ(w0) and for every i ∈ [1, n] where w ever becomes negative, we have ϕi(w′ 0) ≥ max{ϕi(w0), 0} + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The first condition follows from the fact P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 23 that ϕi(w′ 0) = m′(d + i) ≥ m(d + i) = ϕi(w0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the second condition, note that if w ever becomes negative in coordinate i, then ¯ρ must necessarily visit a configuration where in coordinate i, there is an ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In particular, we have d + i ∈ Ω and thus ϕi(w′ 0) = m′(d + i) ≥ m(d + i) + ℓ = ϕi(w0) + maxi∈[1,n]{0, −ϕi(w0)} + k ≥ max{ϕi(w0), 0} + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ ▶ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' There exists a k ∈ N such that L(Vpump) ⊆ L(V) ⊆ L(Vpump) ∪ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The inclusion L(Vpump) ⊆ L(V) is obvious from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the second inclusion, define k ∈ N to be the largest number occurring in the states of KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that then L(V) ⊆ Pk ∪ L(Vpump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let w ∈ L be accepted by a run ρ in V and suppose w /∈ Pi,k for some i ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If u is a prefix of w, then we say that i ∈ [1, n] is crossing at u if ϕi(u) < 0 and ϕi(v) ≥ 0 for every prefix v of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Observe that whenever i is crossing at u, then ϕi(v) > k for some prefix v of u: Otherwise, w would belong to Pi,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This implies that ρ has a corresponding run in KM(¯V): Whenever a counter in [d + 1, d + n] drops below zero, it must have been higher than k before and thus been set to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, w is also accepted by KM(¯V) and thus w ∈ L(Vpump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ C Proof Details for Basic Separators Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' First of all, if Lπ(A) is empty, then Condition (i) trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, in the following we assume that Lπ(A) ̸= ∅ and in particular that the final state qπ associated with the profile π is reachable from A’s initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We want to set up a system of linear inequalities that has a solution x if and only if there is a k such that Lπ(A) ⊆ Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore let us talk about some requirements that are necessary for the above inclusion to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' These requirements will be on cycles of transitions in π, and we make sure that they can be expressed as linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The cycle σπ that contains exactly the transitions in π has to be over a word vπ ∈ Σ∗ n with ⟨x, ϕ(vπ)⟩ ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Otherwise, we can just repeat σπ infinitely often and prepend any prefix leading to qπ from A’s initial state, yielding a word that violates requirement b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') of Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Any primitive cycle σ in A of transitions in π has to be over a word v with ⟨x, ϕ(v)⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Otherwise we repeat σπ infinitely often from some arbitrary prefix reaching qπ like before, and then perform k + 1 insertions of the cycle σ into each copy of σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This yields a word that violates requirement a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=') of Sx,k, and we can do this for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We can now use these requirements on cycles to construct a linear system of inequalities Aπx ≤ b for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the cycle σπ corresponding to word vπ ∈ Σ∗ n, we add the inequality x1ϕ1(vπ) + x2ϕ2(vπ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' + xnϕn(vπ) ≤ −1, and for each primitive cycle σ of transitions in π over a word v ∈ Σ∗ n, we add the inequality x1ϕ1(v) + x2ϕ2(v) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' + xnϕn(v) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us argue that the precise choice of the justifying cycle σπ does not affect the satisfiability of the system Aπx ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To this end we argue that x ∈ Nn is a valid solution to the system if and only if (1) all primitive cycles have an x-weighted balance at most zero, and (2) at least one primitive cycle has a strictly negative x-weighted balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Constraint (1) is clearly equivalent to the inequalities added for each primitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For constraint (2), assume that the inequality ⟨x, ϕ(vπ)⟩ ≤ −1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now observe that any valid choice of σπ is a cycle and therefore can be constructed by inserting primitive cycles into each other a finite number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If all primitive cycles had non-negative x-weighted balance, then the x-weighted balance for σπ could not be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 24 Regular Separability in Büchi VASS For the other implication direction, assume that constraint (2) holds, and let the primitive cycle with negative x-weighted balance be σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since any valid choice of σπ contains each transition in π, its |π|-fold repetition σ| ππ contains each primitive cycle as a (possibly non-contiguous) subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now, if we delete σ′ from σ| ππ, the remaining (possibly not connected) transition sequences still combine to form a collection of cycles, since σ′ is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, the summed-up x-weighted balance of this collection is the sum of x-weighted balances of primitive cycles, and can therefore be at most zero by Condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then adding σ′ back in gives us that ⟨x, ϕ(v| ππ|)⟩ is negative, and therefore ⟨x, ϕ(vπ)⟩ is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since the letter balance can only have integer-values, and weighting by x ∈ N does not change this, it follows that ⟨x, ϕ(vπ)⟩ ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The characterization of solutions x via constraints (1) and (2) is clearly independent of σπ, meaning the precise choice of the latter does not affect satisfiability of the system Aπx ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, the restriction of x ∈ Nd is not a meaningful one, as we can always compute a solution in Qd from one in Nd, as we explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Applying Farkas’ Lemma (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1) to this system of equations, we either obtain a vector x ∈ Qn ≥0 as a suitable solution, or we obtain a vector y ∈ Qm ≥0 with y⊤Aπ ≥ 0⊤ and y⊤b < 0, where m is the number of rows of Aπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the first case, we can multiply the entries of x by their denominators’ least common multiple, say ℓ, to yield a suitable vector ℓ · x = x′ ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore we set k = |Qπ| · h, where |Qπ| ⊇ {qπ} is the set of all states of A adjacent to transitions in π, and h is the length of the longest word appearing as a transition label of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With this we can show that Lπ(A) ⊆ Sx′,k: Each word w ∈ Lπ(A) decomposes into uv with v = v0v1v2 · · · such that u leads to qπ from A’s initial state and each vj corresponds to some cycle σj on qπ, that contains each transition of π at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then we have ⟨x′, ϕ(vj)⟩ = ℓ·⟨x, ϕ(vj)⟩ < ℓ·0 = 0 as required by Sx,k: each cycle σj can be obtained by starting with σπ, which contributes at most −1 to this value, and inserting finitely many primitive cycles, which all add at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, we need to show ⟨x′, ϕ(f)⟩ ≤ k for every infix f of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Towards a contradiction assume there is at least one infix f of v, for which this does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since f fulfils ⟨x′, ϕ(f)⟩ > |Qπ| · h, and h is the maximum length of a transition label, the transition sequence corresponding to f has to be longer than |Qπ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus this sequence repeats a state and therefore has to contain a primitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, all such primitive cycles add at most 0 to the value ⟨x′, ϕ(f)⟩, meaning one could delete the word corresponding to this cycle from f and still fulfil the aforementioned requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' One can repeatedly remove primitive cycles until one obtains a word f ′ of length |f ′| ≤ |Qπ| · h with ⟨x′, ϕ(f ′)⟩ > |Qπ| · h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is a contradiction, therefore infixes such as f cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the other case we also multiply y with the least common multiple of its entries, say ℓ, to yield ℓ · y = y′ ∈ Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, each row of the matrix Aπ essentially contains the ϕ-values of its corresponding cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The requirement y′⊤Aπ = ℓy⊤Aπ ≥ ℓ · 0⊤ = 0⊤ can then be seen as a selection of cycles, whose combined ϕ-values are all 0 or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, the requirement y′⊤b = ℓy⊤b < ℓ · 0 = 0 ensures that σπ is selected at least once, because all other entries of b are 0, meaning we would have y⊤b = 0 if σπ was not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This means we can combine all the selected cycles into one large cycle σ′ via matching states, which is possible because σπ visits all states in Qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since the combined ϕ-values of all the cycles selected by y are 0 or above, we have that σ′ corresponds to a word w′ with ϕ(w′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, σ′ also contains all transitions of π as required, because it contains the cycle σπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ Regarding Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5 We mentioned in 4 that a single value of k is sufficient for a finite union of basic separators Pi,k and Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is because we have Pi,k ⊆ Pi,k+1 and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 25 Sx,k ⊆ Sx,k+1 for each i ∈ [1, n], x ∈ Nn, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore it suffices to show the following: Let A be a Büchi automaton with L(A) = R ⊆ Σω n and R ∩ Dn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then there is a finite set X ⊆ Nn and a number k ∈ N such that R ⊆ � i∈[1,n] Pi,k ∪ � x∈X Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here R is a separator candidate in the sense of the original phrasing of the theorem, because it is ω-regular and disjoint from Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We begin by invoking Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 on A to obtain a Büchi auto- maton Apump, whose language is pumpable, and a number ℓ such that L(Apump) ⊆ L(A) ⊆ L(Apump) ∪ Pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Using the theorem this way is feasible, because Büchi automata can be seen as 0-dimensional Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since L(A) ∩ Dn = ∅ and L(Apump) ⊆ L(A) we have L(Apump) ∩ Dn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It now suffices to show that the basic separators theorem holds for L(Apump): If there are X, k such that L(Apump) ⊆ � i∈[1,n] Pi,k ∪ � x∈X Sx,k then L(A) ⊆ L(Apump) ∪ Pℓ ⊆ � i∈[1,n] Pi,o ∪ � x∈X Sx,o, where o = max(k, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now consider the decomposition L(Apump) = � π∈Π(Apump) Lπ(Apump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If we can show that each language Lπ(Apump) is contained in a finite union of basic separators, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the following let us fix a profile π of Apump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We now invoke Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 on Apump and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If Condition (i) holds, then this already yields x, k such that Lπ(Apump) ⊆ Sx,k, and we need not concern ourselves with the languages Pi,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the other case, Condition (ii) yields a cycle c′ in Apump that contains all transitions in π and is over a word w′ with ϕ(w′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since Condition (i) did not hold, we know that Lπ(Apump) is not empty, which means that all states adjacent to transitions of π are reachable from A’s initial state, including the final state qπ associated with π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let u′ be a word that reaches qπ from Apump’s initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then ˜w = u′(w′)ω ∈ L(Apump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let m be the lowest value of ϕi for any index i and prefix of ˜w, formally m = mini∈[1,n],v∈prefix( ˜ w) ϕi(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since ϕ(w′) ≥ 0 we know that m ∈ Z is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, since L(Apump) is pumpable, there is a decomposition ˜w = u0w1 and a word v0 ∈ Σ∗ n such that v0w1 ∈ L(Apump), ϕ(v0) ≥ ϕ(u0), and ϕi(v0) ≥ ϕi(u0) + |m| for all indices i where there is a v ∈ prefix( ˜w) with ϕi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then swapping u0 for v0 in ˜w can only increase the ϕi-values of its prefixes, and in fact all such values that fell below 0 are now raised above 0 by choice of |m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This means that v0w1 ∈ L(Apump) ∩ Dn, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ D Proof Details for Decidability ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If L ⊆ Σω n is pumpable, then L | Dn if and only if L |limDn, where L |limDn means L ⊆ � x∈X Sx,k for some finite set X ⊆ Nn and some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The “if” direction is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Conversely, let L | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5, we have L ⊆ � i∈[1,n] Pi,k ∪ � x∈X Sx,k for some finite X ⊆ Nn and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We claim that L ⊆ � x∈X Sx,k, which yields L |limDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Indeed, given u ∈ L, pumpability yields a word u′ ∈ L such that u′ ∼ u and u′ /∈ Pi,k for any i ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since u′ ∈ L ⊆ Pk ∪� x∈X Sx,k, we conclude u′ ∈ Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Finally, observe that membership in Sx,k is not affected by changing a finite prefix of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore, we also have u ∈ Sx,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ ▶ Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let π ∈ Π(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If Aπx ≤ b for x ∈ Nn, then Lπ(V) ⊆ Sx,k for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We regard KM(V) as a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then, π is in particular a profile for KM(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, the cycle witnessing that π is a profile is also an admissible cycle for π in KM(V) as a Büchi automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 implies Lπ(V) ⊆ Lπ(KM(V)) ⊆ Sx,k for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ 26 Regular Separability in Büchi VASS E Proof Details for One-dimensional Büchi VASS E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1: PSPACE-hardness We begin with the straightforward reduction from intersection emptiness of finite-word languages of 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Suppose L1, L2 ⊆ Σ∗ are finite-word languages of 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VASS with succinct counter updates, and acceptance by final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Checking whether the intersection L1 ∩ L2 is empty is PSPACE-complete [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We construct Büchi VASS for L1#ω and L2#ω, where # is a fresh letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since L1 and L2 are coverability languages of finitely-branching WSTS, we know from [14, Theorem 7] that L1 ∩ L2 = ∅ if and only if L1 | L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, with a fresh letter #, it is easy to observe that L1 | L2 if and only if L1#ω | L2#ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Hardness for disjoint languages In the PSPACE-hardness proof above, one can notice that the languages L1#ω and L2#ω are regularly separable if and only if they are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In order to further highlight the disparity between the finite-word case of WSTS languages (where disjointness and separability coincide [14]) and the infinite-word case, we want to present a proof that PSPACE-hardness already holds if the input languages are promised to be disjoint: Note that with this promise, separability in the finite-word case becomes trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, we reduce directly from configuration reachability in bounded one-counter automata, which was shown to be PSPACE-hard in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A bounded one-counter automaton B = (VB, b) consists of a 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VASS VB equipped with a bound b ∈ N on its counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This means transitions of B are enabled if and only if they meet the firing restrictions of a VASS and also lead to a configuration (q, m) with m ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, counter values and the bound b are encoded in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In particular, the size of B is that of the underlying VASS plus log b, and the size of a configuration (q, m) is log m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now we want to construct two 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V1 and V2, whose languages are always disjoint, but are ω-regular separable if and only if (q, m) is not reachable from (q0, 0) in B = (VB, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let T be the set of transitions of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We use Σ = T ∪ {#} ∪ Σ1 as the alphabet for V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let VD1 be the 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS accepting D1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' VD1 consists of a single state, both initial and final, with two loops e1|a1 and −e1|¯a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore let VS be the 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS from Figure 1(left) accepting the language S with S ∩D1 = ∅ but S ̸ | D1, which we talked about in Section 3 (see the proof of the first statement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We start constructing V1 by using a copy of VB with all states being non-final and every transition t ∈ T labelled with t itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then we add a copy of VD1 with its only state still being final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To connect the two copies, we add the transition −m|# from state q of VB to the initial state of VD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For V2 we also start with a copy of VB with all non-final states and transitions labeled with themselves, but we also invert every transition effect, changing it from z ∈ Z to −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then we add a new initial state q′ 0 with the same outgoing transitions as the initial state of VB, except we change their original effects z to b − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' These new transitions of q′ 0 are labelled with their original copies from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Additionally, we add a copy of VS with q2 still being a final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The two copies are then connected with a transition m − b|# from q to the initial state of VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If (q, m) = (q0, 0), we also add a transition 0|# from q′ 0 to the initial state of VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now let R1 be the set of all transition sequences over T that cover (q, m) in VB and do not necessarily respect the bound b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Formally, ρ ∈ R1 if ρ leads from (q0, 0) to (q, m′) in VB for some m′ ∈ N with m′ ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover let R2 be the set of all transition sequences over T that reach q with a counter value below m, when respecting the upper bound b, but not P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 27 necessarily the lower bound 0 of VASS counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Formally, ρ ∈ R2 if ρ leads from (q0, 0) to (q, z′) in B′ for some z′ ∈ Z with z′ ≤ m, where B′ = (V′, b) and V′ is just V interpreted as a Z-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We now want to argue that there are languages L1, L2 such that L(V1) = R1#L1 and L(V2) = R2#L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' L(V1) = R1#L1 is easy to see, since V1 simulates VB faithfully, and can only read # if a configuration (q, m) or greater is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For L(V2) = R2#L2 observe that before reading #, V2 essentially simulates VB with inverted counter values, starting with b instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since V can go above b, this essentially simulates going below 0 in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The # can also only be read if in q the counter is valued at least m − b, which corresponds to at most m before inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let us now show that L(V1) ∩ L(V2) = ∅, and furthermore L(V1) | L(V2) if and only if (q0, 0) → (q, m) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The configuration (q, m) not being reachable in B is equivalent to R1 and R2 being disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In this case L(V1) and L(V2) also have to be disjoint, since the prefixes before the ’#’ of their words cannot coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' They are however ω-regular separable: With Q being the states of B, an exponential size Büchi automaton A with states Q × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , b} can simulate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To make A accept all words with prefixes in R1, we add a final state with loops on all input letters, that is reachable by every transition that would make the counter value go above b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now L(V1) ⊆ L(A) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since transition sequences that do not respect the bound b cannot be prefixes of elements of R2, L(A) ∩ L(V2) = ∅ immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, L(A) is an ω regular separator for L(V1) and L(V2), which also means that they are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If (q, m) is reachable in B, we have a finite transition sequence ρ ∈ R1 ∩ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Reading ρ then leads to (q, m) in V1, respectively to (q, b − m) in V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore if # is read right after, the counter value of either Büchi VASS would be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This implies that ρ#D1 ⊆ L(V1) and ρ#S ⊆ L(V1), as the second component of either VASS would be simulated faithfully after this prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A regular separator A for L(N1) and L(N2) would therefore have to accept all words in ρ#D1 but no words in ρ#S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By adding a new initial state qinit to A and adding all outgoing transitions of states reachable via ρ# in the original A to qinit, we obtain an ω-regular separator for D1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is a contradiction, since we established earlier that these languages are not ω-regular separable as shown in the proof of the first half of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It remains to show that L(V1) ∩ L(V2) = ∅ in the case where (q, m) is reachable in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For transition sequences ρ over T, we know that ρ ∈ R1 ∩ R2 if and only if ρ reaches (q, m) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Therefore two words w1 ∈ L(V1) and w2 ∈ L(V2) can only agree on a prefix ρ#, if ρ has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, in this case w1 = ρ#w′ 1 for some w′ 1 ∈ D1 and w2 = ρ#w′ 2 for some w′ 2 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This yields w1 ̸= w2 since D1 and S are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2 ▶ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a Büchi VASS with L(V) ⊆ Σω n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then L(V) ̸ | Dn if and only if KM(¯V) has an inseparability flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We first invoke Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='7 to obtain Vpump with L(Vpump) ̸ | Dn if and only if L(V) ̸ | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that Vpump was constructed as the product of V and KM(¯V), which means that every cycle of Vpump is also a cycle of KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the if direction, we get that KM(Vpump) contains an inseparability flower by The- orem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Its three cycles then correspond to three cycles of Vpump, which then also appear in KM(¯V), where they still fulfill the requirements of an inseparability flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For the only if direction, observe that KM(Vpump) is essentially the product construction of KM(V) and KM(¯V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Furthermore, any transition sequence permitted by ¯V is also permitted 28 Regular Separability in Büchi VASS by V, as the former only added restrictions in the form of more counters, but did not remove any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus, each cycle of KM(¯V) (including the ones that make up its inseparability flower) also appears as a cycle in KM(Vpump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This implies that KM(Vpump) also has an inseparability flower, and by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 it follows that L(Vpump) ̸ | Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 ▶ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V1, V2 with binary updates, there is a a 1-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS V with L(V1) ∩ L(V2) = ∅ iff L(V) ∩ D1 = ∅, L(V1) | L(V2) iff L(V) | D1, and we can construct in time polynomial in |V1| + |V2| the 2-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Büchi VASS ¯V (binary updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Recall that ¯V is constructed from a Büchi VASS V over alphabet Σn by adding n additional counters and for each transition t replacing its label w ∈ Σ∗ n with ε and instead adding to t an effect of ϕ(w) on the n additional counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A precise definition can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To now proof Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 we have to argue that we can modify Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 to directly construct ¯V instead of V, and that the modified version is feasible in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If we analyze the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 in Appendix A then we obtain exponential time complexity for this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The bottleneck here is Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, we already mentioned in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2, that its associated complexity shrinks from exponential to polynomial time, if we can somehow compress the exponentially long transition labels that we end up with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' An adequate compression for this is replacing an exponentially long string w ∈ Σ∗ n by its effect on the letter balance δ(w), which is exactly what we do when going from V to ¯V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since we encode δ(w) in binary, this is an exponential compression, and therefore the time complexity of constructing ¯V directly is only polynomial, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that for two 1-dimensional Büchi VASS as input, we have n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' But our proof shows that constructing ¯V in polynomial time would still be feasible for Büchi VASS of arbitrary dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='4 Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5 ▶ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The constrained runs problem for 2-VASS is solvable in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We show that if there is a constrained run, then there is one where all counters have at most exponential values along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For this, we rely on a result from [5] about linear path schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A linear path scheme (LPS) for a 2-dimensional VASS V is a regular expression of the form S = σ0λ1σ1 · · · λnσn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Its alphabet is the set T of transition of V, and each infix λi corresponds to a cycle of transitions in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Each LPS S induces a reachability relation →S over configurations of V, where (q, x, y) →S (q′, x′, y′) if and only if there are numbers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , xn ∈ N such that σ0λx1 1 σ1 · · · λxn m σn is a run of V from (q, x, y) to (q′, x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1], it is shown that for any two states q, q′ in a 2-VASS V, there exists a set S of LPSs, each of which is of polynomial length, such that for x, y, x′, y′ ∈ N, (q′, x′, y′) is reachable from (q, x, y) if and only if (q, x, y) →S (q′, x′, y′) for some S from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In [5], this yields a PSPACE algorithm for configuration reachability in 2-dimensional VASS: If there is run reaching a certain configuration, then there is one of the form σ0λx1 1 σ1 · · · λxn n σn for some LPS σ0λ1σ1 · · · λnσn of polynomial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now the fact that σ0λx1 1 σ1 · · · λxn n σn is a run between two given configurations can be expressed using a set of linear inequalities P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 29 over x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Since each solvable polynomial-sized set of linear inequalities has a solution with at most exponential entries, this yields a run where all counters are at most exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We only need to extend this argument from [5] slightly: First, we want to guess a system of linear inequalities, whose solutions would satisfy the Presburger formula ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To this end, we view ψ as a propositional formula by treating each atomic formula as a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For Presburger arithmetic, an atomic formula is either an equality t1 = t2 or an inequality t1 < t2, where t1, t2 are additive terms over variables and/or the constants 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With this propositional view of ψ, we can guess an assignment to its propositions, and verify that its a satisfying assignment, feasible in polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If this assignment sets an atomic formula of the form t1 = t2 to false, this means that t1 < t2 or t2 < t1 has to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Similarly if t1 < t2 is set to false then t1 = t2 or t2 < t1 has to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In both cases, we simply guess one of the two atomic formulas that have to hold instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With these guesses together with the unchanged atomic formulas that were set to true, we obtain a system of equalities and inequalities, whose solutions would satisfy ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Formally, this system is comprised of matrices A ∈ Zℓ×m, C ∈ Zk×m and vectors b ∈ Zℓ, d ∈ Zk with entries encoded in binary, such that x ∈ Nm is a solution if and only if Ax < b and Cx = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In fact unary encodings would suffice for our definition of Presburger, since an entry of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 3 would have come from a term of the form y + y + y for a variable y, meaning all entries are polynomial in the size of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, we do not require unary encodings and can also work with binary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now we only need to check that there is a constrained run (q0, 0, 0) ∗−→ (q1, x1, y1) ∗−→ · · ∗−→ (qm, xm, ym), whose counter values indeed satisfy these equalities and inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This is the case if Az < b and Cz = d, where z = (x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , xm, ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If such a constrained run exists, then for each i ∈ [1, m], there is an LPS for the part (qi−1, xi−1, yi−1) ∗−→ (qi, xi, yi) such that said run conforms to each of these LPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By imposing (a) the linear inequalities of [5], which make sure that all counters stay non-negative, and (b) our linear inequalities Az < b and equalities Cx = d, we obtain a new (poynomial-size) system of linear inequalities over the exponents in the LPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' By [50] systems like these have minimal solutions with at most exponential entries, yielding an overall run with at most exponential counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Binary encoding then means that these solutions only take up polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' More specifically, this implies that we can simply guess configurations (q1, x1, y1) to (qm, xm, ym) of the constrained run in PSPACE, and then check that equalities and inequalities of our system hold for them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' that they are actual solutions to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This concludes the description of our decision procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' As a final remark, note that [50] assumes inequalities of the form t1 ≤ t2 rather than t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' However, since we seek solutions in Nm, we can simply express t1 < t2 as t1 + 1 ≤ t2 to circumvent this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀ F Regular Separability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Intersection In this section we prove the second part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To this end we present a class of WSTS such that, for their ω-languages, intersection is decidable whereas regular separability is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A d-dimensional reset Büchi VASS over alphabet Σ is a tuple V = (Q, q0, T, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The only difference to Büchi VASS is in the finite set of transitions which, besides adding a vector, may reset a counter, T ⊆ Q×(Zd ∪{r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , rd})×Σ∗ ×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The configurations are defined like for Büchi VASS, but the transition relation has to be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We have (q, m) w −→ (q′, m′) if there is a transition (q, x, w, q′) such that either (i) x ∈ Zd and m′ = m + x or (ii) x = ri for some i ∈ [1, d] and m′(j) = m(j) for j ∈ [1, d] \\ {i} and m′(i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Acceptance is defined as before, and so is the language (of infinite words) L(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' 30 Regular Separability in Büchi VASS q0 q q1 q2 q3 ε 0|a1 0|¯a1 ε 0|ε 0|¯a1 −e1 + ed+1|a1 e1 − ed+1|¯a1 Vε Figure 4 Weak Büchi reset VASS V′ in the proof of Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For general Büchi reset VASS, emptiness and intersection are undecidable [37, Theorem 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We consider a slight restriction of the model that makes the problems decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A Büchi reset VASS is weak if there is no path from a final state to a reset transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In particular, an accepting run can only perform finitely many resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that the usual product construction of V1 and V2 to yield a Büchi reset VASS for L(V1) ∩ L(V2) preserves weakness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For weak Büchi reset VASS, emptiness (hence intersection) is decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Here, emptiness can be decided using standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We order the configurations Q×Nd in the usual way: We have (q, m) ≤ (q′, m′) if q = q′ and m ≤ m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' First, one observes that for any Büchi VASS V, the set U(V) ⊆ Q × Nd of all configurations (q, m) from which an infinite accepting run can start, is upward closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, using a saturation procedure, we can effectively compute the finitely many minimal elements (q1, m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , (qℓ, mℓ) of U(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The details can be found in Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3 at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Then, for a weak Büchi reset VASS V, we do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We construct the Büchi VASS V′, which is obtained from V by deleting all reset transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Now L(V) is non-empty if and only if V, as a reset VASS, can cover any of the configurations (q1, m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' , (qℓ, mℓ) of U(V′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Whether the latter is the case can be decided because coverability is decidable in reset VASS [19, 25, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For weak B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' reset VASS over Σ1, regular separability from D1 is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We reduce from the place boundedness problem for reset VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' A reset VASS is a Büchi reset VASS without input words and without final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' For k ∈ N, we say that a reset VASS V is k-place bounded if for every reachable configurations (q, m), we have m(1) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Moreover, we call V place bounded if V is k-place bounded for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The place boundedness problem then asks whether a given reset VASS is place bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The place boundedness problem (more generally, the boundedness problem) for reset VASS is known to be undecidablei [19, Theorem 8] (for a simpler proof, see [37, Theorem 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Given a d-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' reset VASS V, we build a (d + 1)-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' weak Büchi reset VASS V′ with L(V′) = {w ∈ S1,k | k ∈ N, V can reach some (q, m) ∈ Q × Nd with m(1) ≥ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' (2) Before we describe V′, observe that L(V′) | D1 iff V is place bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If V is k-place bounded, then L(V′) ⊆ S1,k and thus L(V′) | D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' On the other hand, if V is not place bounded, then L(V′) = � k∈N S1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' As for the Büchi VASS in Figure 1 (left), one can show L(V′) ̸ | D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The construction is depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The dashed box contains Vε, which is obtained from V by changing every transition (p, u, q) into (p, (u, 0), ε, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the figure, q stands for arbitrary states of Vε, meaning for every state q in Vε, we have a transition (q, 0, ε, q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Observe that in the states q1, q2, q3, V′ behaves exactly like the Büchi VASS in Figure 1(left), except that the additional counter ensures that for each infix the balance on letter ai is bounded by k from configurations (q1, (k, u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Thus the accepted language from (q1, (k, u)) is exactly S1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This shows that V′ accepts the language (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ▶ Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Let V be a Büchi VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We can compute the set U(V) of minimal configura- tions from which there is an infinite accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Baumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Meyer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Zetzsche 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' It is decidable whether a given Büchi VASS has an accepting run [21, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We strengthen this result to checking whether a given Büchi VASS V has an accepting run starting in a downward-closed set of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The downward-closed set is given as a finite union I of ideals, each represented by a generalized configuration (q, m) ∈ Q×(N∪{ω})d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The algorithm is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We construct an instrumented Büchi VASS VI from V and I that starts in a gadget for I from which it moves to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This gadget selects one of the ideals, say (q, m), and increments each counter c to at most m(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Note that m(c) may be ω, in which case we may put an arbitrary value to this counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' After this initial phase, VI moves to state q of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The states in the gadget are not accepting, so VI will eventually move to V to obtain an infinite accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' To be precise, we have in V an accepting run from a configuration in I if and only if VI has an accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' With this, we can saturate a set of markings S, initially S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We repeatedly ask for an accepting run starting in a downward-closed set of configurations represented by a set of ideals I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' Initially, we just ask for any run, I = Q × {ωd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If such a run does not exist, we return S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' If such a run exists, we can reconstruct a configuration (q, m) ∈ I, m ∈ Nd, with which VI moved from the gadget for I to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' This can be done with an enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' We add (q, m) to S and refine the downward-closed set represented by I by subtracting the upward-closure of the new S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The subtraction can be computed effectively and yields a new set of ideals with which we repeat the check of an accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' The process terminates: the set S represents an upward-closed set of configurations, and every infinite sequence of such sets becomes stationary due to the wqo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' In the moment when the set becomes stationary, we will no longer find an accepting run and return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FIT4oBgHgl3EQfXCt6/content/2301.11242v1.pdf'} +page_content=' ◀' metadata={'source': 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Not. R. Astron. Soc. 000, 1–5 (2019) +Printed 10 January 2023 +(MN LATEX style file v2.2) +Angular Momentum Transfer in QPEs from Galaxy Nuclei +Andrew King1,2,3,⋆ +1 Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK +2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands +3 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, Netherlands +⋆ E-mail: ark@astro.le.ac.uk +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +A suggested model for quasi–periodic eruptions (QPEs) from galaxy nuclei invokes a +white dwarf in an eccentric orbit about the central massive black hole. I point out +that the extreme mass ratio allows the presence of strong Lindblad resonances in +the accretion disc. These are important for the stability of mass transfer, and may +trigger the eruptions themselves by rapidly transferring angular momentum from the +accretion disc (which is likely to be eccentric itself) to the orbiting WD companion at +pericentre. I consider the effects of von Zeipel–Lidov–Kozai (ZLK) cycles caused by +a perturber on a more distant orbit about the central black hole. I show that ZLK +cycles can change the orbital periods of QPE systems on timescales much shorter than +the mass transfer time, as seen in ASASSN-14ko, and produce correlated short–term +variations of their mass transfer rates and orbital periods, as recently observed in +GSN 069. Further monitoring of these sources should constrain the parameters of any +perturbing companions. This in turn may constrain the nature of the events creating +QPE systems, and perhaps give major insights into how the central black holes in +low–mass galaxies are able to grow. +Key words: galaxies: active: black hole physics: X–rays: galaxies +1 +INTRODUCTION +X–ray observations of the nuclei of low–mass galaxies +show that several of them produce quasi–periodic eruptions +(QPEs: Miniutti et al., 2019; Giustini et al. 2020; Song et al. +2020; Arcodia et al. 2021; Chakraborty et al. 2021; Payne +et al., 2021, 2022). Typically these sources have outbursts +by factors ∼ 100 in X–rays, which recur in roughly pe- +riodic fashion. The recurrence times currently range from +a few hours up to ∼ 100 d or ∼ 1 yr, and it is likely +that these limits will widen as data accumulate from di- +rect observations and archival searches. The X–rays have +ultrasoft blackbody spectra and luminosities implying typ- +ical radii ≳ Rg = GM1/c2 = 6 × 1010m5 cm, of order the +gravitational (Rg) and ISCO radii of a black hole of mass +M1 ∼ 105m5M⊙. These are consistent with the massive +black holes (MBHs) one might expect in these low–mass +galaxy nuclei. +In (King, 2020) I suggested a simple model for the first +known QPE source GSN 069. This postulated a low–mass +star (with mass M2 ≪ M1) – found from the requirement +of internal consistency with observational selection to be a +white dwarf (WD) – on a highly eccentric orbit about the +central MBH, transferring mass to it at pericentre passage +via an accretion disc. In a second paper (King, 2022, here- +after K22) I used the parametrization introduced by Chen et +al. (2022), extended to the full WD mass range, to show that +this model was consistent with the data for 5 of the 6 known +QPE sources, together with a previously unassociated sys- +tem HLX-1, where the periodicity is ∼ 1 yr. (I discuss the +‘missing’ system ASSASN–14ko in Note 1 to Table 1 at the +end of the paper.) +In all cases, loss of orbital energy E and angular mo- +mentum J via gravitational radiation (GR) is the ultimate +driver of mass transfer, and observational selection effects +mean that the orbiting star is in practice a low–mass white +dwarf (WD) in detectable QPE systems (K22). This is the +likely explanation for the presence of CNO–processed mate- +rial found in GSN 069 (Sheng et al., 2021). +This model requires that mass transfer is dynamically +stable; the Roche lobe and the WD radius must move in +step as GR reduces the orbital semi–major axis a and the +eccentricity e on the same timescale. Since the WD expands +as it loses mass, it must gain angular momentum from the +accretion disc and move in a wider orbit, implying a larger +tidal radius. +This stability has recently been questioned, so I dis- +cuss it further in Section 2. Orbital resonances within the +accretion disc are likely to cause the required enhanced an- +gular momentum transfer to the WD. The resonances may +© 2019 RAS + +2 +Andrew King +also directly cause the quasiperiodic eruptions themselves +as the orbiting WD passes pericentre – a similar origin has +been suggested for the superoutbursts of the stellar–mass +SU UMa binaries (Osaki & Kato, 2013). +During the orbital evolution of a QPE binary the peri- +centre separation p = a(1−e) remains almost constant. This +is reasonable, since significant GR effects only occur in an +effective point interaction at pericentre. Unusually, the inter- +action in QPE systems is strong enough that the resulting +instantaneous mass transfer rate is close to the long–term +evolutionary mean driven by GR. This is very different from +the situation in accreting stellar–mass binaries. There the +instantaneous rate oscillates widely around the evolution- +ary mean because of unrelated short–term effects. It only +converges to this mean when averaged on timescales long +compared with that taken for the driving process (here GR) +to move the critical (Roche) lobe by a distance of order the +density scaleheight of the donor star. This means for ex- +ample that it is generally unsafe to try to deduce the mass +transfer rate of a stellar–mass binary by measuring the rate +of change of its orbital period – see King & Lasota (2021) +for a recent discussion – whereas QPE sources appear to be +constrained to stay close to this mean mass transfer rate. +Although the simple model of King (2020) and K22 +works well for QPE systems, there is clearly more complexity +in the structure of these sources. The QPE system ASASSN- +14ko (Payne et al., 2021, 2022) has a very nearly strict period +of P = 114 days, and its mass transfer rate and luminosity +agree with the predictions of GR driving. But the predicted +GR period derivative ˙P ≃ −1.3 × 10−6 is in flat contradic- +tion with the measured value ˙P = −1.7 × 10−3 (Payne et +al., 2021). +A second deviation from the expectations of K22 +emerges from the recent thorough observational study by +Minutti et al. (2022) of the first QPE source, GSN 069, +which gives a historical X–ray light curve. This shows an +episode where the orbital period appears to increase by a +factor ∼ 1.3 over a timescale of order 3000 d, and the mass +transfer rate declines significantly on the same timescale. +In this paper I suggest that the underlying cause of the +unusual behaviour of both ASASSN-14ko and GSN 069 is +that in each of these two galaxies the central MBH–WD bi- +nary system is not isolated, but gravitationally influenced by +a perturber which itself is in a wider orbit about the MBH. +This object may be a star (or star cluster) which was part +of the infall event causing the formation of the inner QPE +‘binary’ itself. The interaction between the inner and outer +binaries induces a variety of effects, now collectively known +as von Zeipel–Lidov–Kozai (ZLK) cycles1, usually studied in +the case where the secondary component in the inner binary +(here the WD) has negligible mass compared with the pri- +mary (MBH) and the perturber. In many cases one can treat +the problem by expanding the combined gravitational poten- +tial only to quadrupole order, effectively modelling the two +binaries as wire loops with the masses spread around their +1 The author ordering ZLK I adopt here follows the historical +sequence in which the authors studied the interaction between +two binaries in the contexts of the Solar System (von Zeipel 1910; +Kozai, 1962), and artificial satellites in the Earth–Moon system +(Lidov, 1961, 1962). +orbits. As I explicitly noted in K22, it is inherently plausible +that QPE binaries should be accompanied by other more dis- +tant orbiting stars (or star clusters), as such satellites of the +MBH are likely additional results of the tidal capture events +which probably produce QPE binaries (cf Cufari, Coughlin +& Nixon, 2022). +I show here that ZLK cycles can account for large or- +bital period derivatives, as seen in ASASSN-14ko, which are +unrelated to the mass transfer process, and can also cause +correlated rapid changes change of orbital period and mass +transfer rate, as in GSN 069. In general the parameter space +open to a perturber is large in any given case, but the pos- +sibility of narrowing it down should encourage continued +monitoring of QPE sources, as this may give insight into +the capture events forming them. +2 +MASS TRANSFER STABILITY IN QPE +BINARIES, AND THE ORIGIN OF THE +ERUPTIONS +The QPE model discussed here requires that that mass +transfer is dynamically stable, i.e. that the tidal lobe and +the donor radius move in step as mass is transferred. Sug- +gestions to the contrary have appeared in the literature, but +several of these neglected the orbital expansion driven by +the tidal coupling of the accretion disc angular momentum +when mass is transferred from the less massive star (here +the WD) to a more massive accretor (the MBH); see K22 +for a discussion. More recently, Lu & Quataert (2022) have +argued that in a highly eccentric system such as the QPE +binaries considered here, the tidal coupling would actually +transfer angular momentum from the donor to the accretion +disc. This would cause mass transfer from a donor which +expands on mass loss (as here) to be dynamically unstable, +and the binary would coalesce in a few orbits. +The argument of Lu & Quataert (2022) implicitly as- +sumes that the accretion disc is circular. In that case mat- +ter at the outer edge of the disc moves more slowly than +the donor in an eccentric binary, and indeed the angular +momentum transfer is from the donor to the disc, leading +to instability. But it is rather unlikely that the disc is at +all circular in QPE systems. Since the mass ratio M2/M1 +is ≪ 1, the disc can easily grow large enough to contain +strong Lindblad resonances (see e.g. Fig. 1 of Whitehurst & +King, 1991). These make the disc very eccentric and cause +prograde apsidal precession, and are the cause of the super- +hump modulations at periods slightly longer than orbital in +superoutbursts of the short–period (P ≲ 100 min) SU UMa +cataclysmic binaries (Lubow, 1991). Early Lagrangian (e.g. +SPH) simulations had readily found these effects, beginning +with Whitehurst (1988). Eulerian simulations took longer to +achieve this, as the use of axisymmetric coordinates tends +to suppress the disc eccentricity which is the basis of super- +humps, but with attention to this problem superhumps now +appear in these simulations too (Wienkers & Ogilvie, 2018, +and references therein). +SU UMa superhumps are driven by the 3:1 commen- +surability. In QPE binaries the much smaller mass ratios +mean that the stronger 2:1 resonance is also accessible, so +we can expect their discs to be strongly eccentric and pre- +cessing. In the SU UMa systems, superhumps appear during +© 2019 RAS, MNRAS 000, 1–5 + +Angular Momentum Transfer in QPEs +3 +superoutbursts, when the discs undergo outbursts which are +longer and brighter than their usual dwarf nova outbursts. A +suggestive possibility (Osaki & Kato, 2013) is that the tidal +effects themselves actually cause the increased disc accretion +of superoutbursts. By analogy, the presence of resonances in +eccentric QPE sources may trigger the eruptions themselves +when the donor is near pericentre. In addition, the preces- +sion of the eccentric disc would naturally cause deviations +from strict periodicity, particularly in short–period systems. +(I note that in long–period QPE systems such as ASSASN- +14ko and HLX–1 the eruptions tend to be more regular; K22 +discusses why in HLX–1 the disc may occasionally not re– +form in time for the next periastron passage, and so miss an +entire cycle). In this picture the angular momentum lost by +the rapidly–accreting disc material is transferred to the WD +orbit, maintaining orbital stability. +Simulations (which are presumably easier with SPH) +are needed to check these suggestions, and in particular to +determine the size and direction of angular momentum ex- +change between the eccentric binary orbit and the eccentric +precessing disc. The presence of very significant CNO en- +hancement in at least one QPE source (Sheng et al., 2021) +strongly supports the suggestion of WD donors in QPE +sources, and so the idea that mass transfer is stable in them. +3 +ZLK CYCLES +As remarked above, there is good reason to suspect the ac- +tion of ZLK effects in QPE sources. The characteristic fea- +ture of ZLK cycles is that the inner binary (the QPE system +in our case) continuously exchanges its eccentricity e with +the inclination i of its orbit to that of the outer (perturb- +ing) binary. (The plane of the latter is almost fixed in many +cases of interest, as the outer binary has the largest compo- +nent of the whole system’s total angular momentum.) The +exchanges are subject to the constraint +(1 − e2)1/2 cos i ≃ C, +(1) +where C is a constant set by the initial conditions. This +asserts that ZLK cycles have no effect on the angular mo- +mentum component of the inner binary orthogonal to its +instantaneous plane. This is precisely the angular momen- +tum J being depleted by GR to drive mass transfer. +For given initial conditions the inner binary plane ei- +ther librates (oscillates between two fixed inclinations i) or +circulates (revolves continuously wrt the outer binary). The +characteristic timescale for these motions is +tZLK ≃ +8 +15π +� +1 + M1 +M3 +� �P 2 +out +P +� +(1 − e2 +out)3/2, +(2) +(Antognini, 2015), where M1, M3 are the MBH and per- +turber masses P, Pout are the periods of the inner (QPE) +binary and the outer one respectively, and eout is the eccen- +tricity of the outer binary. We see that this timescale tends +to infinity in the limit of vanishing perturber mass M3, as +expected. +ZLK cycles are quickly washed out if the binary pre- +cesses too rapidly, as this gradually destroys the near- +resonance allowing the exchange of inclination and eccen- +tricity. The strongest precession in QPE systems with WD +donors (cf Willems, Deloye & Kalogera, 2010) is the general– +relativistic advance of pericentre, with period +PGR = 4.27M −2/3 +5 +P 5/3 +4 +(1 − e2) yr, +(3) +so that +PGR +P += 1.28 × 104M −2/3 +5 +P 2/3 +4 +(1 − e2), +(4) +where e is the eccentricity, M5 = M1/105M⊙ with M1 the +black hole mass, and P4 is the orbital period P in units of +104 s. This ratio is given for the current known systems in +Table 1, and is always significantly larger than unity, al- +though only of order 22 and 36 for two systems. This sug- +gests that ZLK cycles can appear stably in most QPE sys- +tems, but may (if they appear at all) be fairly shortlived in +some cases. I discuss this point further below. +4 +EVOLUTION OF THE INNER BINARY +DURING ZLK CYCLES +ZLK cycles modulate the eccentricity e of the inner (QPE) +binary. But we see from (1) that they have essentially no +direct effect on its orbital angular momentum J. Since mass +transfer is stable in a QPE binary, this system must evi- +dently respond to ZLK changes of e by holding constant its +tidal lobe R2: this must remain equal to the current radius +of the WD donor, whose mass is unchanged. The lobe radius +R2 is proportional to the pericentre separation (see K22) +p = a(1 − e), +(5) +so that the ZKL effect makes the semi–major axis a change +as +a ∝ (1 − e)−1. +(6) +This in turn implies that ZLK cycles cause the period of a +stably mass–transferring binary system to evolve as +P ∝ a3/2 ∝ (1 − e)−3/2. +(7) +The GR–driven mass transfer rate must evolve in response +to the changes in e and P as +− ˙M2 ∝ P −8/3(1 − e)−5/2 ∝ P −1, +(8) +where I have used eqn (15) of K22 together with (6, 7). +The constraint (1) implies that during a ZKL cycle the +eccentricity e reaches a maximum as the inner binary plane +crosses the plane of the perturbing outer binary at i = 0. +From (7) the inner binary period reaches a maximum at this +point. Logarithmically differentiating (1) we get +e ˙e +1 − e2 ≃ − tan idi +dt. +(9) +From this equation and (7) we have +˙P +P = 3 +2 +˙e +1 − e = −31 + e +2e +tan idi +dt. +(10) +In all cases we expect di/dt ∼ ±1/tZLK. +© 2019 RAS, MNRAS 000, 1–5 + +4 +Andrew King +5 +COMPARISON WITH OBSERVATIONS +5.1 +Period Changes +We have seen that ZLK cycles can produce very rapid pe- +riod changes in QPE binaries (cf eqn 10), which may be +accompanied by significant changes in the accretion lumi- +nosity (cf eqn 8). Because ZLK cycles produce these changes +by altering the eccentricity affecting the GR losses driving +mass transfer, there is no paradox in changes more rapid +than given by the timescale tGR for the latter. The appar- +ently discordantly large period derivative of ASASSN–14ko +is then a potential signature of this effect. There are sev- +eral ways to explain values of order the ˙P = −1.7 × 10−3 +observed there as a result of ZLK cycles. +If tan i ∼ 1 (i.e. the QPE plane is not close to the +perturber plane) we must have e significantly smaller than +unity. We see from Table 1 that this explanation cannot work +for ASSASN–14ko itself, or any of the known QPE systems, +which all have a much higher eccentricities. +Future observations may reveal QPE systems with lower +e, and these would have − ˙P ∼ 3P/2etZLK, so from (2) we +find a value ˙P ∼ 10−3 would result if the perturber mass +and period are connected by +M3 ≃ 13 em5 +1 + e +�Pout +P +�2 +(1 − e2 +out)3/2M⊙, +(11) +where m5 = M1/105M⊙. We need Pout > P = 114 d for con- +sistency in ASSASN–14ko. This is evidently possible with +perturber having a normal stellar mass, as e is very close to +unity (see Table 1). +So we must look to other candidates for the perturbers +in known QPE systems. Other possibilities are that the per- +turber is a star cluster rather than a single star, or that it is a +single star with extreme eout approaching unity2. This latter +case may be more likely if the QPE binary and its perturber +result from the same tidal capture event. This is important +in highlighting the potential the QPE sources have in sig- +nalling these events, and their possible role in promoting +black hole growth. Clearly, only further observational mon- +itoring of the known QPE systems can distinguish between +all these possibilities. +5.2 +Light Curve and Correlated Period Changes +Equation (8) shows that ZKL cycles can continuously mod- +ify the mass transfer rate and so the luminosity of a QPE +binary, as recently observed in GSN 069 by Miniutti et al. +(2022). As the period is increasing here, and the luminos- +ity decreasing, we must have increasing eccentricity, so the +plane of the QPE binary is approaching the perturber plane. +These events should eventually appear in time–reversed or- +der. The 3000 d timescale is easy to accomodate (cf eqn 2) +with a stellar–mass perturber and an outer period not much +longer than that of the QPE binary. +Presumably systems showing little change between +eruptions, and no very rapid period changes, must either +2 In this case we have the eccentric ZLK effect: this becomes +considerably more complicated, as now there are octupole con- +tributions to the gravitational potential of similar order to the +quadrupole ones considered so far. See Naoz (2016) for a review. +have no associated perturber, or a perturber period which +is very long. Orbital changes induced by ZKL cycles might +trigger other light curve effects, e.g. by cyclically altering +disc accretion. These would add to the effect already noted +by K22 that for systems with periods P ≳ 1 yr the accretion +disc may have to re–form after a few outbursts, which may +account for the missing outburst in HLX-1. +6 +CONCLUSIONS +I have argued that the presence of resonances in the ac- +cretion disc makes it likely that in systems where a white +dwarf orbits a massive black hole, mass transfer driven by +the loss of gravitational wave energy is stable on a dynam- +ical timescale. The resonances may also promote the rapid +loss of disc angular momentum to the WD, and so directly +cause the quasiperiodic eruptions. +I have considered some of the effects that may appear +in QPE systems because of von Zeipel–Lidov–Kozai (ZLK) +cycles triggered by a perturber on a more distant orbit about +the central massive black hole. The presence of perturbers +of this kind appears likely, as they may be products of the +same tidal capture events that formed the QPE binaries +themselves. Evidently more observations are needed to check +the validity of the ZLK idea. If it is tenable, the parameter +space available to the perturbers is currently very large, and +still more observations would be needed to narrow it down. +QPE +systems +showing +orbital +period +changes +on +timescales much shorter than the mass transfer time are +obvious candidates for ZLK effects, and are very likely to re- +ward further monitoring or archival searches. For example, +the predicted timescale for the disappearance of the ZLK +cycles in ASSASN–14ko is only of order a decade. Similarly, +correlated short–term variations of mass transfer rates and +orbital periods in QPE systems may result from ZLK cy- +cles. Here we can expect the data and interpretation to be +more complex than for period changes, because other effects +can also modulate the mass transfer rates. But this kind +of study can potentially give major insights into how the +central black holes in low–mass galaxies are able to grow. +DATA AVAILABILITY +No new data were generated or analysed in support of this +research. +ACKNOWLEDGMENTS +I thank Giovanni Miniutti for giving me early insight into +important observational data and for many helpful and con- +tinuing discussions, and Chris Nixon and the anonymous +referee for very helpful comments. +REFERENCES +Antognini, J.M.O., 2015, MNRAS, 452, 3610 +Arcodia, R., Merloni, A., Nandra, K., et al., 2021, Nature, 592, +704 Arcodia et al. 2021 +© 2019 RAS, MNRAS 000, 1–5 + +Angular Momentum Transfer in QPEs +5 +Table 1. Parameters of the Current QPE Sample +Source +P4 +m5 +(L∆t)45 +m2 +1 − e +PGR/P +eRO – QPE2 +0.86 +2.5 +0.8 +0.18 9.9 × 10−2 +1250 +XMMSL1 +0.90 +0.85 +0.34 +0.18 9.9 × 10−2 +2503 +RXJ1301.9 +1.65 +18 +1.7 +0.15 7.2 × 10−2 +36 +GSN 069 +3.16 +4.0 +10 +0.32 2.8 × 10−2 +130 +eRO– QPE1 +6.66 +9.1 +0.045 +0.46 1.4 × 10−2 +291 +ASSASN–14ko +937 +700 +3388 +0.56 9.0 × 10−3 +22 +HLX–1 +2000 [0.5] +1000 +1.43 1.2 × 10−4 +774 +Note 1 This table is adapted from Table 1 of K22, but now ordered by period P . We note the general tendency that the eccentricity +e is smaller for shorter periods, consistent with the effects of GR losses. The very bright QPE source ASSASN-14ko (Payne et al., 2021, +2022) was missing from Table 1 of K22. The difficulty in modelling it arose because it is (uniquely) extremely close to the limit +m5(L∆t)1/3 +45 +≲ 104 +(12) +required to avoid the model formally predicting that the WD pericentre distance a(1 − e) is larger than the innermost stable orbital +radius, which is itself ≃ Rg = GM1/c2, where Rg is the black hole’s gravitational radius. Here m5 = M1/105M⊙, and (L∆t)45 is the +total energy radiated at pericentre passage in units of 1045 erg. Equation (12) is the condition +a(1 − e)3 > +� GM1 +c2 +�3 +(13) +written using the parametrization of Chen et al. (2022) followed in K22. Evidently the form (12) expresses the facts that the radiated +energy is increased in a tighter orbit, but that a larger black hole mass increases Rg. For ASSASN-14ko we have m5 ≃ 700, requiring +(L∆t)45 ≲ 3388, as compared with rough observational estimates (L∆t)45 ≃ 4000. Here I adopt the extreme value (L∆t)45 ≲ 3388 for +this system. For all other currently known QPE systems the constraint (12) is very easily satisfied. +Note 2 There is no secure mass estimate for the black hole in HLX–1. Here I adopt the minimum value m5 = 0.5 allowing the +donor to be below the Chandrasekhar mass (i.e. m2 ≃ 1.4; see K22). Larger m5 values allow smaller m2 (see K22). +Note 3 The Table also gives the values of PGR/P specifying the stability of possible ZLK cycles. +Chakraborty, J., Kara, E., Masterson, M., et al., 2021, ApJL, 921, +L40 +Chen, X., Qiu,Y., Li, S., Liu, F.K. 2022, ApJ, 930, 122 +Cufari, M., Coughlin, E.R., Nixon, C.J., 2022, ApJ, 929, L20 +Giustini, M., Miniutti, G., & Saxton, R. D., 2020, A&A, 636, L2 +King, A.R., 2020, MNRAS, 493, L120 +King, A.R., 2022, MNRAS, 515, 4344 (K22) +King, A.R., Lasota, J.P., 2021, arXiv211203779K +Kozai, Y. 1962, AJ, 67, 591 +Lidov, M.L., 1961, Artificial Earth Satellites, 8, 5–45 +Lidov, M. L. 1962, Planetary and Space Science, 9, 719 +Lu, W. & Quataert, E., 2022, arXiv:2210.08023 +Lubow, S., 1991, ApJ, 381, 259 +Miniutti, G., Giustini, M., Arcodia, R., Saxton, R. D., Read, A. +M., Bianchi, S., Alexander, K. D., 2022, arXiv:2207.07511 +Miniutti, G., Saxton, R. D., Giustini, M., et al. 2019, Nature, 573, +381 +Naoz, S., 2016, Ann. Rev. Astron. Astrophys. 54, 441 +Osaki, Y., Kato, T., 2013, PASJ, 65, 50 +Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2021, ApJ, 910, +125 +Payne, A. V., Shappee, B. J., Hinkle, J. T., et al. 2022, ApJ, 926, +142 +Sheng, Z., Wang, T., Ferland, G., et al., 2021, ApJ 920L, 25 +Song, J. R., Shu, X. W., Sun, L. M., et al., 2020, A&A, 644, L9 +Whitehurst, R., 1988, MNRAS, 232, 35 +Whitehurst, R. & King, A., 1991, MNRAS, 249, 25 +Wienkers, A.F., & Ogilvie, G.I., 2018, MNRAS, 477, 4838 +Willems, B., Deloye, C.J., Kalogera, V., 2010, ApJ 713, 239 +Xian, J., Zhang, F., Dou, L., et al., 2021, ApJ 921L, 32 +von Zeipel, H. 1910, Astron. Nachr., 183, 345 +This paper has been typeset from a TEX/ LATEX file pre- +pared by the author. +© 2019 RAS, MNRAS 000, 1–5 + diff --git a/H9E1T4oBgHgl3EQf_gbR/content/tmp_files/load_file.txt b/H9E1T4oBgHgl3EQf_gbR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a546a1d2a1fd6f4ee575034958f190a3d5934e4 --- /dev/null +++ b/H9E1T4oBgHgl3EQf_gbR/content/tmp_files/load_file.txt @@ -0,0 +1,342 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf,len=341 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='03582v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='HE] 9 Jan 2023 Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 000, 1–5 (2019) Printed 10 January 2023 (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='2) Angular Momentum Transfer in QPEs from Galaxy Nuclei Andrew King1,2,3,⋆ 1 Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK 2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands 3 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, Netherlands ⋆ E-mail: ark@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='uk Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' in original form ZZZ ABSTRACT A suggested model for quasi–periodic eruptions (QPEs) from galaxy nuclei invokes a white dwarf in an eccentric orbit about the central massive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I point out that the extreme mass ratio allows the presence of strong Lindblad resonances in the accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' These are important for the stability of mass transfer, and may trigger the eruptions themselves by rapidly transferring angular momentum from the accretion disc (which is likely to be eccentric itself) to the orbiting WD companion at pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I consider the effects of von Zeipel–Lidov–Kozai (ZLK) cycles caused by a perturber on a more distant orbit about the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I show that ZLK cycles can change the orbital periods of QPE systems on timescales much shorter than the mass transfer time, as seen in ASASSN-14ko, and produce correlated short–term variations of their mass transfer rates and orbital periods, as recently observed in GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Further monitoring of these sources should constrain the parameters of any perturbing companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This in turn may constrain the nature of the events creating QPE systems, and perhaps give major insights into how the central black holes in low–mass galaxies are able to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Key words: galaxies: active: black hole physics: X–rays: galaxies 1 INTRODUCTION X–ray observations of the nuclei of low–mass galaxies show that several of them produce quasi–periodic eruptions (QPEs: Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Giustini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Typically these sources have outbursts by factors ∼ 100 in X–rays, which recur in roughly pe- riodic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The recurrence times currently range from a few hours up to ∼ 100 d or ∼ 1 yr, and it is likely that these limits will widen as data accumulate from di- rect observations and archival searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The X–rays have ultrasoft blackbody spectra and luminosities implying typ- ical radii ≳ Rg = GM1/c2 = 6 × 1010m5 cm, of order the gravitational (Rg) and ISCO radii of a black hole of mass M1 ∼ 105m5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' These are consistent with the massive black holes (MBHs) one might expect in these low–mass galaxy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In (King, 2020) I suggested a simple model for the first known QPE source GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This postulated a low–mass star (with mass M2 ≪ M1) – found from the requirement of internal consistency with observational selection to be a white dwarf (WD) – on a highly eccentric orbit about the central MBH, transferring mass to it at pericentre passage via an accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In a second paper (King, 2022, here- after K22) I used the parametrization introduced by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (2022), extended to the full WD mass range, to show that this model was consistent with the data for 5 of the 6 known QPE sources, together with a previously unassociated sys- tem HLX-1, where the periodicity is ∼ 1 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (I discuss the ‘missing’ system ASSASN–14ko in Note 1 to Table 1 at the end of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=') In all cases, loss of orbital energy E and angular mo- mentum J via gravitational radiation (GR) is the ultimate driver of mass transfer, and observational selection effects mean that the orbiting star is in practice a low–mass white dwarf (WD) in detectable QPE systems (K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is the likely explanation for the presence of CNO–processed mate- rial found in GSN 069 (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This model requires that mass transfer is dynamically stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' the Roche lobe and the WD radius must move in step as GR reduces the orbital semi–major axis a and the eccentricity e on the same timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Since the WD expands as it loses mass, it must gain angular momentum from the accretion disc and move in a wider orbit, implying a larger tidal radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This stability has recently been questioned, so I dis- cuss it further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Orbital resonances within the accretion disc are likely to cause the required enhanced an- gular momentum transfer to the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The resonances may © 2019 RAS 2 Andrew King also directly cause the quasiperiodic eruptions themselves as the orbiting WD passes pericentre – a similar origin has been suggested for the superoutbursts of the stellar–mass SU UMa binaries (Osaki & Kato, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' During the orbital evolution of a QPE binary the peri- centre separation p = a(1−e) remains almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is reasonable, since significant GR effects only occur in an effective point interaction at pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Unusually, the inter- action in QPE systems is strong enough that the resulting instantaneous mass transfer rate is close to the long–term evolutionary mean driven by GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is very different from the situation in accreting stellar–mass binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' There the instantaneous rate oscillates widely around the evolution- ary mean because of unrelated short–term effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' It only converges to this mean when averaged on timescales long compared with that taken for the driving process (here GR) to move the critical (Roche) lobe by a distance of order the density scaleheight of the donor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This means for ex- ample that it is generally unsafe to try to deduce the mass transfer rate of a stellar–mass binary by measuring the rate of change of its orbital period – see King & Lasota (2021) for a recent discussion – whereas QPE sources appear to be constrained to stay close to this mean mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Although the simple model of King (2020) and K22 works well for QPE systems, there is clearly more complexity in the structure of these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The QPE system ASASSN- 14ko (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021, 2022) has a very nearly strict period of P = 114 days, and its mass transfer rate and luminosity agree with the predictions of GR driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' But the predicted GR period derivative ˙P ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='3 × 10−6 is in flat contradic- tion with the measured value ˙P = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='7 × 10−3 (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' A second deviation from the expectations of K22 emerges from the recent thorough observational study by Minutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (2022) of the first QPE source, GSN 069, which gives a historical X–ray light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This shows an episode where the orbital period appears to increase by a factor ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='3 over a timescale of order 3000 d, and the mass transfer rate declines significantly on the same timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In this paper I suggest that the underlying cause of the unusual behaviour of both ASASSN-14ko and GSN 069 is that in each of these two galaxies the central MBH–WD bi- nary system is not isolated, but gravitationally influenced by a perturber which itself is in a wider orbit about the MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This object may be a star (or star cluster) which was part of the infall event causing the formation of the inner QPE ‘binary’ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The interaction between the inner and outer binaries induces a variety of effects, now collectively known as von Zeipel–Lidov–Kozai (ZLK) cycles1, usually studied in the case where the secondary component in the inner binary (here the WD) has negligible mass compared with the pri- mary (MBH) and the perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In many cases one can treat the problem by expanding the combined gravitational poten- tial only to quadrupole order, effectively modelling the two binaries as wire loops with the masses spread around their 1 The author ordering ZLK I adopt here follows the historical sequence in which the authors studied the interaction between two binaries in the contexts of the Solar System (von Zeipel 1910;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Kozai, 1962), and artificial satellites in the Earth–Moon system (Lidov, 1961, 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' As I explicitly noted in K22, it is inherently plausible that QPE binaries should be accompanied by other more dis- tant orbiting stars (or star clusters), as such satellites of the MBH are likely additional results of the tidal capture events which probably produce QPE binaries (cf Cufari, Coughlin & Nixon, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I show here that ZLK cycles can account for large or- bital period derivatives, as seen in ASASSN-14ko, which are unrelated to the mass transfer process, and can also cause correlated rapid changes change of orbital period and mass transfer rate, as in GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In general the parameter space open to a perturber is large in any given case, but the pos- sibility of narrowing it down should encourage continued monitoring of QPE sources, as this may give insight into the capture events forming them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2 MASS TRANSFER STABILITY IN QPE BINARIES, AND THE ORIGIN OF THE ERUPTIONS The QPE model discussed here requires that that mass transfer is dynamically stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' that the tidal lobe and the donor radius move in step as mass is transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Sug- gestions to the contrary have appeared in the literature, but several of these neglected the orbital expansion driven by the tidal coupling of the accretion disc angular momentum when mass is transferred from the less massive star (here the WD) to a more massive accretor (the MBH);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' see K22 for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' More recently, Lu & Quataert (2022) have argued that in a highly eccentric system such as the QPE binaries considered here, the tidal coupling would actually transfer angular momentum from the donor to the accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This would cause mass transfer from a donor which expands on mass loss (as here) to be dynamically unstable, and the binary would coalesce in a few orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The argument of Lu & Quataert (2022) implicitly as- sumes that the accretion disc is circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In that case mat- ter at the outer edge of the disc moves more slowly than the donor in an eccentric binary, and indeed the angular momentum transfer is from the donor to the disc, leading to instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' But it is rather unlikely that the disc is at all circular in QPE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Since the mass ratio M2/M1 is ≪ 1, the disc can easily grow large enough to contain strong Lindblad resonances (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 1 of Whitehurst & King, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' These make the disc very eccentric and cause prograde apsidal precession, and are the cause of the super- hump modulations at periods slightly longer than orbital in superoutbursts of the short–period (P ≲ 100 min) SU UMa cataclysmic binaries (Lubow, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Early Lagrangian (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' SPH) simulations had readily found these effects, beginning with Whitehurst (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Eulerian simulations took longer to achieve this, as the use of axisymmetric coordinates tends to suppress the disc eccentricity which is the basis of super- humps, but with attention to this problem superhumps now appear in these simulations too (Wienkers & Ogilvie, 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' SU UMa superhumps are driven by the 3:1 commen- surability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In QPE binaries the much smaller mass ratios mean that the stronger 2:1 resonance is also accessible, so we can expect their discs to be strongly eccentric and pre- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In the SU UMa systems, superhumps appear during © 2019 RAS, MNRAS 000, 1–5 Angular Momentum Transfer in QPEs 3 superoutbursts, when the discs undergo outbursts which are longer and brighter than their usual dwarf nova outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' A suggestive possibility (Osaki & Kato, 2013) is that the tidal effects themselves actually cause the increased disc accretion of superoutbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' By analogy, the presence of resonances in eccentric QPE sources may trigger the eruptions themselves when the donor is near pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In addition, the preces- sion of the eccentric disc would naturally cause deviations from strict periodicity, particularly in short–period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (I note that in long–period QPE systems such as ASSASN- 14ko and HLX–1 the eruptions tend to be more regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' K22 discusses why in HLX–1 the disc may occasionally not re– form in time for the next periastron passage, and so miss an entire cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' In this picture the angular momentum lost by the rapidly–accreting disc material is transferred to the WD orbit, maintaining orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Simulations (which are presumably easier with SPH) are needed to check these suggestions, and in particular to determine the size and direction of angular momentum ex- change between the eccentric binary orbit and the eccentric precessing disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The presence of very significant CNO en- hancement in at least one QPE source (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021) strongly supports the suggestion of WD donors in QPE sources, and so the idea that mass transfer is stable in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 3 ZLK CYCLES As remarked above, there is good reason to suspect the ac- tion of ZLK effects in QPE sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The characteristic fea- ture of ZLK cycles is that the inner binary (the QPE system in our case) continuously exchanges its eccentricity e with the inclination i of its orbit to that of the outer (perturb- ing) binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (The plane of the latter is almost fixed in many cases of interest, as the outer binary has the largest compo- nent of the whole system’s total angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=') The exchanges are subject to the constraint (1 − e2)1/2 cos i ≃ C, (1) where C is a constant set by the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This asserts that ZLK cycles have no effect on the angular mo- mentum component of the inner binary orthogonal to its instantaneous plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is precisely the angular momen- tum J being depleted by GR to drive mass transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' For given initial conditions the inner binary plane ei- ther librates (oscillates between two fixed inclinations i) or circulates (revolves continuously wrt the outer binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The characteristic timescale for these motions is tZLK ≃ 8 15π � 1 + M1 M3 � �P 2 out P � (1 − e2 out)3/2, (2) (Antognini, 2015), where M1, M3 are the MBH and per- turber masses P, Pout are the periods of the inner (QPE) binary and the outer one respectively, and eout is the eccen- tricity of the outer binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' We see that this timescale tends to infinity in the limit of vanishing perturber mass M3, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' ZLK cycles are quickly washed out if the binary pre- cesses too rapidly, as this gradually destroys the near- resonance allowing the exchange of inclination and eccen- tricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The strongest precession in QPE systems with WD donors (cf Willems, Deloye & Kalogera, 2010) is the general– relativistic advance of pericentre, with period PGR = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='27M −2/3 5 P 5/3 4 (1 − e2) yr, (3) so that PGR P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='28 × 104M −2/3 5 P 2/3 4 (1 − e2), (4) where e is the eccentricity, M5 = M1/105M⊙ with M1 the black hole mass, and P4 is the orbital period P in units of 104 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This ratio is given for the current known systems in Table 1, and is always significantly larger than unity, al- though only of order 22 and 36 for two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This sug- gests that ZLK cycles can appear stably in most QPE sys- tems, but may (if they appear at all) be fairly shortlived in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I discuss this point further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 4 EVOLUTION OF THE INNER BINARY DURING ZLK CYCLES ZLK cycles modulate the eccentricity e of the inner (QPE) binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' But we see from (1) that they have essentially no direct effect on its orbital angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Since mass transfer is stable in a QPE binary, this system must evi- dently respond to ZLK changes of e by holding constant its tidal lobe R2: this must remain equal to the current radius of the WD donor, whose mass is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The lobe radius R2 is proportional to the pericentre separation (see K22) p = a(1 − e), (5) so that the ZKL effect makes the semi–major axis a change as a ∝ (1 − e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (6) This in turn implies that ZLK cycles cause the period of a stably mass–transferring binary system to evolve as P ∝ a3/2 ∝ (1 − e)−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (7) The GR–driven mass transfer rate must evolve in response to the changes in e and P as − ˙M2 ∝ P −8/3(1 − e)−5/2 ∝ P −1, (8) where I have used eqn (15) of K22 together with (6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The constraint (1) implies that during a ZKL cycle the eccentricity e reaches a maximum as the inner binary plane crosses the plane of the perturbing outer binary at i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' From (7) the inner binary period reaches a maximum at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Logarithmically differentiating (1) we get e ˙e 1 − e2 ≃ − tan idi dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (9) From this equation and (7) we have ˙P P = 3 2 ˙e 1 − e = −31 + e 2e tan idi dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (10) In all cases we expect di/dt ∼ ±1/tZLK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' © 2019 RAS, MNRAS 000, 1–5 4 Andrew King 5 COMPARISON WITH OBSERVATIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='1 Period Changes We have seen that ZLK cycles can produce very rapid pe- riod changes in QPE binaries (cf eqn 10), which may be accompanied by significant changes in the accretion lumi- nosity (cf eqn 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Because ZLK cycles produce these changes by altering the eccentricity affecting the GR losses driving mass transfer, there is no paradox in changes more rapid than given by the timescale tGR for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The appar- ently discordantly large period derivative of ASASSN–14ko is then a potential signature of this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' There are sev- eral ways to explain values of order the ˙P = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='7 × 10−3 observed there as a result of ZLK cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' If tan i ∼ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' the QPE plane is not close to the perturber plane) we must have e significantly smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' We see from Table 1 that this explanation cannot work for ASSASN–14ko itself, or any of the known QPE systems, which all have a much higher eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Future observations may reveal QPE systems with lower e, and these would have − ˙P ∼ 3P/2etZLK, so from (2) we find a value ˙P ∼ 10−3 would result if the perturber mass and period are connected by M3 ≃ 13 em5 1 + e �Pout P �2 (1 − e2 out)3/2M⊙, (11) where m5 = M1/105M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' We need Pout > P = 114 d for con- sistency in ASSASN–14ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is evidently possible with perturber having a normal stellar mass, as e is very close to unity (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' So we must look to other candidates for the perturbers in known QPE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Other possibilities are that the per- turber is a star cluster rather than a single star, or that it is a single star with extreme eout approaching unity2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This latter case may be more likely if the QPE binary and its perturber result from the same tidal capture event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' This is important in highlighting the potential the QPE sources have in sig- nalling these events, and their possible role in promoting black hole growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Clearly, only further observational mon- itoring of the known QPE systems can distinguish between all these possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='2 Light Curve and Correlated Period Changes Equation (8) shows that ZKL cycles can continuously mod- ify the mass transfer rate and so the luminosity of a QPE binary, as recently observed in GSN 069 by Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' As the period is increasing here, and the luminos- ity decreasing, we must have increasing eccentricity, so the plane of the QPE binary is approaching the perturber plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' These events should eventually appear in time–reversed or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The 3000 d timescale is easy to accomodate (cf eqn 2) with a stellar–mass perturber and an outer period not much longer than that of the QPE binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Presumably systems showing little change between eruptions, and no very rapid period changes, must either 2 In this case we have the eccentric ZLK effect: this becomes considerably more complicated, as now there are octupole con- tributions to the gravitational potential of similar order to the quadrupole ones considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' See Naoz (2016) for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' have no associated perturber, or a perturber period which is very long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Orbital changes induced by ZKL cycles might trigger other light curve effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' by cyclically altering disc accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' These would add to the effect already noted by K22 that for systems with periods P ≳ 1 yr the accretion disc may have to re–form after a few outbursts, which may account for the missing outburst in HLX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 6 CONCLUSIONS I have argued that the presence of resonances in the ac- cretion disc makes it likely that in systems where a white dwarf orbits a massive black hole, mass transfer driven by the loss of gravitational wave energy is stable on a dynam- ical timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The resonances may also promote the rapid loss of disc angular momentum to the WD, and so directly cause the quasiperiodic eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' I have considered some of the effects that may appear in QPE systems because of von Zeipel–Lidov–Kozai (ZLK) cycles triggered by a perturber on a more distant orbit about the central massive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The presence of perturbers of this kind appears likely, as they may be products of the same tidal capture events that formed the QPE binaries themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Evidently more observations are needed to check the validity of the ZLK idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' If it is tenable, the parameter space available to the perturbers is currently very large, and still more observations would be needed to narrow it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' QPE systems showing orbital period changes on timescales much shorter than the mass transfer time are obvious candidates for ZLK effects, and are very likely to re- ward further monitoring or archival searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' For example, the predicted timescale for the disappearance of the ZLK cycles in ASSASN–14ko is only of order a decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Similarly, correlated short–term variations of mass transfer rates and orbital periods in QPE systems may result from ZLK cy- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Here we can expect the data and interpretation to be more complex than for period changes, because other effects can also modulate the mass transfer rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' But this kind of study can potentially give major insights into how the central black holes in low–mass galaxies are able to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' DATA AVAILABILITY No new data were generated or analysed in support of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' ACKNOWLEDGMENTS I thank Giovanni Miniutti for giving me early insight into important observational data and for many helpful and con- tinuing discussions, and Chris Nixon and the anonymous referee for very helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' REFERENCES Antognini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2015, MNRAS, 452, 3610 Arcodia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', Merloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', Nandra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021, Nature, 592, 704 Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' 2021 © 2019 RAS, MNRAS 000, 1–5 Angular Momentum Transfer in QPEs 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Parameters of the Current QPE Sample Source P4 m5 (L∆t)45 m2 1 − e PGR/P eRO – QPE2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='9 × 10−2 1250 XMMSL1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='9 × 10−2 2503 RXJ1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='65 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='2 × 10−2 36 GSN 069 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='8 × 10−2 130 eRO– QPE1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='66 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='4 × 10−2 291 ASSASN–14ko 937 700 3388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='0 × 10−3 22 HLX–1 2000 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='5] 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='2 × 10−4 774 Note 1 This table is adapted from Table 1 of K22, but now ordered by period P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' We note the general tendency that the eccentricity e is smaller for shorter periods, consistent with the effects of GR losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The very bright QPE source ASSASN-14ko (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', 2021, 2022) was missing from Table 1 of K22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' The difficulty in modelling it arose because it is (uniquely) extremely close to the limit m5(L∆t)1/3 45 ≲ 104 (12) required to avoid the model formally predicting that the WD pericentre distance a(1 − e) is larger than the innermost stable orbital radius, which is itself ≃ Rg = GM1/c2, where Rg is the black hole’s gravitational radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Here m5 = M1/105M⊙, and (L∆t)45 is the total energy radiated at pericentre passage in units of 1045 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Equation (12) is the condition a(1 − e)3 > � GM1 c2 �3 (13) written using the parametrization of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' (2022) followed in K22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Evidently the form (12) expresses the facts that the radiated energy is increased in a tighter orbit, but that a larger black hole mass increases Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' For ASSASN-14ko we have m5 ≃ 700, requiring (L∆t)45 ≲ 3388, as compared with rough observational estimates (L∆t)45 ≃ 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Here I adopt the extreme value (L∆t)45 ≲ 3388 for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' For all other currently known QPE systems the constraint (12) is very easily satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Note 2 There is no secure mass estimate for the black hole in HLX–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Here I adopt the minimum value m5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='5 allowing the donor to be below the Chandrasekhar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' m2 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' see K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Larger m5 values allow smaller m2 (see K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Note 3 The Table also gives the values of PGR/P specifying the stability of possible ZLK cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=' Chakraborty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', Kara, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'} +page_content=', Masterson, M.' metadata={'source': 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+Jungwon Byun, Maggie Appleton, Andreas Stuhlmüller +Ought +ABSTRACT +Language models (LMs) can perform complex reasoning either end-to-end, with hidden latent state, or +compositionally, with transparent intermediate state. Composition offers benefits for interpretability +and safety, but may need workflow support and infrastructure to remain competitive. We describe +iterated decomposition, a human-in-the-loop workflow for developing and refining compositional +LM programs. We improve the performance of compositions by zooming in on failing components +and refining them through decomposition, additional context, chain of thought, etc. To support +this workflow, we develop ICE, an open-source tool for visualizing the execution traces of LM +programs. We apply iterated decomposition to three real-world tasks and improve the accuracy +of LM programs over less compositional baselines on held-out test sets: describing the placebo +used in a randomized controlled trial (25% → 65%), evaluating participant adherence to a medical +intervention (53% → 70%), and answering NLP questions on the QASPER dataset (38% → 69%). +These applications serve as case studies for a workflow that, if automated, could keep ML systems +interpretable and safe even as they scale to increasingly complex tasks. +1 +Introduction +Language models are often trained using feedback on outcomes, leveraging reward models that imitate human +evaluations provided as pairwise comparisons (Christiano et al., 2017; Ziegler et al., 2020). This works well for basic +question-answering, summarization, simple code generation, and general short-form instruction-following (Stiennon +et al., 2020; Wu et al., 2021; Ouyang et al., 2022). For these tasks, good outputs can readily be distinguished from +bad ones, especially with model-supported evaluation (Saunders et al., 2022). And with rare exceptions like WebGPT +(Nakano et al., 2021), there is no difference between the model’s output and the relevant outcomes, so evaluations +relatively directly inform the model’s behavior. +However, as model capabilities and task complexities scale up, outcome-based evaluation may run into alignment +problems: +First, for some important applications the process used to generate the output matters as much as the output itself. +Consider long-range forecasting, and policy decisions informed by such forecasts: The quality of a forecast or decision +depends on the assumptions, evidence, and reasoning used to produce it. If feedback doesn’t inform the process used to +generate outputs, we may get results that look good (because they are optimized for favorable evaluation), and may +even look systematic, but are worse in exactly the ways that can’t easily be measured (Stuhlmüller and Byun, 2022). +Second, outcome-based feedback may create incentives for language models to deceive or manipulate their users by +exploiting gaps or biases in the feedback signal (Amodei et al., 2016). In extreme cases, this could lead to models +behaving as intended during training, hiding their true intentions and capabilities, but defecting at deployment (Cotra, +2022; Ngo et al., 2022). +∗Correspondence to justin@ought.org and andreas@ought.org. +arXiv:2301.01751v1 [cs.CL] 4 Jan 2023 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Supervise outcomes +? +Input +Did this study on +drowning use a +placebo? +output +Yes, a video +about dog bites +Supervise process +Find sections +about trial +arms +Extract trial +arms from +sections +Classify & +name +placebo +5 paragraphs +found +Drowning video,
 +dog bite video +Input +Did this study on +drowning use a +placebo? +output +Yes, a video +about dog bites +observable Interim outputs +Figure 1: Process supervision breaks black-box language model computation into human-understandable reasoning +steps without end-to-end optimization. This increases transparency and trust because it allows inspection of intermediate +results and reduces risks from imperfect feedback signals and deceptive alignment. +Process supervision is an alternative to outcome-based training that uses language models to execute human- +understandable task decompositions, either by imitating human steps or by rewarding human-endorsed steps (Christiano, +2016; Stuhlmüller and Byun, 2022; Uesato et al., 2022). This paradigm promises increased interpretability, trust, and +alignment by reducing the reliance on black-box computation and enabling users to inspect and intervene in the model’s +reasoning process. +Right now, process supervision is ahead of outcome-based training: Language model capabilities are weak, so complex +tasks require a composition of multiple calls. Indeed, engineered multi-step pipelines have been a staple of NLP for a +long time. However, without scalable infrastructure to support it, we expect that outcome-based training will eventually +crush process supervision performance-wise, leading to the alignment problems above. +This paper describes our experience applying process supervision to academic question-answering in the context of +Elicit2, the AI research assistant developed by Ought. Our contributions are: +1. A review of the literature on process supervision, highlighting gaps in workflows and tooling that contribute to +making real-world use cases rare +2. Iterated decomposition, a human-in-the-loop workflow for developing compositional language model programs +3. ICE, an open-source visualizer for language model execution traces +4. Case studies that use this workflow to improve performance over baselines on three real-world tasks: +(a) extracting placebo information from randomized controlled trials (RCTs), +(b) analyzing participant flow in RCTs, and +(c) answering questions about natural language processing papers. +As models advance and become more reliable at completing component tasks, we expect that process supervision +will become more feasible, and that eventually the iterated decomposition process will be automated. By sharing +our workflow and tooling now when even basic tasks are still challenging, we hope to accelerate a future where LM +deployments are controllable and interpretable. +2 +Process Supervision +Process supervision refers to approaches to LM training and deployment that rely on human-understandable intermediate +steps. We review the literature and highlight gaps, then explain the iterated decomposition workflow and ICE visualizer. +2.1 +Prior work on process supervision +Over the past year, there have been significant advances in techniques for process supervision as well as frameworks, +interfaces, and libraries for implementing it. At the same time, real-world use cases are still rare. We review prior work +and highlight gaps in the literature that may be contributing to this. This is a rapidly growing field, so we are only able +to review a sample of the work (Table 1). +2https://elicit.org +2 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Authors +Title +Decomposition +Training +Workflow +Survey +Tool +Theory +Saha et al. (2022) +Summarization Programs: Interpretable Abstractive Summarization [..] +✓ +✓ +✓ +Wu et al. (2021) +Recursively Summarizing Books with Human Feedback +✓ +✓ +✓ +Yao et al. (2022) +ReAct: Synergizing Reasoning and Acting in Language Models +✓ +✓ +Kumar and Talukdar (2020) +NILE : Natural Language Inference with Faithful Natural Language Explanations +✓ +✓ +Nakano et al. (2021) +WebGPT: Browser-assisted question-answering with human feedback +✓ +✓ +Creswell et al. (2022) +Selection-Inference: Exploiting LLMs for Interpretable Logical Reasoning +✓ +✓ +Jung et al. (2022) +Maieutic Prompting: Logically Consistent Reasoning with Recursive Explanations +✓ +✓ +Sanyal et al. (2022) +FaiRR: Faithful and Robust Deductive Reasoning over Natural Language +✓ +✓ +Creswell and Shanahan (2022) +Faithful Reasoning Using Large Language Models +✓ +✓ +Bostrom et al. (2022) +Natural Language Deduction through Search over Statement Compositions +✓ +✓ +Wang et al. (2022) +Locate Then Ask: Interpretable Stepwise Reasoning for Multi-hop Question Answering +✓ +✓ +Fu et al. (2021) +Decomposing Complex Questions Makes Multi-Hop QA Easier and More Interpretable +✓ +✓ +Dua et al. (2022) +Successive Prompting for Decomposing Complex Questions +✓ +✓ +Guo et al. (2022) +Complex Reading Comprehension Through Question Decomposition +✓ +✓ +Shridhar et al. (2022) +Distilling Multi-Step Reasoning Capabilities of LLMs into Smaller Models [..] +✓ +✓ +Zhou et al. (2022) +Learning to Decompose: Hypothetical Question Decomposition [..] +✓ +✓ +Khattab et al. (2022) +Demonstrate-Search-Predict: Composing retrieval and language models [..] +✓ +✓ +✓ +Press et al. (2022) +Measuring and narrowing the compositionality gap in language models +✓ +✓ +Khot et al. (2022) +Decomposed Prompting: A Modular Approach for Solving Complex Tasks +✓ +Ozturkler et al. (2022) +ThinkSum: Probabilistic reasoning over sets using large language models +✓ +Yang et al. (2022) +Re3: Generating Longer Stories With Recursive Reprompting and Revision +✓ +Gao et al. (2022) +PAL: Program-aided Language Models +✓ +Drozdov et al. (2022) +Compositional Semantic Parsing with Large Language Models +✓ +Trivedi et al. (2022) +Interleaving Retrieval with Chain-of-Thought Reasoning for [..] Multi-Step Questions +✓ +Kazemi et al. (2022) +LAMBADA: Backward Chaining for Automated Reasoning in Natural Language +✓ +Zelikman et al. (2022) +STaR: Bootstrapping Reasoning With Reasoning +✓ +✓ +Xie et al. (2022) +Calibrating Trust of Multi-Hop Question Answering Systems with Decompositional Probes +✓ +✓ +Uesato et al. (2022) +Solving math word problems with process- and outcome-based feedback +✓ +Dohan et al. (2022) +Language Model Cascades +✓ +✓ +✓ +Stuhlmüller et al. (2022) +Factored Cognition Primer +✓ +✓ +Wu et al. (2022a) +PromptChainer: Chaining Large Language Model Prompts through Visual Programming +✓ +✓ +Wu et al. (2022b) +AI Chains: Transparent and Controllable Human-AI Interaction by Chaining LLM Prompts +✓ +✓ +Chase (2022) +LangChain +✓ +✓ +Polu (2022) +Dust +✓ +Wies et al. (2022) +Sub-Task Decomposition Enables Learning in Sequence to Sequence Tasks +✓ +Table 1: A sample of prior work on process supervision, categorized into work that primarily contributes (1) new task +decompositions, (2) training and finetuning techniques, (3) workflows and tutorials, (4) surveys and frameworks for +organizing prior work, (5) tools, and (6) theory. While work on decompositions and training techniques is rapidly +growing, there is little investigation of workflows, tooling, and theory. +Task decompositions +There is a quickly growing literature on how to compose multiple language model calls to improve performance or +accomplish more difficult tasks: +• Creswell et al. (2022) and Creswell and Shanahan (2022) generate reasoning steps by alternating between +selection and inference. They show that their approach outperforms other prompting methods on multi-step +logical deduction and scientific QA tasks, and generates interpretable reasoning traces. +• Kazemi et al. (2022) apply backward-chaining to simple logic tasks, starting with a goal proposition and +recursively decomposing it into sub-goals until the sub-goals can be proved or disproved. +• Wu et al. (2021) and Saha et al. (2022) apply recursive summarization to generate summaries of long texts, such +as books or articles. They use LMs to summarize small sections of the text and then recursively summarize +these summaries to produce a summary of the entire text. They show that recursive summarization improves +the quality and coherence of the summaries, and enables human feedback and evaluation. +• Yang et al. (2022) generate long stories by first creating a story plan, generating passages b yprompting a +model with contextual information from the plan and the current story state, and then revising the passages by +reranking and editing them. +• ReAct (Yao et al., 2022) interleaves generating chain-of-thought reasoning and actions pertaining to a task +(e.g., search, lookup). +• WebGPT (Nakano et al., 2021) finetunes LMs to answer long-form questions using a text-based web-browsing +environment. +3 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +• Gao et al. (2022) offload computation to Python interpreters. +• Trivedi et al. (2022) and Khattab et al. (2022) interleave chain-of-thought with knowledge retrieval steps. +• Khot et al. (2022) study decomposition in general, letting the language model push subtasks to task-specific +handlers. +• Ozturkler et al. (2022) aggregate language model probabilities using mathematical combinators like sum and +product. +• Jung et al. (2022) recursively generate a tree of explanations for a statement, then determine the truth of the +statement by treating the inference as a satisfiability problem over these explanations and their logical relations. +• Various works, including Fu et al. (2021), Dua et al. (2022), Guo et al. (2022), and Zhou et al. (2022), explore +decomposition of questions into subquestions, often under the name multi-hop question-answering. +Training and finetuning techniques +The most relevant work is Uesato et al. (2022) who directly compare process-based and outcome-based feedback +for solving math word problems with LMs. They find that process-based feedback improves the correctness and +interpretability of reasoning steps, but requires more label supervision than outcome-based feedback. +Wu et al. (2021) finetune GPT-3 using behavioral cloning and reward modeling to do summarization recursively. Zhou +et al. (2022) train a decomposition model based on a parallel news corpus. Nakano et al. (2021) compare behavior +cloning, RL and rejection sampling for training a web browsing model, and find that a combination of behavior +cloning and rejection sampling against a reward model worked best. Zelikman et al. (2022) generate chains-of-thought, +finetuning on the ones that lead to correct answers. +Most other work under “training” in Table 1 trains small models from scratch for particular composition steps. +Theory and conceptual advances +Wies et al. (2022) show some benefits of process supervision: When concatenating intermediate supervision to the +input and training a sequence-to-sequence model on this modified input, unlearnable composite problems can become +learnable. +Press et al. (2022) coin the name “compositionality gap” for the fraction of questions that the model answers incorrectly +out of the questions for which the model answers all of the sub-questions correctly. They find this number to be around +40%. They show that chain of thought can narrow the gap, and that generating and answering follow-up questions +further narrows it. +Workflows, tutorials, and tools +Dohan et al. (2022) introduce language model cascades, which are probabilistic programs that compose LMs with +random variables and control flow. They formalize several existing techniques, such as scratchpads, verifiers, STaR, +selection-inference, and tool use, as instances of language model cascades. They provide an open-source probabilistic +programming system. The Factored Cognition Primer (Stuhlmüller et al., 2022) is a tutorial that explains how to write +compositional LM programs, including recursive question-answering, debate, search, and verification. Xie et al. (2022) +show in human participant studies that letting users probe a language model with subquestions helps them calibrate +when the model is correct. Wu et al. (2022a) and Wu et al. (2022b) describe a closed-source visual programming +interfaces for making compositional language model programs. Dust (Polu, 2022) is a web service and Rust library for +designing and deploying LM apps. LangChain (Chase, 2022) is a Python library that assists in the development of LM +applications that involve chaining LMs with each other or with other experts. +Relation to this work +While there is a quickly growing literature on task decompositions, there is effectively no work on theory and only a +few tools, tutorials, and workflows for building real-world process-based systems. We see this paper as demonstrating a +real-world application (science Q&A) as well as a description of the workflow (iterated decomposition) and tooling +(ICE) used. +The prior work above varies in what exactly is meant by “process” and “supervision”. We provide a taxonomy in +Appendix 9.3. Briefly, in this work, we focus on decompositions designed by a human developer, with occasional +choices made by the language model. We balance pragmatic decompositions that improve task performance with +decompositions that reflect ideal reasoning and facilitate better supervision. Our decompositions mostly improve +performance by helping the model use long context more effectively, although some also apply multiple lines of +4 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Task +decompose into Subtasks +Papers +Results +1 +Decompose the task into subtasks +Subtasks should be independently meaningful so
 +that they can be evaluated on their own +2 +Run the decomposition +across multiple inputs +For example, research papers for +which gold-standard results are +available +Gold standards +VS. +3 +Evaluate the results +Either manually or +automatically using +metrics such as accuracy +and F1-score +inputs +outputs +evaluate failed Subtasks +4 +Inspect failing subtasks +Zoom in on failures using the ICE +trace visualizer to inspect the inputs +and outputs of each subtask +custom +code edits +decomposition +5 +Improve failing subtasks +Using further decomposition +(e.g. adding chain-of-thought, +retrieval steps) or custom code +(e.g. prompt edits) +Repeat +Figure 2: Iterated decomposition is a workflow for human-in-the-loop language model programming. We start with a +trivial decomposition, evaluate it against gold standards, diagnose the source of failures using the ICE visualizer, refine +the failing subtasks through further decomposition or other adjustments, and repeat. +reasoning to subtasks. When we talk about “supervision“, we mean checking the outputs or behavior of individual steps +in the composition. We supervise the process to improve the LM program, the supervisor is the human developer, and +we always have access to the correct answer. +2.2 +Iterated decomposition +We study process supervision via iterated decomposition, a human-in-the-loop workflow that incrementally improves a +task decomposition through error diagnosis and amendment (illustrated in Figure 2): +1. Start with a minimal decomposition, breaking the task into subtasks that can be performed by a LM. For +example, the task of extracting the placebo used in academic studies can be decomposed into first finding the +most relevant section, then generating the placebo (if any) given that section. +2. Apply the decomposition to multiple test inputs with gold standard answers. For example, this could be a +dataset of academic papers from various domains, such as medicine, psychology, and economics, and their +corresponding placebo descriptions, if applicable. +5 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Figure 3: The Interactive Composition Explorer (ICE) is an open-source debugger for language model execution traces. +It visualizes task decompositions and supports zooming in on failing subcomponents. The top left pane in the screenshot +shows the trace of function calls in the original execution hierarchy. The bottom pane supports sorting and filtering +of calls, with each row showing values that were recorded at execution time. The right pane shows function inputs, +outputs, recorded intermediate values, and source code of the call selected in the main pane. +3. Evaluate the results automatically (using LM) or manually, using metrics such as accuracy, F1-score, or other +metrics. For example, the result of the placebo extraction task can be compared to a gold standard dataset of +placebo descriptions from academic papers. +4. Identify failures using the ICE trace visualizer or other tools to inspect the intermediate inputs and outputs +of each subtask. For example, given that the generated placebo is incorrect, one can determine whether the +wrong section was found or the right section was found but the wrong answer was extracted. +5. Improve failing components using further decomposition (e.g., semantic decomposition into subtasks, chain- +of-thought, retrieval), or by making custom edits (e.g. prompt tweaks, language model inference parameters). +For example, the task can be split into first finding each experiment mentioned in the paper, then asking for +each experiment whether it used a placebo or not. +The process repeats steps 2-5 until it achieves good performance or exhausts the relevant resources (time, compute +budget for experiments). +2.3 +Interactive Composition Explorer +To support the error identification step in iterated decomposition, we have developed the Interactive Composition +Explorer (ICE), an open-source3 debugger and execution trace visualizer for language model programs. +ICE records and visualizes the execution of all async functions in a Python program, including all inputs (prompts, +parameters) and outputs (language model responses, intermediate values), in a way that is suitable even for executions +3https://github.com/oughtinc/ice +6 + +Debate +8 +OpenAlAgent: Complete +Should we legalize all drugs? → +speaker +Generate an answer to a question given some context. +Turn +Inputs and Outputs +Source +QuestionShould we legalize all drugs? → + Alice Legalizing drugs would reduce the b.. +OpenAlAgent: Complete +Inputs +You are . There are turns left in the debate. You are..... +Legalizing drugs would redu +Prompt: } You are Alice. There are 8 turns left in the +debate. You are trying to win the debate using reason and +Turn +evidence. Don't repeat yourself. No more than 1-2 +sentences per turn. +OpenAlAgent: Complete +Question: "Should we legalize all drugs?" +You are . There are turns left in the debate. You are.... +Legalizing drugs would crea1 +You: "I'm in favor." +Bob: "'m against." +Turn +You: " +QuestionShould we legalize all drugs? →Alice The evidence shows that when drugs... +Stop: +OpenAlAgent: Complete +Verbose: false +You are . There are turns left in the debate. You are... +The evidence shows that wh +Default: empty +Max tokens: 256 +Turn +QuestionShould we legalize all drugs? → +BobWhile regulation of the drug market... +Outputs +Agent +Turns_left +Legalizing drugs would reduce the burden on our criminal +Answer +Alice +Legalizing drugs would reduce the burden on our criminal justice system, freein... +8 +serious crimes. Additionally, it would reduce the risk of +overdose and other health risks associated with the +Bob +Legalizing drugs would create a new set of problems, such as increased drug u... +7 +unregulated drug market." +Alice +The evidence shows that when drugs are legalized, drug use does not necessa... +6 +Bob +While regulation of the drug market may be possible, it would be difficult to enf... +5 +Alice +Regulation of the drug market can be achieved through a variety of measures, ... +4 +Bob +While there may be potential benefits to legalizing drugs, the risks associated .. +3Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Placebo classification
 +and description +EXAMPLE input +Did Shen et al. 2015 use a +placebo in their experiment on +using testimonial videos about +drowning to reduce drowning +risk? If so, what was it? +Participant flow +Example input +What were the experiments, trial +arms, and adherence for each +arm in Bloom et al. 2016? +Qasper NLP Q&A +Example input +Which retrieval system was used +for the baselines? +Figure 4: We apply iterated decomposition to three real-world case studies: Extracting placebos from academic papers +about randomized controlled trials, extracting participant flow, and NLP question-answering. The tasks are part of our +work on automated literature review for the AI Research Assistant Elicit. +with hundreds to thousands of LM calls. ICE provides a decorator that can be applied to any async function that should +be recorded, as well as utilities for recording all top-level async functions defined in a given module, any custom values +of interest within a function, and the structure of each interpolated string. This data can then be visualized in a browser +for interactive exploration, as shown in Figure 3. +ICE provides three views into this data: +1. An expandable tree of function calls shown in order of execution +2. A sortable, filterable table of function calls and their custom values +3. A function call detail pane containing inputs, custom values, outputs, and source code +The tree can be browsed to a particular function call which can then be inspected in the detail pane. Often, it is helpful to +compare many calls to the same function, so ICE provides a dropdown menu of all recorded functions in the execution +trace and their respective call counts. If a function is selected, all calls to it will be shown in the table and highlighted in +the tree. +Since prompts are of particular interest in LM programs, the detail pane includes special support for rendering +interpolated strings (f-strings). Each interpolated value is shown in an alternating color, and the interpolating source +code is shown in a tooltip. +These functions all serve the purpose of understanding, analyzing, and debugging task decompositions, going from the +high-level structure of the call tree to seeing all functions of a particular type in the table, to seeing what exactly the +prompt was when a particular instance of that type generated an unexpected result. +3 +Real-World Context of Case Studies +We used iterated decomposition and ICE to improve performance over simple baselines on three real-world case studies +we encountered in our work on Elicit (Figure 4). +A primary use case for Elicit is to find out what the academic literature knows about a research question. Many of +Elicit’s users work in biomedicine, psychology, experimental economics, and other fields where randomized controlled +trials (RCTs) are an important source of evidence. When these users do literature review or metanalyses, they often +roughly follow this process: +1. What are the RCTs that are potentially most relevant to answering my research question? +2. For each RCT: +(a) Does this study actually address my research question? +(b) What is the risk of bias? Can I take the findings at face value, or do I need to discount it or ignore it? +3. What does the aggregate of RCTs, weighted by evidence quality, say about my research question? +7 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +input +Did Shen et al. 2015 use a +placebo in their experiment on +using testimonial videos about +drowning to reduce drowning +risk? If so, what was it? +methods +Baseline: Select-and-generate +like Elicit +Trial arm classification +Paragraph-wise aggregation +Rank & answer for description +output +Yes, a testimonial video about dog +bites +Figure 5: In the placebo case study, the goal is to determine if a randomized controlled trial used a placebo, and if so, +what it was. We start with the select-and-generate baseline used in Elicit and improve it by ensembling classification +based on trial arms and paragraphs. +The Placebo Classification & Description task is designed to help researchers evaluate risk of bias. Information about +the placebo helps researchers assess the risk of bias in the study and decide how much to trust the results – comparing +an intervention to a placebo control is usually stronger evidence than comparing to a no-treatment control, especially if +the placebo successfully blinded the participants. +The first step of the Participant Flow in RCTs task is to identify the trials and trial arms within each study. This helps +the user understand what the researchers did in the study, to determine whether it addresses their research question. +The second step, evaluating participant adherence to an intervention, helps the researcher assess risk of bias and +contextualize the study’s findings: Was a small effect driven by poor adherence to the treatment? +These tasks represent a much broader class of tasks in Elicit: answering questions about papers. Elicit currently supports +about 20 pre-specified questions, but also allows users to enter their own questions. Ultimately, we want to find highly +generalizable strategies to improve performance. The QASPER NLP Q&A task tests this by generalizing the participant +flow decomposition to a different domain. +4 +Case Study: Placebo Classification & Description +In this case study, we focus on domain-specific decompositions. Through iteration on the classification and description +steps, accuracy of the generated placebo description improved from 25% to 65% on a held-out test set. +4.1 +Setup +Given the full text of an RCT, the task is to answer “Did this RCT use a placebo?” (classification) and “If so, what was +it?” (description). For both classification and description we find strong agreement among human raters. +Appendix 9.5.1 has a detailed description of the task. Table 9.5.2 in the Appendix shows examples of the most relevant +quotations from two papers with corresponding classifications and descriptions. +4.2 +Iterations +4.2.1 +Baseline: Select-and-generate like Elicit +The baseline is the “select-then-generate” algorithm currently deployed for paper question-answering in Elicit. Elicit +uses monoT54 to rank paragraphs from the paper against the question (Lin et al., 2021). The top-ranking paragraph is +fed to text-davinci-002 with the following prompt: +Answer the question "{{ question}} " based on the excerpt from a research paper. +Try to answer, but say "The answer to the question is not mentioned in the excerpt" if you +don't know how to answer. +Include everything that the paper excerpt has to say about the answer. Make sure everything you +say is supported by the excerpt. +The excerpt may cite other papers; answer about the paper you're reading the excerpt from, not +the papers that it cites. +Answer in one phrase or sentence: +4https://huggingface.co/castorini/monot5-base-msmarco-10k +8 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Paper title: {{ title}} +Paper excerpt: {{ paragraph}} +Question: {{ question}} +Answer: +On a held-out test set, this approach has 71% accuracy for classification, 25% for description (Table 2). +4.2.2 +Improving classification +Diagnosis +For each of the 14 trials that the baseline classified incorrectly, there was in fact a placebo. In all cases, the selection +step failed to find the part of the paper that most clearly stated that there was a placebo. For 10/14 trials, selection found +no evidence that there was a placebo. For the remaining 4/14 trials, selection found some mention of the placebo but +not a full description, then generation failed to conclude that there was a placebo (see Appendix 9.5.3 for examples). +To improve classification, we needed a way to more robustly find evidence that there was a placebo without introducing +false positives. +Solution +We created a decomposition that mirrors our own reasoning process for determining placebo classification and +description. When we read the papers in our validation set, we noticed two things: First, almost all papers clearly +describe their trial arms. Then, if there was a placebo in the study, one of the arms will usually be clearly identified as +the placebo arm. So, we identify and describe each trial arm and classify whether any of the arms is a placebo. Second, +often many paragraphs in the paper provide evidence about whether the paper used a placebo, and sometimes this +evidence is contradictory. So, we could check what each paragraph tells us about whether the paper used a placebo, +then aggregate those answers. +In outline, the decomposition looks like this: +1. Classify based on trial arms +(a) What were the trial arms? +i. Rank paragraphs by relevance to this question through pairwise comparisons +ii. Answer based on the most relevant paragraphs +(b) Describe each trial arm: Rank and Answer +(c) Do any of the arms look like placebos? +(d) If so, could participants tell which arm they were in? (If so, there’s not really a placebo.) +2. Classify based on each paragraph +(a) Was there a placebo, according to each paragraph? +(b) Aggregate answers from paragraphs +3. Ensemble the classification based on trial arms and paragraphs +Appendix 9.5.4 shows the full decomposition. +Results +This approach substantially improves on the baseline (71% correct → 98%; p=0.0004), achieving near-perfect perfor- +mance (Table 2). +Reviewing the execution traces in ICE showed that the improved performance is almost entirely due to answering +based on each paragraph, not due to answering based on trial arms. Whenever the arms-based classification tentatively +classified one of the trial arms as a placebo, it also found that participants might be able to tell which arm they were in, +and so it ultimately said that it was unclear whether there was a placebo. +4.2.3 +Improving description +Diagnosis +14 of the 15 trials where the baseline failed to describe the placebo correctly are ones that it failed to classify correctly. +The baseline does not even attempt to describe the placebo if the classification is “no placebo”, so it fails all of these by +9 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Method +Classification (n=48) +Description (n=20) +Elicit select-then-generate baseline +71% +25% +Select-then-generate with “not mentioned" perplexity selection +65% +10% +Stuff paper in prompt +69% +30% +Final decomposition +98% +65% +Keyword decision tree +94% +10% +Table 2: Through iteration on the task decomposition, accuracy of the placebo classification improved from 71% to +98% and description accuracy improved from 25% to 65%, both on a held-out test set. +construction. However, in each of these cases selection failed to find the excerpts needed for description, so any attempt +at description would have failed. +Solution +We noticed that the information required to describe the placebo is often dispersed throughout the paper. So, we rank the +most relevant paragraphs in the paper and then use them to generate an answer. We re-use the same Rank and Answer +technique that we used for classification. This generalizable subtask decomposition works well for both purposes. +Results +This approach results in a big improvement on the Elicit baseline (25% correct → 65%; p = 0.025—see Table 2). +4.2.4 +A keyword baseline +The strong paragraph-based classification results raise the question whether can we encode our understanding of how to +figure out whether a trial used a placebo in a much simpler algorithm. +We created a simple regex keyword-matching algorithm: +1. Classify as no placebo if the paper contains words like "open-label" +2. Then classify as placebo is the paper contains words like "placebo", and take the first matching sentence as a +description +3. Then classify as no placebo if none of these words are present +This algorithm does about as well as our task decomposition on classification. However, a regex keyword approach +doesn’t generalize to harder and more ambiguous tasks—for example, adherence is discussed using a variety of +wordings, so keywords don’t help as much to select relevant passages (Appendix 9.6.2). +5 +Case Study: Participant Flow in Randomized Controlled Trials +In this case study, we mostly focused on simple generalizable decompositions for long-form question-answering. +Through iteration on the selection and generation steps, accuracy on all subtasks improved substantially on a held-out +test set: Extracting experiments improved from 40% to 70%, trial arms from 55% to 86%, and adherence from 53% to +70%. +5.1 +Setup +Randomized controlled trials often include a standardized diagram that helps the reader understand what happened in +a given trial: How did participants journey through the study, from enrollment to final analysis? These diagrams are +called CONSORT diagrams (see Appendix 9.6.1 for an example). In our experience, they appear in about half of recent +RCTs are very often incomplete. If we could generate complete CONSORT diagrams for all RCTs, we could provide +valuable information to readers. +In this case study we consider a limited version of this task: We name the trials in the paper, the arms in each trial, and +describe the participant adherence rate for each arm. Figure 6 shows an example. +10 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +input +What were the experiments, trial +arms, and adherence for each +arm in Bloom et al. 2016? +methods +Baseline: Select-and-generate like +Elicit +Improving selection +Fine-tuned classifiers +Task-specific prompting +Zero-shot perplexity classifier +Pruning +Improving generation +Decontextualization +Auto-few-shot demonstrations +output +Experiments +Fall experiment +Arms +Park walking +Adherence: 72% of the +participants in the +intervention groups +engaged in relaxation or +park walking during their +lunch break at least 8 times. +On average, they engaged +in relaxation/park walking +8.6 out of 10 times during +the intervention period. +Relaxation exercises +Adherence: same as for +park walking +Control +Adherence: not mentioned +Spring experiment +Arms +Park walking +Adherence: 76% of the +participants in the +intervention groups +engaged in relaxation or +park walking during their +lunch break at least 8 times. +On average, they engaged +in relaxation/park walking +8.5 out of 10 times during +the intervention period. +Relaxation exercises +Adherence: same as for +park walking +Control +Adherence: not mentioned +Figure 6: In the participant flow case study, the goal is to analyze a randomized controlled trial and extract the +experiments, the arms for each experiment, and the adherence rate for each arm. Starting again from the Elicit +select-and-generate baseline, we implemented domain-agnostic improvements to both selection and generation. +5.2 +Evaluation +Experiments and arms are relatively easy to evaluate—we can check whether each experiment/arm from the gold +standard is represented and whether there are any additional ones that should not be there. +Adherence tends to require a narrative answer, and to be more subjective. This means that it may benefit more from a +nuanced decomposition, and approaches that work well for adherence may generalize better to other fuzzy tasks. +Each arm has an adherence answer, so there are a total of 135 adherence answers in our test set. Often, no information +about adherence is available in the text of the paper—information about adherence is only available for 56/135 arms +(41%) in our test set. So the adherence task is substantially a classification task (adherence mentioned vs. not mentioned). +5.3 +Iterations +5.3.1 +Baseline: Select-and-generate like Elicit +The baseline is the same “select-then-generate” approach used for general paper question-answering in Elicit as +described above in Section 4.2.1. This is a reasonably strong baseline, scoring 53% on the adherence subtask. +Diagnosis +For adherence, 80% of errors were false negatives, i.e. saying that adherence was not mentioned when it in fact was +(see Table 5). Further, all (51/51) of these false negatives resulted from errors at the selection stage—the answer really +was not mentioned in the top-1 paragraph from the monoT5 ranker. By using an oracle for selection with the same +generation approach, accuracy on the adherence task rose to 77%. So, the baseline fails primarily by failing to make +good use of the long context of the paper. For this reason, we started by iterating on selection. +5.3.2 +Improving selection +Finetuned classifiers +Because our research is aimed at generalizable approaches, we restricted any finetuning to approaches that were either +very sample-efficient (so little labeling effort would be needed to adapt the approach to new tasks) or very general. We +11 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +evaluated small dataset and machine-in-the-loop labeling approaches, favoring approaches using at most a few hours of +subject matter expert labeling effort. +We adapted the approach from Liu et al. (2022) to fine tune classifiers for text classification subtasks. This approach +combines low-rank adaptation (scaling selected activations by learned vectors) with additional regularization terms in +the loss and hyperparameters the authors found to generalize well. This combination enables (a) a very small set of +finetuned weights, enabling multiple finetuned tasks to share a single model backbone at inference time, and (b) true +low-data finetuning, since the approach prescribes hyperparameters, thus obviating the need for separate training and +validation sets. +Liu et al. (2022) suggest that their approach works well with as few as 20 examples. bootstrapped to a larger number of +positive examples and more than 20 "hard negative" examples using weak models: For our tasks, most paragraphs are +not relevant to answering a given question, and it is time-consuming and expensive to collect even a moderate amount +of positive examples by hand. We ensembled multiple weak classifiers such as monoT5, BART-based classifiers, and +T0-3B to identify positive examples from thousands of papers, then had experts moderate the examples identified by +all the weak classifiers as answering the question, leaving in any negatives from this approach as “hard negatives”. +Intuitively, we would expect that this would provide an outsized number of “easy positives” (from the ensemble +consensus) and “hard negatives” (false positives from the ensemble consensus, as corrected by experts). Although we +would expect this approach to have a difficult time identifying “hard positives”, in practice it created a dataset that +worked very well in concert with T-few finetuning, bringing performance on a holdout set on the selection subtask for +adherence from 72% before finetuning to 94% accuracy after. +Using T0-3B and T0pp (11B parameters) as base models, we finetuned using the T-Few approach. We found that the +models generalized better when we left zero-shot prompts in our finetuning data, which can be interpreted as a form of +regularization (in this example, the decoder answer choices are yes and no): +Context: {{ paragraph}} +Section: {{ section}} +Answer yes if the following sentence is about how many participants in the study complied with +the study's protocol, had to drop out, or withdrew; answer no if it is about something else, +such as the study's design, sampling strategy, or results. +Sentence: {{ sentence}} +Answer: +Many failures to get the right answer using finetuned classifiers at the selection stage were rooted in false negatives +from the classifier (for 67% of failures in the development set with n = 9, the classifier missed the gold standard +excerpts), and the generation stage was somewhat robust to “distractors” caused by lower precision. We therefore +explored variations on this technique to favor recall. For instance, we explored classifying sentences (while including the +paragraph as context), then, at inference time, classifying a paragraph as positive if any of its sentences were classified +as positive. We also tried finetuning with multiple zero-shot prompt templates, then, at inference time, classifying a +paragraph as positive if it was classified as positive via any of the prompt templates. +On the adherence subtask, the finetuned classifier significantly outperformed the base monoT5 classifier (paragraph +classification F1 = 0.10 for Elicit top-1 approach versus F1 = 0.77 via T-Few finetuning of T0-3B). +Overall, while this approach led to significant performance improvement, it also had downsides: It requires data and +training, and is less interpretable than some of the classification approaches we explored later. In the future, we would +consider data-efficient finetuning of classifiers where (a) the task was hard to specify but data was easy to collect or (b) +where the inference-time computational advantage of a smaller finetuned classifier was needed. +Task-specific prompting +An alternative to finetuning is crafting prompts for specific subtasks. This reduces the data collection burden of +finetuning, requiring just a few examples, at the cost of involving a different sort of expertise in the craft of writing +effective prompts. +With text-davinci-002, we found that we could get considerable performance gains on specific subtasks by prompt +engineering. Naive zero-shot or few-shot instructions performed performed worse than “simulation” approaches where +the prompt emulated a genre of text that had certain desirable properties, such as containing correct mathematical +12 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +reasoning. Appendix 9.6.4 shows an example of a prompt that frames a task as excerpts from a (nonexistent) textbook +on evaluating scientific papers. Appendix 9.6.5 shows a prompt that simulates a StatsExchange post. +Because we wanted to focus on general-purpose decompositions in this case study, and because we expect simulation- +based approaches to be less necessary with future instruction-tuned models, we did not evaluate this line of approaches +systematically. +Zero-shot perplexity classifier +Ideally, we could improve selection performance over the baseline monoT5 classifier with an approach that requires +neither substantial data collection nor task-specific prompt engineering. +Inspired by the relative strength of the “generation” stage of the baseline approach, we implemented a zero-shot +perplexity-based cross-encoder using text-davinci-002 that performed remarkably well across many questions, +outperforming the monoT5 classifier and competitive with the T-Few low-rank finetuning. +In this paragraph classification approach, we used the following prompt with text-davinci-002, echoing the log +probs back from the OpenAI API in order to measure perplexity: +Answer the question "{{ question}} " based on the excerpt from a research paper. Try to answer, +but say "The answer to the question is not mentioned in the excerpt" if you don't know how to +answer. Include everything that the paper excerpt has to say about the answer. Make sure +everything you say is supported by the excerpt. The excerpt may cite other papers; answer about +the paper you're reading the excerpt from, not the papers that it cites. Answer in one phrase +or sentence: +Paper excerpt: {{ paragraph}} +Question: {{ question}} +Answer: The answer to the question is not mentioned in the excerpt +We then scored paragraphs by the inverse perplexity of the tokens X : {(xi=1, . . . , xt)} in the sequence “The answer to +the question is not mentioned in the excerpt”: +exp +� +1 +t +t +� +i=1 +log pθ(xi|x Paragraph 7 +Which of paragraphs 7 and 12 better answers the question, "What were the +trial arms (subgroups of participants) of the experiment?" +23 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +Paragraph 7: ``Trachoma-control programs have distributed more than 600 +million doses of oral azithromycin in an effort to eliminate the ocular +strains of chlamydia that cause the disease. 1,2 ..." +Paragraph 12: ``In this cluster-randomized trial, we assigned communities in +Malawi, Niger, and Tanzania to four twice-yearly mass distributions of either +oral azithromycin (approximately 20 mg per kilogram of body weight) or +placebo..." +A: Paragraph 12 +ii. Answer: Azithromycin, Placebo +Answer the question "What were the trial arms (subgroups of participants) of the +experiment?" based on the following paragraphs. +Paragraphs: +In this cluster-randomized trial, we assigned communities in Malawi, Niger, and +Tanzania to four twice-yearly mass distributions of either oral azithromycin +(approximately 20 mg per kilogram of body weight) or placebo... +A: Azithromycin, Placebo +(b) Describe the trial arms: Communities assigned to receive the oral antibiotic azithromycin, Communities +assigned to receive the vehicle of the azithromycin suspension in an identical bottle +Rank and answer, with relevance determined based on the trial arm we’re describing +(c) Do any of the arms look like placebos?: Yes +Classify whether the paper used a placebo or not. Err on the side of caution: If +you are unsure, answer "Unclear". +Arm 1: Azithromycin +Description of arm 1: communities assigned to receive the oral antibiotic +azithromycin +Arm 2: Placebo +Description of arm 2: communities assigned to receive the vehicle of the +azithromycin suspension in an identical bottle +Does the paper use a placebo? Give your reasoning, then answer. +A: Yes +(d) Could participants tell which arm they were in?: Yes +...Followup-question: can participants tell which arm they're in? +A: Yes +(e) Was a placebo used? (based on arms): Unclear +Give the answer from "Naively classify placebo from arms" unless participants can tell which arm they’re +in, in which case say "Unclear" +2. Was a placebo used, based on the paragraphs? Yes +(a) Was there a placebo, according to each paragraph? Yes, Unclear, ... +i. Was there a placebo, according to this paragraph? Yes +Paragraph from paper: The placebo contained the vehicle of the oral azithromycin +suspension and was bottled and labeled identically to azithromycin. +Based on the paragraph, did the paper use a placebo? Give your reasoning step by +step, then answer. +A: Yes +24 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +(b) Was a placebo used, aggregate answers from paragraphs? Yes +If any paragraph indicated there was no placebo return "No". Then, if any paragraph indicated a placebo, +return "Yes". Otherwise, return "No". +3. Was a placebo used, combining answers from arms and paragraphs? Yes +In short: say "Yes" if either arms or paragraphs say yes and the other doesn’t contradict, else say "No" +Description +For description we again used rank-and-answer. In the example above, this return “The placebo group received the +vehicle of the oral azithromycin suspension. The suspension was bottled and labeled identically to azithromycin.” +9.5.5 +Additional baselines +Stuff paper in prompt +The most obvious language-model baseline is to just stuff as much of the paper as possible in the prompt. +Decomposition outline: +1. Stuff the prompt with as much of the paper as will fit (a text-davinci-002 prompt plus completion can be 4096 +tokens total) and ask the model “Was this a placebo-controlled study? Let’s think step by step:". Then: +2. Classification: Add a classification question after the model’s answer to the original prompt: “So, to sum up, +was this a placebo-controlled study? Answer "Yes", "No", or "Unclear"." +3. Placebo description: Add a description question after the model’s answer to the classification prompt: “Got it! +Please describe the placebo (in one sentence)." +This method does about as well as the Elicit baseline for both classification and description, and seems worse than the +task decomposition for both (see Table 2; vs. the decomposition, p=0.0002 for classification and p=0.06 for description). +It’s interesting to see that this simple language model approach performs worse than the keyword matching algorithm. +Another big weakness of this method is that we don’t know what part of the paper the answer is based on, so we can’t +easily check if it’s correct. +Classify by Elicit Prompt “Not Mentioned" Perplexity then Answer with Elicit Prompt +This approach is discussed in more detail in Section 5.3.2. We include this as an alternative, potentially stronger +alternative to the Elicit baseline. In this case, it performs similarly to the Elicit baseline. +25 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +9.6 +Participant flow case study details +9.6.1 +CONSORT diagram example +CONSORT diagrams document the flow of participants through a study. The figure below shows an excerpt of the +diagram from Fowler et al. (2019). +9.6.2 +Dataset examples +The table below shows examples of the most relevant quotes for evaluating adherence in the participant flow task. +Paper +Most relevant quotes +Answer +1 +"All adolescents who were recruited completed all +3 study visits, and all intervention arm participants +initiated gameplay." +All participants in the intervention group initiated +gameplay. +2 +"In the Imipramine trial, 49 of the 63 randomized +participants completed 6 weeks of treatment (for +details, see previous papers 14,15) and 50 of the 63 +randomized participants provided ESM data at base- +line and post-intervention and thus were included in +the present analyses." +49 of the 63 randomized participants across both +groups completed 6 weeks of treatment +26 + +170Patientsrandomized +86Randomized to receive vitaminC +84Randomizedtoreceiveplacebo +84Receivedinterventionas +83Received interventionas +randomized +randomized +2Did not receive intervention +1Did not receive intervention +asrandomized(dueto +asrandomized(duetoacute +alveolarhemorrhage) +eosinophilicpneumonia) +84 Included in primaryanalysis +83IncludedinprimaryanalysisIterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +9.6.3 +Participant flow error analysis +Task: +Adherence +Best performance +Method: +Baseline +Best performance +n +% +n +% +False negative +51 +80% +18 +45% +False positive +6 +9% +12 +30% +Hallucination +4 +6% +0 +0% +Missing info +3 +5% +0 +0% +Includes irrelevant info +1 +2% +6 +15% +Answered the wrong question +0 +0% +4 +10% +Total +64 +100% +40 +100% +Table 5: Failure mode counts for adherence +Failure Mode +Decomposition Answer +Gold Answer +False negative +Not mentioned. +52 out of 53 allocated endometrial +resections were performed +False positive +149 children were entered into the +study, and each child was followed +up for two years. +Not mentioned +Hallucination +The paper says that the subjects ad- +hered to the treatment regimens. +Across all groups, 1 man dropped out +because of compliance problems. +Missing info +The paper says that both groups had +good compliance with all products in +each nursing home. +Compliance with oral supplementa- +tion was "good", and daily intake av- +eraged about 400 kcal +Includes irrelevant info +Three of the 43 men dropped out dur- +ing the treatment phase because of +compliance problems (one subject), +use of illicit drugs detected on rou- +tine drug screen (one subject), and an +automobile accident (one subject). +Across all groups, 1 man dropped out +because of compliance problems. +Answered the wrong question +All participants were sober at the +start of the experiment, and they +were given their drinks and they +watched a documentary about South +Africa for half an hour to allow the +BAC level to reach its maximum. +Adherence was probably 100%, +since the study lasted two hours and +was conducted immediately after ran- +domization. +Table 6: Failure mode examples for adherence +27 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +9.6.4 +Adherence subtask prompt, textbook style +We framed some tasks as excerpts from a (nonexistent) textbook on evaluating scientific papers, framing the in-context +examples as exercises from this textbook. We found that this discouraged hallucination (since it is unlikely for answers +to exercises to go beyond what is available in the supplied text) and allowed us to naturally and didactically specify +subtle properties of the task, since textbook examples often build on one another, and it is unlikely for a textbook answer +to go beyond what is present in the “problem”. +From the textbook, "Critically Evaluating Interventional Studies," Chapter 3: +When evaluating the quality of a randomized controlled trial, you should also consider whether any participants dropped +out of the study or failed to follow its protocols correctly. This is sometimes called "adherence," "attrition," or +"compliance". If too many participants failed to receive the intervention or perform it correctly, for whatever reason, +this may damage the internal validity of the study's results. +Unfortunately, papers are often not as clear as they should be when discussing adherence. For simple interventions that +are accomplished in one shot (e.g., having a group of college students complete a test in a lab that takes 30 minutes), +the study doesn't discuss adherence unless something unusual happened, and we can safely assume that everyone in the +sample completed the study. Sometimes studies provide specific numbers or percentages of people who dropped out +(attrited), and sometimes they only provide qualitative descriptions, such as saying that adherence was "generally +good." Often, papers are genuinely unclear, and we can only conclude that there is not enough information in the paper +for us to know anything about adherence or compliance. +Let's look at excerpts from five different papers to see what information, if any, they provide about the study's +adherence, attrition, or compliance. We'll have to identify what each extract tells us about adherence (some extracts +may only discuss methodology or results, telling us nothing about adherence), and for some, we may have to conclude that +the attrition or compliance is simply unclear. +First, consider these three excerpts from a paper studying the Tumaini game: +1. Intervention arm participants completed a 45-minute informational onboarding session, including instructions on the +interface, technology, and game content. They were instructed to play at least 1 hour per day for the 16 days of the +study and asked not to share their own gameplay profile with others. The game interface allows for 5 additional players' +profiles so that others may play without compromising the enrolled player's data. Intervention participants were +provided with a phone with the game preloaded and used it at their own pace for the duration of the intervention. +Control participants received standard of care, namely no additional intervention beyond any existing sex education from +family, school, and peers. No specific data on the content or source of this education were collected from participants. +All study smartphones were returned by the participants at the end of the intervention period. +2. Preliminary cleaning of survey data was conducted in MS Excel, with additional cleaning and all analyses completed +using SAS version 9.4 (SAS Institute Inc., Cary, NC, USA). All control arm participants were included in analyses. One +participant from the intervention arm was removed from analyses of effect at T2 due to delayed completion of the T2 +survey. His data were retained for T1-T3 analyses, as he completed all other study activities on time. Descriptive +statistics on demographic questions and game feedback questions were computed. +3. We recruited and enrolled 60 adolescent participants. Half of the participants were allocated to the intervention arm. +All adolescents who were recruited completed all 3 study visits, and all intervention arm participants initiated +gameplay. Participant demographics are presented in Table 3 . There were no significant demographic differences between +the two arms. Preliminary calculations of exposure indicate that the intervention arm played Tumaini a mean of +approximately 27 hours over the 16 days of the intervention. +Let's think about what each excerpt tells us, if anything, about adherence, attrition or compliance: The first excerpt +describes the study's methodology, but does not tell us how many or how well participants followed the instructions, so +it does not inform us about adherence. The second excerpt tells us that all control arm participants were included in +analysis, but one intervention arm participant was removed from the analysis of effect at T2 but included in the T3 +analysis; this is attrition information. The third excerpt says that all participants completed all visits and that all +intervention arm participants initiated gameplay; this is adherence information. +Here's all the information in this paper about adherence, attrition, and compliance: All participants completed all +visits, and all intervention arm participants initiated gameplay. One intervention arm participant was not included in +the T2 analysis but was included in the T3 analysis. +Second, consider these three excerpts from a paper studying the Preschool Situational Self-Regulation Toolkit (PRSIST) +Program: +28 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +1. All children in their final prior-to-school year in these centers, who attended at least one of the 1-2 assessment +days, were invited to participate in this study. There were no further exclusion criteria. Parental consent to +participate was provided for 547 3-5-year old children, all of whom were identified as likely to be attending school in +the subsequent year. The flow of participants throughout the study is depicted in Figure 1 . At baseline, 473 of these +children were assessed (86.5%), with non-participation largely due to absence on the day of assessment. The mean age of +this sample was 4.44 years (SD = 0.38, range = 3.20-5.33), with a relative balance of boys and girls (48.2% girls). +Children who were identified as of Aboriginal or Torres Strait Islander descent comprised 7.2% of the sample, which is +in line with population estimates for this age group (Australian Institute of Health and Welfare (AIHW), 2012). Family +income was diverse: 11.9% of families qualified for full childcare benefit subsidies (low income); 65.5% of families +qualified for some childcare benefit (low-middle to middle-high income); and 22.7% of families did not qualify for any +childcare benefit subsidy (high income). Maternal education levels were also diverse: 9.5% did not complete high school; +9.3% completed only high school; 30.6% had completed a diploma, trade, certificate; 34.6% completed a tertiary degree; +and 16.0% a post-graduate qualification. At follow-up, 426 children were assessed, which corresponded to a 90.1% +retention rate. Nonparticipation at follow-up was due to the child having left the center or absence on the day of +assessment. +2. Based on these patterns of participation, 20 services (80%) were deemed to have met or exceeded the minimum threshold +of participation (i.e., completed the professional development modules and met the minimum of three child activities per +week). Those that did not participate in the program were a result of: preparations for government assessment and rating +(n = 1); substantial illness, maternity leave or turnover of key staff that precluded participation (n = 2); or low-or +non-participation for undisclosed reasons (n = 2). Two of these five centers did not participate in any program elements. +The other three centers did not engage with professional development modules or induction teleconference call yet +completed child activities. Overall, there were good levels of adherence to the program, especially amongst those +centers without significant sector-imposed impediments to participation. +3. Inability to conclusively and exclusively provide evidence for one of these possibilities, however, highlights +limitations within the current study. That is, although the evaluation was rigorously designed and executed according to +CONSORT guidelines, funding considerations limited the roll-out and intervention period to only 6 months. It is possible +that a full year of program implementation would yield stronger program effects (see, for example, Schachter, 2015). It +is also possible that program effects would be strengthened with stricter adherence to highquality program +implementation. While fidelity data indicate good compliance in the frequency and timing of program elements, data are +insufficient to evaluate the integrity with which program elements were implemented. While in-person or video fidelity +checks were not possible in the current study, this would help monitor adherence. As a researcher-implemented model of +delivery would violate our aspiration for a lowcost and barrier-free resource for educators, a plausible middle ground +might be a coaching model that supports educators in implementation and adaptation of the program in their context. +Lastly, the program was designed with the intention to foster selfregulation in all children, and thus did not focus on +instances of dysregulation. However, it is clear that child dysregulation remains a significant concern for educators +(Neilsen-Hewett et al., 2019), and future iterations of the program would do well to more explicitly provide support for +these children. In guiding such an expansion of the program, there is evidence that children with frequent and severe +dysregulation require a different approach to fostering self-regulation, as demonstrated successfully in trauma-informed +practice approaches (Holmes et al., 2015). Future studies would also do well to consider implications of differing +educator qualifications and experience, whereby different types and levels of support may be needed at varying levels of +behavior challenges and educators' skills to address these. +Let's think about what each excerpt tells us, if anything, about adherence, attrition or compliance: The first excerpt +includes demographic information about the participants but also reveals that at baseline, 473 of the total sample of +547 children were assessed (with non-participation mostly due to absence), and at follow-up, 426 children were assessed +(with non-participation mostly due to the child having left the center or absence), corresponding to a 90.1% retention +rate. The second excerpt describes compliance with protocols: 20 of the 25 intervention centers met or exceeded the +minimum threshold of participation. The third excerpt describes compliance in the frequency and timing of program +elements as "good" but also says that the study did not monitor adherence with in-person or video checks, which would +have helped provide a better picture of compliance with the study design. +Here's all the information in this paper about adherence, attrition, and compliance: Of the initial sample of 547 +children, 473 were assessed at baseline and 426 at follow-up. While 20 of 25 intervention centers met or exceeded the +minimum threshold of participation and the frequency and timing of program elements was good, the study did not monitor +adherence with in-person or video checks. +Third, consider these four excerpts from a paper studying Study 2 on depression and psychosis: +1. The intervention was a single session that lasted approximately one hour for participants to provide informed consent, +complete a demographic form, watch videos relevant to their study arm, complete the assessments, and be debriefed. +Participants in either of the video groups stayed for the full hour, but participants in the control condition who did +not watch the video finished in about 50 min. In Study 2, which included two 8 min videos with diagnostic accuracy for +both conditions, the protocol required an additional 15 min. Survey data were collected using SurveyCTO (Ver 2.30, +Dobility, Inc., Cambridge, MA, USA), an android application, on tablets (www.surveycto.com/accessed on: 19 June 2017). +In Study 1, after completion of the video session, participants were invited to participate in the optional qualitative +interview to be held within one week. +29 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +2. After review of 2nd and 3rd year MBBS student rosters, 18 students were excluded prior to randomization because of +being international students not speaking Nepali or having already completed their psychiatry rotation. Among the +remaining students, 100 were selected for randomization to one of the three arms. No potential participants refused to +participate in this study. An additional six students were excluded at the time of analysis because information on their +demographic forms revealed that they were international students whose native language was not Nepali or they had +completed their clinical psychiatry rotation; this information had not been up to date in the class rosters at the time +of randomization (Figure 1 ). One participant in the service user arm was excluded because of both being an +international non-Nepali student and having completed a psychiatry rotation. Demographic characteristics of these +participants are in Table 2 . Of note, only three participants indicated that they were primarily interested psychiatry +as a specialty (see Figure 2 ). Participants were randomized into one the three conditions: the control group with no +video (n = 31, 33%), the didactic video group (n = 31, 33%), and the service user recovery testimonial video group (n = +32; 34%). +3. Due to limited time availability on the part of the researchers and students as well as the exploratory nature of the +interviews, only six participants completed interviews. Qualitative results were analyzed from a subset of six students, +two women and four men in their third year, who participated in in-depth interviews. +4. For the second study, 248 students were enrolled in first-and second-year MBBS program across the two institutions +participating. From roster, 28 students were excluded because of being international or having completed a psychiatry +clinical rotation. The remaining 220 students were randomized; however, seven students declined to participate or were +unavailable during data collection periods. Therefore, 213 participants were randomly allocated to the following arms: +didactic video condition (n = 73), the service user video condition (n = 72), and the no video control condition (n = 75) +(Figure 3 ). At the analysis phase, there were additional exclusions because of missing data or identification of +exclusion criteria that was not recorded in the school registers. Participant characteristics for each condition are +shown in Table 4 . +Let's think about what each excerpt tells us, if anything, about adherence, attrition or compliance. The first excerpt +describes the methodology, describing the intervention as taking place in a single one-hour session. This does not tell +us anything explicitly about adherence, but it does make it more likely that adherence was high, since participants only +had to attend the single session, which is easy to do. The second excerpt says that 18 students were excluded prior to +randomization; since this took place before sampling, it is not relevant to adherence. It also says that six students +were excluded at the time of analysis because it turned out that they met exclusion criteria. Although this is not +adherence strictly speaking, it is important to note when thinking about the validity of the results. The third excerpt +says that only six participants completed interviews. The fourth excerpt says that in Study 2, seven students declined +to participate or were not available during data collection after randomization of 220 students, and there were +additional exclusions at analysis phase because of missing data or identification of exclusion criteria. +Here's all the information in this paper about adherence, attrition, and compliance: This paper does not discuss +adherence explicitly. For the video study, six of the 100 randomized students were excluded from analysis, and in the +second study, seven of the 220 randomized students declined to participate or were unavailable during data collection +periods, with additional students excluded from the analysis because of missing data or identification of exclusion +criteria. Only six participants completed interviews. +Fourth, consider these three excerpts from a paper studying antioxidant/anti-inflammatory supplement containing lemon +verbena extract and omega-3 fatty acid: +1. Flow chart showing the dropout rate at different timepoints in the study. +2. Forty-eight (48) participants were enrolled for screening evaluation (Fig. 1 ) and after 3 exclusions, 45 +participants were randomly assigned either to placebo or nutritional supplement groups, n = 22 and n = 23, respectively. +Of these, 14 participants were withdrawn during the study for different reasons; there were 10 dropouts in the placebo +group and 4 dropouts in the supplement group (treatment refusal, irregular treatment, starting on medication, or +occurrence of an adverse event [AE]). Finally, 31 participants completed the study (12 in the placebo and 19 in the +supplement group; Fig. 1 ). +3. Only 1 patient reported an AE (i.e., a heartburn sensation). The subject, who was in the placebo group, stopped the +treatment immediately and was excluded from the study (Table 1 ). No major complications were reported by this subject. +Let's think about what each excerpt tells us, if anything, about adherence, attrition or compliance: The first excerpt +refers to a flow chart showing the dropout rate, but since we do not have the figure here, we cannot conclude anything +from this about the study's attrition. The second excerpt says that there were 10 dropouts in the placebo group of 22 +participants and 4 dropouts in the supplement group of 23 participants, meaning that 31 participants out of the initial +45 participants after randomization completed the study. The third excerpt provides more detail for one patient in the +placebo group who dropped out, stopping treatment after experiencing a heartburn sensation. +Here's all the the information in this paper about adherence, attrition, and compliance: Ten of the 22 participants in +the placebo group dropped out, and 4 of the 23 participants in the supplement group dropped out. +Fifth, consider these {{ paragraph_n}} excerpt{{ "s" if len(paragraphs) > 1 else ""}} from a paper studying +{{ intervention}} : +{{ paragraphs_to_numbered_list(paragraphs).strip()}} +Let's think about what {{ "each" if len(paragraphs) > 1 else "this"}} excerpt tells us, if anything, about adherence, +attrition or compliance: +30 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +9.6.5 +Adherence subtask prompt, Stats Exchange style +We experimented with attempting to calculate a final adherence rate where it was available. One possible approach to +such calculations is to, e.g., generate Python code then offload the calculations to a Python interpreter. We also found +that we could get stronger in-context mathematical reasoning in a zero-shot setting by simulating a StatsExchange post. +A user on Statistics Stack Exchange needs help with this problem: +Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention +or at what rate they completed the intervention. Ignore irrelevant excerpts. If there is not enough information to +determine the answer, answer "Unclear". +The intervention is Intracytoplasmic sperm injection (ICSI). +You must decide which excerpts are relevant to your reasoning. Some may be very relevant, and most will be irrelevant. +Excerpts: Thus, -90% of the couples had an embryo transfer and the viable pregnancy rate was 21% for ejaculated, 22% for +epididymal and 19% for testicular spermatozoa (with 25-30% multiple pregnancies)....Data forms were sent to all centres +that had already taken part in the previous ICSI Task Force survey, as well as to new ICSI centres known from the +literature or from national registries....Up to 31 December 1995, 101 centres had submitted their clinical results based +on 2 to 2507 cycles (see Table I )....Hence, only 17 of the 101 centres that have submitted ICSI results for 1995 are +performing a prospective follow-up of the children and only nine as a part of a special project, while another 46 +centres are trying to collect information by contacting the infertility specialist, the paediatrician, or the nurses. +Before giving your answer describe your reasoning in detail. Think about how many people participated in the +Intracytoplasmic sperm injection (ICSI). Explain how to deduce an answer the question "What is the adherence rate?" from +the relevant excerpts step by step. State your final answer. +The best response from Statistics Stack Exchange was the following: +The quote "Hence, only 17 of the 101 centres that have submitted ICSI results for 1995 are performing a prospective +follow-up of the children and only nine as a part of a special project, while another 46 centres are trying to collect +information by contacting the infertility specialist, the paediatrician, or the nurses." Talks about the number of +centres that performed follow-ups but makes no mention of the number of patients who completed, adhered or complied with +the intervention. Therefore it is impossible to tell. +Final answer: Unclear +A user on Statistics Stack Exchange needs help with this problem: +Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention +or at what rate they completed the intervention. Ignore irrelevant excerpts. If there is not enough information to +determine the answer, answer "Unclear". +The intervention is malaria chemoprevention (SMC) and long-acting artemisinin-based combination therapies (ACTs). +You must decide which excerpts are relevant to your reasoning. Some may be very relevant, and most will be irrelevant. +Excerpts: A study including approximately 800 children in each group, allowing for a 10% loss to follow-up, would have +90% power to detect a reduction of 20% in the malaria incidence rate between children given a long-acting ACT and a +short-acting ACT....A study with 800 children in each group would have greater than 90% power to detect a difference of +25% between children given a long-acting ACT and those given SMC, and very high power to detect larger differences +between the SMC group and the short-acting ACT....Caregivers were followed up by CHWs between 4 and 7 days after each +SMC administration to ask about adverse events and adherence to the SMC regimen using a standard form....However, it is +unlikely that adherence was in reality this high, as during the qualitative component of this study, some caregivers +were found to have tablets remaining that had not been administered....Adherence data were collected but implausibly +high values of adherence were reported, as seen elsewhere [23]. +Before giving your answer describe your reasoning in detail. Think about how many people participated in the malaria +chemoprevention (SMC) and long-acting artemisinin-based combination therapies (ACTs). Explain how to deduce an answer +the question "What is the adherence rate?" from the relevant excerpts step by step. State your final answer. +The best response from Statistics Stack Exchange was the following: +A quote says "However, it is unlikely that adherence was in reality this high, as during the qualitative component of +this study, some caregivers were found to have tablets remaining that had not been administered." This tells us that the +follow-up system was not completely accurate, and that some people did not adhere to the SMC regimen. However, we are +not given any numbers or percentages, so it is impossible to say what the adherence rate was quantitatively. But this +along with "Adherence data were collected but implausibly high values of adherence were reported, as seen elsewhere". +Leads us to believe that authors found the adherence to be high but thought is was implausible. +Final answer: The adherence was stated to be high. But authors thought is was implausible. +A user on Statistics Stack Exchange needs help with this problem: +Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention +or at what rate they completed the intervention. Ignore irrelevant excerpts. If there is not enough information to +determine the answer, answer "Unclear". +31 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +The intervention is pulsed electromagnetic field (PEMF) therapy. +You must decide which excerpts are relevant to your reasoning. Some may be very relevant, and most will be irrelevant. +Excerpts: A total of 286 subjects indicated their interest in the study, and they were sent a health questionnaire with +a prepaid return envelope to assess their preliminary eligibility to participate in the study....Twenty-three +respondents declined to take part in the trial and 263 respondents returned the questionnaire, 133 of which were +excluded from the study due to exclusion criteria....Fourteen patients did not meet the eligibility criteria during the +clinical examinations, and eight participants withdrew from the study, thus leaving a sample of 108 participants for the +study....Power calculations indicated that a sample of 110 patients (55 in each treatment), assuming a dropout rate of +approximately 10%, would provide an 80% (b ¼ 0.20) chance of detecting a 40% (a ¼ 0.05) difference in improvement +between the active and sham treatments....Of the 108 patients who started the treatments, nine dropped out during the +study....Four patients dropped out during the active treatment period....Five patients dropped out during the sham +treatment period....The reasons for those dropouts were amplified overall pain and worsened irritable bowel syndrome (n +¼ 1), lumbar herniated disc (n ¼ 1), cervical herniated disc surgery (n ¼ 1), chronic pneumonia (n ¼ 1), and unwilling +to continue (n ¼1; Fig. 1 )....Of the enrolled patients, 92% completed the study....Mean treatment compliance, measured +as attendance at all 168 planned, 8-min treatment sessions, was 134 AE 41 (80%) when using the active devices and 131 AE +44 (78%) when using the sham devices....The treatment adherence was high ($80%)....Fourth, considering that high dropout +rates are a major issue with crossover design [Mills et al., 2009], in this study the dropout rate was low (8%) despite +the relatively long duration of the study (28 weeks). +Before giving your answer describe your reasoning in detail. Think about how many people participated in the pulsed +electromagnetic field (PEMF) therapy. Explain how to deduce an answer the question "What is the adherence rate?" from +the relevant excerpts step by step. State your final answer. +The best response from Statistics Stack Exchange was the following: +In context of this study the adherence could mean either the completion or compliance rate. We will consider both +individually. +For the completion rate we know that of the 108 patient 9 dropout during the course of the study. So 108-9 = 99 complete +the study. Hence the completion rate is 99/108 = 91.7%. +For the compliance rate we are told "Mean treatment compliance, measured as attendance at all 168 planned, 8-min +treatment sessions, was 134 AE 41 (80%) when using the active devices and 131 AE 44 (78%) when using the sham devices." +and that "The treatment adherence was high ($80%)." So the compliance rate is 80% for the active devices and 78% for the +sham devices. +Final answer: The completion rate is 91.7%. The compliance rate is 80% for the active devices and 78% for the sham +devices. +A user on Statistics Stack Exchange needs help with this problem: +Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention +or at what rate they completed the intervention. Ignore irrelevant excerpts. If there is not enough information to +determine the answer, answer "Unclear". +The intervention is {{ intervention}} . +You must decide which excerpts are relevant to your reasoning. Some may be very relevant, and most will be irrelevant. +Excerpts: {{ "...".join(excerpts)}} +Before giving your answer describe your reasoning in detail. Think about how many people participated in the +{intervention}. Explain how to deduce an answer the question "What is the adherence rate?" from the relevant excerpts +step by step. State your final answer. +The best response from Statistics Stack Exchange was the following: +32 + +Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes +9.6.6 +Decontextualization few-shot prompt +Instructions: Enrich each Passage with the Context. +Context: Lisa loves to play practical jokes. +Passage: But sometimes she goes too far. +Rewrite: But sometimes she [Lisa] goes too far. +--- +Context: The Super Bowl XLI halftime show took place on February 4, 2007. +Passage: It was headlined by Prince. +Rewrite: It [The Super Bowl XLI halftime show] was headlined by Prince. +--- +Context: More than one fifth of the world’s population lives on less than Purchasing Power Parity (PPP) US$1.25 a day, +and there is an emerging international consensus that this share should (and can) be driven close to zero by 2030 (1, +2). +Passage: Reaching this objective will require enabling the poorest families, who are often the most marginalized within +their villages, to shift from insecure and fragile sources of income to more sustainable income-generating activities. +Rewrite: Reaching this objective [driving the share of the world’s population that lives on less than Purchaing Power +Parity (PPP) US$1.25 a day from more than one fifth of the to zero by 2030] will require enabling the poorest families, +who are often the most marginalized within their villages, to shift from insecure and fragile sources of income to more +sustainable income-generating activities. +--- +Context: We present results from randomized control trials (RCTs) in six countries of a particular approach to foster +self-employment activities amongst the very poor. Originally designed and implemented by BRAC, a large Bangladeshi NGO +that runs several country-wide programs, the ``Graduation" program provides a holistic set of services, including the +grant of a productive asset, to the poorest households in a village (referred to by BRAC as the ``ultra-poor"). The +beneficiaries [of the Graduation program, the poorest housholds in a village, or the "ultra-poor"] are identified +through a participatory process in a village meeting, followed by a verification visit by the organization’s [the +implmenter of the "Graduation" program] staff. Selected beneficiaries [among the poorest housholds in a village, or the +"ultra-poor"] are then given a productive asset [by the implementer of the "Graduation Program"] that they choose from a +list, training and support for the asset they have chosen, as well as general life skills coaching, weekly consumption +support for some fixed period, and typically access to savings accounts and health information or services. +Passage: These different activities (plus regular interactions with the households over the course of a year) are +designed to complement each other in helping households to start a productive self-employment activity. +Rewrite: These different activities [training and support for the assest they have chosen and received, general life +skills coaching, weekly consumption support, and typically access to savings accounts and health information or +services] (plus regular interactions with the households over the course of a year) are designed to complement each +other in helping households [beneficiaries selected for the "Graduation" program from among the poorest housholds in a +village, or the "ultra-poor"] to start a productive self-employment activity. +--- +Context: {{ context}} +Passage: {{ passage}} +Rewrite: +33 + diff --git a/J9AzT4oBgHgl3EQfyP6v/content/tmp_files/load_file.txt b/J9AzT4oBgHgl3EQfyP6v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df8b44dbe8ebce6f6a685849c62f7ad63fd80a10 --- /dev/null +++ b/J9AzT4oBgHgl3EQfyP6v/content/tmp_files/load_file.txt @@ -0,0 +1,1385 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf,len=1384 +page_content='ITERATED DECOMPOSITION: IMPROVING SCIENCE Q&A BY SUPERVISING REASONING PROCESSES Justin Reppert∗, Ben Rachbach, Charlie George, Luke Stebbing, Jungwon Byun, Maggie Appleton, Andreas Stuhlmüller Ought ABSTRACT Language models (LMs) can perform complex reasoning either end-to-end, with hidden latent state, or compositionally, with transparent intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Composition offers benefits for interpretability and safety, but may need workflow support and infrastructure to remain competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We describe iterated decomposition, a human-in-the-loop workflow for developing and refining compositional LM programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We improve the performance of compositions by zooming in on failing components and refining them through decomposition, additional context, chain of thought, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' To support this workflow, we develop ICE, an open-source tool for visualizing the execution traces of LM programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We apply iterated decomposition to three real-world tasks and improve the accuracy of LM programs over less compositional baselines on held-out test sets: describing the placebo used in a randomized controlled trial (25% → 65%), evaluating participant adherence to a medical intervention (53% → 70%), and answering NLP questions on the QASPER dataset (38% → 69%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' These applications serve as case studies for a workflow that, if automated, could keep ML systems interpretable and safe even as they scale to increasingly complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 1 Introduction Language models are often trained using feedback on outcomes, leveraging reward models that imitate human evaluations provided as pairwise comparisons (Christiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This works well for basic question-answering, summarization, simple code generation, and general short-form instruction-following (Stiennon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For these tasks, good outputs can readily be distinguished from bad ones, especially with model-supported evaluation (Saunders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' And with rare exceptions like WebGPT (Nakano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2021), there is no difference between the model’s output and the relevant outcomes, so evaluations relatively directly inform the model’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' However, as model capabilities and task complexities scale up, outcome-based evaluation may run into alignment problems: First, for some important applications the process used to generate the output matters as much as the output itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Consider long-range forecasting, and policy decisions informed by such forecasts: The quality of a forecast or decision depends on the assumptions, evidence, and reasoning used to produce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If feedback doesn’t inform the process used to generate outputs, we may get results that look good (because they are optimized for favorable evaluation), and may even look systematic, but are worse in exactly the ways that can’t easily be measured (Stuhlmüller and Byun, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Second, outcome-based feedback may create incentives for language models to deceive or manipulate their users by exploiting gaps or biases in the feedback signal (Amodei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In extreme cases, this could lead to models behaving as intended during training, hiding their true intentions and capabilities, but defecting at deployment (Cotra, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ∗Correspondence to justin@ought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='org and andreas@ought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='01751v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='CL] 4 Jan 2023 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Supervise outcomes ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Input Did this study on drowning use a placebo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' output Yes, a video about dog bites Supervise process Find sections about trial arms Extract trial arms from sections Classify & name placebo 5 paragraphs found Drowning video, dog bite video Input Did this study on drowning use a placebo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' output Yes, a video about dog bites observable Interim outputs Figure 1: Process supervision breaks black-box language model computation into human-understandable reasoning steps without end-to-end optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This increases transparency and trust because it allows inspection of intermediate results and reduces risks from imperfect feedback signals and deceptive alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Process supervision is an alternative to outcome-based training that uses language models to execute human- understandable task decompositions, either by imitating human steps or by rewarding human-endorsed steps (Christiano, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Stuhlmüller and Byun, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Uesato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This paradigm promises increased interpretability, trust, and alignment by reducing the reliance on black-box computation and enabling users to inspect and intervene in the model’s reasoning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Right now, process supervision is ahead of outcome-based training: Language model capabilities are weak, so complex tasks require a composition of multiple calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Indeed, engineered multi-step pipelines have been a staple of NLP for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' However, without scalable infrastructure to support it, we expect that outcome-based training will eventually crush process supervision performance-wise, leading to the alignment problems above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This paper describes our experience applying process supervision to academic question-answering in the context of Elicit2, the AI research assistant developed by Ought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Our contributions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A review of the literature on process supervision, highlighting gaps in workflows and tooling that contribute to making real-world use cases rare 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Iterated decomposition, a human-in-the-loop workflow for developing compositional language model programs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ICE, an open-source visualizer for language model execution traces 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Case studies that use this workflow to improve performance over baselines on three real-world tasks: (a) extracting placebo information from randomized controlled trials (RCTs), (b) analyzing participant flow in RCTs, and (c) answering questions about natural language processing papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' As models advance and become more reliable at completing component tasks, we expect that process supervision will become more feasible, and that eventually the iterated decomposition process will be automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' By sharing our workflow and tooling now when even basic tasks are still challenging, we hope to accelerate a future where LM deployments are controllable and interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2 Process Supervision Process supervision refers to approaches to LM training and deployment that rely on human-understandable intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We review the literature and highlight gaps, then explain the iterated decomposition workflow and ICE visualizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 Prior work on process supervision Over the past year, there have been significant advances in techniques for process supervision as well as frameworks, interfaces, and libraries for implementing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' At the same time, real-world use cases are still rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We review prior work and highlight gaps in the literature that may be contributing to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This is a rapidly growing field, so we are only able to review a sample of the work (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2https://elicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='org 2 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Authors Title Decomposition Training Workflow Survey Tool Theory Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Summarization Programs: Interpretable Abstractive Summarization [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.] ✓ ✓ ✓ Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) Recursively Summarizing Books with Human Feedback ✓ ✓ ✓ Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) ReAct: Synergizing Reasoning and Acting in Language Models ✓ ✓ Kumar and Talukdar (2020) NILE : Natural Language Inference with Faithful Natural Language Explanations ✓ ✓ Nakano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) WebGPT: Browser-assisted question-answering with human feedback ✓ ✓ Creswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Selection-Inference: Exploiting LLMs for Interpretable Logical Reasoning ✓ ✓ Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Maieutic Prompting: Logically Consistent Reasoning with Recursive Explanations ✓ ✓ Sanyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) FaiRR: Faithful and Robust Deductive Reasoning over Natural Language ✓ ✓ Creswell and Shanahan (2022) Faithful Reasoning Using Large Language Models ✓ ✓ Bostrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Natural Language Deduction through Search over Statement Compositions ✓ ✓ Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Locate Then Ask: Interpretable Stepwise Reasoning for Multi-hop Question Answering ✓ ✓ Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) Decomposing Complex Questions Makes Multi-Hop QA Easier and More Interpretable ✓ ✓ Dua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Successive Prompting for Decomposing Complex Questions ✓ ✓ Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Complex Reading Comprehension Through Question Decomposition ✓ ✓ Shridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Distilling Multi-Step Reasoning Capabilities of LLMs into Smaller Models [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.] ✓ ✓ Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Learning to Decompose: Hypothetical Question Decomposition [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.] ✓ ✓ Khattab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Demonstrate-Search-Predict: Composing retrieval and language models [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.] ✓ ✓ ✓ Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Measuring and narrowing the compositionality gap in language models ✓ ✓ Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Decomposed Prompting: A Modular Approach for Solving Complex Tasks ✓ Ozturkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) ThinkSum: Probabilistic reasoning over sets using large language models ✓ Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Re3: Generating Longer Stories With Recursive Reprompting and Revision ✓ Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) PAL: Program-aided Language Models ✓ Drozdov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Compositional Semantic Parsing with Large Language Models ✓ Trivedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Interleaving Retrieval with Chain-of-Thought Reasoning for [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.] Multi-Step Questions ✓ Kazemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) LAMBADA: Backward Chaining for Automated Reasoning in Natural Language ✓ Zelikman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) STaR: Bootstrapping Reasoning With Reasoning ✓ ✓ Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Calibrating Trust of Multi-Hop Question Answering Systems with Decompositional Probes ✓ ✓ Uesato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Solving math word problems with process- and outcome-based feedback ✓ Dohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Language Model Cascades ✓ ✓ ✓ Stuhlmüller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Factored Cognition Primer ✓ ✓ Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022a) PromptChainer: Chaining Large Language Model Prompts through Visual Programming ✓ ✓ Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022b) AI Chains: Transparent and Controllable Human-AI Interaction by Chaining LLM Prompts ✓ ✓ Chase (2022) LangChain ✓ ✓ Polu (2022) Dust ✓ Wies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) Sub-Task Decomposition Enables Learning in Sequence to Sequence Tasks ✓ Table 1: A sample of prior work on process supervision, categorized into work that primarily contributes (1) new task decompositions, (2) training and finetuning techniques, (3) workflows and tutorials, (4) surveys and frameworks for organizing prior work, (5) tools, and (6) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' While work on decompositions and training techniques is rapidly growing, there is little investigation of workflows, tooling, and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Task decompositions There is a quickly growing literature on how to compose multiple language model calls to improve performance or accomplish more difficult tasks: Creswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) and Creswell and Shanahan (2022) generate reasoning steps by alternating between selection and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They show that their approach outperforms other prompting methods on multi-step logical deduction and scientific QA tasks, and generates interpretable reasoning traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Kazemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) apply backward-chaining to simple logic tasks, starting with a goal proposition and recursively decomposing it into sub-goals until the sub-goals can be proved or disproved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) and Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) apply recursive summarization to generate summaries of long texts, such as books or articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They use LMs to summarize small sections of the text and then recursively summarize these summaries to produce a summary of the entire text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They show that recursive summarization improves the quality and coherence of the summaries, and enables human feedback and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) generate long stories by first creating a story plan, generating passages b yprompting a model with contextual information from the plan and the current story state, and then revising the passages by reranking and editing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ReAct (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022) interleaves generating chain-of-thought reasoning and actions pertaining to a task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', search, lookup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' WebGPT (Nakano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2021) finetunes LMs to answer long-form questions using a text-based web-browsing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 3 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) offload computation to Python interpreters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Trivedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) and Khattab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) interleave chain-of-thought with knowledge retrieval steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) study decomposition in general, letting the language model push subtasks to task-specific handlers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ozturkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) aggregate language model probabilities using mathematical combinators like sum and product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) recursively generate a tree of explanations for a statement, then determine the truth of the statement by treating the inference as a satisfiability problem over these explanations and their logical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Various works, including Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021), Dua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022), Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022), and Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022), explore decomposition of questions into subquestions, often under the name multi-hop question-answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Training and finetuning techniques The most relevant work is Uesato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) who directly compare process-based and outcome-based feedback for solving math word problems with LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They find that process-based feedback improves the correctness and interpretability of reasoning steps, but requires more label supervision than outcome-based feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) finetune GPT-3 using behavioral cloning and reward modeling to do summarization recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) train a decomposition model based on a parallel news corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Nakano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2021) compare behavior cloning, RL and rejection sampling for training a web browsing model, and find that a combination of behavior cloning and rejection sampling against a reward model worked best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Zelikman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) generate chains-of-thought, finetuning on the ones that lead to correct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Most other work under “training” in Table 1 trains small models from scratch for particular composition steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Theory and conceptual advances Wies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) show some benefits of process supervision: When concatenating intermediate supervision to the input and training a sequence-to-sequence model on this modified input, unlearnable composite problems can become learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) coin the name “compositionality gap” for the fraction of questions that the model answers incorrectly out of the questions for which the model answers all of the sub-questions correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They find this number to be around 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They show that chain of thought can narrow the gap, and that generating and answering follow-up questions further narrows it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Workflows, tutorials, and tools Dohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) introduce language model cascades, which are probabilistic programs that compose LMs with random variables and control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They formalize several existing techniques, such as scratchpads, verifiers, STaR, selection-inference, and tool use, as instances of language model cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' They provide an open-source probabilistic programming system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The Factored Cognition Primer (Stuhlmüller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2022) is a tutorial that explains how to write compositional LM programs, including recursive question-answering, debate, search, and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) show in human participant studies that letting users probe a language model with subquestions helps them calibrate when the model is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022a) and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022b) describe a closed-source visual programming interfaces for making compositional language model programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Dust (Polu, 2022) is a web service and Rust library for designing and deploying LM apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' LangChain (Chase, 2022) is a Python library that assists in the development of LM applications that involve chaining LMs with each other or with other experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Relation to this work While there is a quickly growing literature on task decompositions, there is effectively no work on theory and only a few tools, tutorials, and workflows for building real-world process-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We see this paper as demonstrating a real-world application (science Q&A) as well as a description of the workflow (iterated decomposition) and tooling (ICE) used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The prior work above varies in what exactly is meant by “process” and “supervision”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We provide a taxonomy in Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Briefly, in this work, we focus on decompositions designed by a human developer, with occasional choices made by the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We balance pragmatic decompositions that improve task performance with decompositions that reflect ideal reasoning and facilitate better supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Our decompositions mostly improve performance by helping the model use long context more effectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' although some also apply multiple lines of 4 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Task decompose into Subtasks Papers Results 1 Decompose the task into subtasks Subtasks should be independently meaningful so that they can be evaluated on their own 2 Run the decomposition across multiple inputs For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' research papers for which gold-standard results are available Gold standards VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 3 Evaluate the results Either manually or automatically using metrics such as accuracy and F1-score inputs outputs evaluate failed Subtasks 4 Inspect failing subtasks Zoom in on failures using the ICE trace visualizer to inspect the inputs and outputs of each subtask custom code edits decomposition 5 Improve failing subtasks Using further decomposition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' adding chain-of-thought, retrieval steps) or custom code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' prompt edits) Repeat Figure 2: Iterated decomposition is a workflow for human-in-the-loop language model programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We start with a trivial decomposition, evaluate it against gold standards, diagnose the source of failures using the ICE visualizer, refine the failing subtasks through further decomposition or other adjustments, and repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' reasoning to subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' When we talk about “supervision“, we mean checking the outputs or behavior of individual steps in the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We supervise the process to improve the LM program, the supervisor is the human developer, and we always have access to the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 Iterated decomposition We study process supervision via iterated decomposition, a human-in-the-loop workflow that incrementally improves a task decomposition through error diagnosis and amendment (illustrated in Figure 2): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Start with a minimal decomposition, breaking the task into subtasks that can be performed by a LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For example, the task of extracting the placebo used in academic studies can be decomposed into first finding the most relevant section, then generating the placebo (if any) given that section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Apply the decomposition to multiple test inputs with gold standard answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For example, this could be a dataset of academic papers from various domains, such as medicine, psychology, and economics, and their corresponding placebo descriptions, if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Figure 3: The Interactive Composition Explorer (ICE) is an open-source debugger for language model execution traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' It visualizes task decompositions and supports zooming in on failing subcomponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The top left pane in the screenshot shows the trace of function calls in the original execution hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The bottom pane supports sorting and filtering of calls, with each row showing values that were recorded at execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The right pane shows function inputs, outputs, recorded intermediate values, and source code of the call selected in the main pane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Evaluate the results automatically (using LM) or manually, using metrics such as accuracy, F1-score, or other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For example, the result of the placebo extraction task can be compared to a gold standard dataset of placebo descriptions from academic papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Identify failures using the ICE trace visualizer or other tools to inspect the intermediate inputs and outputs of each subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For example, given that the generated placebo is incorrect, one can determine whether the wrong section was found or the right section was found but the wrong answer was extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Improve failing components using further decomposition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', semantic decomposition into subtasks, chain- of-thought, retrieval), or by making custom edits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' prompt tweaks, language model inference parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For example, the task can be split into first finding each experiment mentioned in the paper, then asking for each experiment whether it used a placebo or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The process repeats steps 2-5 until it achieves good performance or exhausts the relevant resources (time, compute budget for experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3 Interactive Composition Explorer To support the error identification step in iterated decomposition, we have developed the Interactive Composition Explorer (ICE), an open-source3 debugger and execution trace visualizer for language model programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ICE records and visualizes the execution of all async functions in a Python program, including all inputs (prompts, parameters) and outputs (language model responses, intermediate values), in a way that is suitable even for executions 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='com/oughtinc/ice 6 Debate 8 OpenAlAgent: Complete Should we legalize all drugs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' → speaker Generate an answer to a question given some context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Turn Inputs and Outputs Source QuestionShould we legalize all drugs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' → Alice Legalizing drugs would reduce the b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='. OpenAlAgent: Complete Inputs You are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' There are turns left in the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Legalizing drugs would redu Prompt: } You are Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' There are 8 turns left in the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You are trying to win the debate using reason and Turn evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=" Don't repeat yourself." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' No more than 1-2 sentences per turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' OpenAlAgent: Complete Question: "Should we legalize all drugs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" You are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' There are turns left in the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='. Legalizing drugs would crea1 You: "I\'m in favor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" Bob: "\'m against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" Turn You: " QuestionShould we legalize all drugs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' →Alice The evidence shows that when drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Stop: OpenAlAgent: Complete Verbose: false You are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' There are turns left in the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The evidence shows that wh Default: empty Max tokens: 256 Turn QuestionShould we legalize all drugs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' → BobWhile regulation of the drug market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Outputs Agent Turns_left Legalizing drugs would reduce the burden on our criminal Answer Alice Legalizing drugs would reduce the burden on our criminal justice system, freein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 8 serious crimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Additionally, it would reduce the risk of overdose and other health risks associated with the Bob Legalizing drugs would create a new set of problems, such as increased drug u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 7 unregulated drug market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" Alice The evidence shows that when drugs are legalized, drug use does not necessa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 6 Bob While regulation of the drug market may be possible, it would be difficult to enf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5 Alice Regulation of the drug market can be achieved through a variety of measures, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4 Bob While there may be potential benefits to legalizing drugs, the risks associated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='. 3Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Placebo classification and description EXAMPLE input Did Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2015 use a placebo in their experiment on using testimonial videos about drowning to reduce drowning risk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If so, what was it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Participant flow Example input What were the experiments, trial arms, and adherence for each arm in Bloom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2016?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Qasper NLP Q&A Example input Which retrieval system was used for the baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Figure 4: We apply iterated decomposition to three real-world case studies: Extracting placebos from academic papers about randomized controlled trials, extracting participant flow, and NLP question-answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The tasks are part of our work on automated literature review for the AI Research Assistant Elicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' with hundreds to thousands of LM calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ICE provides a decorator that can be applied to any async function that should be recorded, as well as utilities for recording all top-level async functions defined in a given module, any custom values of interest within a function, and the structure of each interpolated string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This data can then be visualized in a browser for interactive exploration, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' ICE provides three views into this data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' An expandable tree of function calls shown in order of execution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A sortable, filterable table of function calls and their custom values 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A function call detail pane containing inputs, custom values, outputs, and source code The tree can be browsed to a particular function call which can then be inspected in the detail pane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Often, it is helpful to compare many calls to the same function, so ICE provides a dropdown menu of all recorded functions in the execution trace and their respective call counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If a function is selected, all calls to it will be shown in the table and highlighted in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Since prompts are of particular interest in LM programs, the detail pane includes special support for rendering interpolated strings (f-strings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Each interpolated value is shown in an alternating color, and the interpolating source code is shown in a tooltip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' These functions all serve the purpose of understanding, analyzing, and debugging task decompositions, going from the high-level structure of the call tree to seeing all functions of a particular type in the table, to seeing what exactly the prompt was when a particular instance of that type generated an unexpected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 3 Real-World Context of Case Studies We used iterated decomposition and ICE to improve performance over simple baselines on three real-world case studies we encountered in our work on Elicit (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A primary use case for Elicit is to find out what the academic literature knows about a research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Many of Elicit’s users work in biomedicine, psychology, experimental economics, and other fields where randomized controlled trials (RCTs) are an important source of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' When these users do literature review or metanalyses, they often roughly follow this process: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' What are the RCTs that are potentially most relevant to answering my research question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For each RCT: (a) Does this study actually address my research question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (b) What is the risk of bias?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Can I take the findings at face value, or do I need to discount it or ignore it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' What does the aggregate of RCTs, weighted by evidence quality, say about my research question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 7 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes input Did Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2015 use a placebo in their experiment on using testimonial videos about drowning to reduce drowning risk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If so, what was it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' methods Baseline: Select-and-generate like Elicit Trial arm classification Paragraph-wise aggregation Rank & answer for description output Yes, a testimonial video about dog bites Figure 5: In the placebo case study, the goal is to determine if a randomized controlled trial used a placebo, and if so, what it was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We start with the select-and-generate baseline used in Elicit and improve it by ensembling classification based on trial arms and paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The Placebo Classification & Description task is designed to help researchers evaluate risk of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Information about the placebo helps researchers assess the risk of bias in the study and decide how much to trust the results – comparing an intervention to a placebo control is usually stronger evidence than comparing to a no-treatment control, especially if the placebo successfully blinded the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The first step of the Participant Flow in RCTs task is to identify the trials and trial arms within each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This helps the user understand what the researchers did in the study, to determine whether it addresses their research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The second step, evaluating participant adherence to an intervention, helps the researcher assess risk of bias and contextualize the study’s findings: Was a small effect driven by poor adherence to the treatment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' These tasks represent a much broader class of tasks in Elicit: answering questions about papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Elicit currently supports about 20 pre-specified questions, but also allows users to enter their own questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ultimately, we want to find highly generalizable strategies to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The QASPER NLP Q&A task tests this by generalizing the participant flow decomposition to a different domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4 Case Study: Placebo Classification & Description In this case study, we focus on domain-specific decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Through iteration on the classification and description steps, accuracy of the generated placebo description improved from 25% to 65% on a held-out test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 Setup Given the full text of an RCT, the task is to answer “Did this RCT use a placebo?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (classification) and “If so, what was it?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For both classification and description we find strong agreement among human raters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 has a detailed description of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 in the Appendix shows examples of the most relevant quotations from two papers with corresponding classifications and descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 Iterations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 Baseline: Select-and-generate like Elicit The baseline is the “select-then-generate” algorithm currently deployed for paper question-answering in Elicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Elicit uses monoT54 to rank paragraphs from the paper against the question (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The top-ranking paragraph is fed to text-davinci-002 with the following prompt: Answer the question "{{ question}} " based on the excerpt from a research paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Try to answer, but say "The answer to the question is not mentioned in the excerpt" if you don\'t know how to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Include everything that the paper excerpt has to say about the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Make sure everything you say is supported by the excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The excerpt may cite other papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=" answer about the paper you're reading the excerpt from, not the papers that it cites." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Answer in one phrase or sentence: 4https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='co/castorini/monot5-base-msmarco-10k 8 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Paper title: {{ title}} Paper excerpt: {{ paragraph}} Question: {{ question}} Answer: On a held-out test set, this approach has 71% accuracy for classification, 25% for description (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 Improving classification Diagnosis For each of the 14 trials that the baseline classified incorrectly, there was in fact a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In all cases, the selection step failed to find the part of the paper that most clearly stated that there was a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For 10/14 trials, selection found no evidence that there was a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For the remaining 4/14 trials, selection found some mention of the placebo but not a full description, then generation failed to conclude that there was a placebo (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3 for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' To improve classification, we needed a way to more robustly find evidence that there was a placebo without introducing false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Solution We created a decomposition that mirrors our own reasoning process for determining placebo classification and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' When we read the papers in our validation set, we noticed two things: First, almost all papers clearly describe their trial arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Then, if there was a placebo in the study, one of the arms will usually be clearly identified as the placebo arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So, we identify and describe each trial arm and classify whether any of the arms is a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Second, often many paragraphs in the paper provide evidence about whether the paper used a placebo, and sometimes this evidence is contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So, we could check what each paragraph tells us about whether the paper used a placebo, then aggregate those answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In outline, the decomposition looks like this: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Classify based on trial arms (a) What were the trial arms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Rank paragraphs by relevance to this question through pairwise comparisons ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Answer based on the most relevant paragraphs (b) Describe each trial arm: Rank and Answer (c) Do any of the arms look like placebos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (d) If so, could participants tell which arm they were in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (If so, there’s not really a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Classify based on each paragraph (a) Was there a placebo, according to each paragraph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (b) Aggregate answers from paragraphs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ensemble the classification based on trial arms and paragraphs Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='4 shows the full decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Results This approach substantially improves on the baseline (71% correct → 98%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='0004), achieving near-perfect perfor- mance (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Reviewing the execution traces in ICE showed that the improved performance is almost entirely due to answering based on each paragraph, not due to answering based on trial arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Whenever the arms-based classification tentatively classified one of the trial arms as a placebo, it also found that participants might be able to tell which arm they were in, and so it ultimately said that it was unclear whether there was a placebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3 Improving description Diagnosis 14 of the 15 trials where the baseline failed to describe the placebo correctly are ones that it failed to classify correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The baseline does not even attempt to describe the placebo if the classification is “no placebo”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' so it fails all of these by 9 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes Method Classification (n=48) Description (n=20) Elicit select-then-generate baseline 71% 25% Select-then-generate with “not mentioned" perplexity selection 65% 10% Stuff paper in prompt 69% 30% Final decomposition 98% 65% Keyword decision tree 94% 10% Table 2: Through iteration on the task decomposition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' accuracy of the placebo classification improved from 71% to 98% and description accuracy improved from 25% to 65%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' both on a held-out test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' However, in each of these cases selection failed to find the excerpts needed for description, so any attempt at description would have failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Solution We noticed that the information required to describe the placebo is often dispersed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So, we rank the most relevant paragraphs in the paper and then use them to generate an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We re-use the same Rank and Answer technique that we used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This generalizable subtask decomposition works well for both purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Results This approach results in a big improvement on the Elicit baseline (25% correct → 65%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='025—see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='4 A keyword baseline The strong paragraph-based classification results raise the question whether can we encode our understanding of how to figure out whether a trial used a placebo in a much simpler algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We created a simple regex keyword-matching algorithm: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Classify as no placebo if the paper contains words like "open-label" 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Then classify as placebo is the paper contains words like "placebo", and take the first matching sentence as a description 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Then classify as no placebo if none of these words are present This algorithm does about as well as our task decomposition on classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' However, a regex keyword approach doesn’t generalize to harder and more ambiguous tasks—for example, adherence is discussed using a variety of wordings, so keywords don’t help as much to select relevant passages (Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5 Case Study: Participant Flow in Randomized Controlled Trials In this case study, we mostly focused on simple generalizable decompositions for long-form question-answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Through iteration on the selection and generation steps, accuracy on all subtasks improved substantially on a held-out test set: Extracting experiments improved from 40% to 70%, trial arms from 55% to 86%, and adherence from 53% to 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 Setup Randomized controlled trials often include a standardized diagram that helps the reader understand what happened in a given trial: How did participants journey through the study, from enrollment to final analysis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' These diagrams are called CONSORT diagrams (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In our experience, they appear in about half of recent RCTs are very often incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If we could generate complete CONSORT diagrams for all RCTs, we could provide valuable information to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In this case study we consider a limited version of this task: We name the trials in the paper, the arms in each trial, and describe the participant adherence rate for each arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Figure 6 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 10 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes input What were the experiments, trial arms, and adherence for each arm in Bloom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 2016?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' methods Baseline: Select-and-generate like Elicit Improving selection Fine-tuned classifiers Task-specific prompting Zero-shot perplexity classifier Pruning Improving generation Decontextualization Auto-few-shot demonstrations output Experiments Fall experiment Arms Park walking Adherence: 72% of the participants in the intervention groups engaged in relaxation or park walking during their lunch break at least 8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' On average, they engaged in relaxation/park walking 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6 out of 10 times during the intervention period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Relaxation exercises Adherence: same as for park walking Control Adherence: not mentioned Spring experiment Arms Park walking Adherence: 76% of the participants in the intervention groups engaged in relaxation or park walking during their lunch break at least 8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' On average, they engaged in relaxation/park walking 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5 out of 10 times during the intervention period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Relaxation exercises Adherence: same as for park walking Control Adherence: not mentioned Figure 6: In the participant flow case study, the goal is to analyze a randomized controlled trial and extract the experiments, the arms for each experiment, and the adherence rate for each arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Starting again from the Elicit select-and-generate baseline, we implemented domain-agnostic improvements to both selection and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 Evaluation Experiments and arms are relatively easy to evaluate—we can check whether each experiment/arm from the gold standard is represented and whether there are any additional ones that should not be there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Adherence tends to require a narrative answer, and to be more subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This means that it may benefit more from a nuanced decomposition, and approaches that work well for adherence may generalize better to other fuzzy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Each arm has an adherence answer, so there are a total of 135 adherence answers in our test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Often, no information about adherence is available in the text of the paper—information about adherence is only available for 56/135 arms (41%) in our test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So the adherence task is substantially a classification task (adherence mentioned vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' not mentioned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3 Iterations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1 Baseline: Select-and-generate like Elicit The baseline is the same “select-then-generate” approach used for general paper question-answering in Elicit as described above in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This is a reasonably strong baseline, scoring 53% on the adherence subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Diagnosis For adherence, 80% of errors were false negatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' saying that adherence was not mentioned when it in fact was (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Further, all (51/51) of these false negatives resulted from errors at the selection stage—the answer really was not mentioned in the top-1 paragraph from the monoT5 ranker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' By using an oracle for selection with the same generation approach, accuracy on the adherence task rose to 77%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So, the baseline fails primarily by failing to make good use of the long context of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For this reason, we started by iterating on selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='2 Improving selection Finetuned classifiers Because our research is aimed at generalizable approaches, we restricted any finetuning to approaches that were either very sample-efficient (so little labeling effort would be needed to adapt the approach to new tasks) or very general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We 11 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes evaluated small dataset and machine-in-the-loop labeling approaches, favoring approaches using at most a few hours of subject matter expert labeling effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We adapted the approach from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) to fine tune classifiers for text classification subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This approach combines low-rank adaptation (scaling selected activations by learned vectors) with additional regularization terms in the loss and hyperparameters the authors found to generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This combination enables (a) a very small set of finetuned weights, enabling multiple finetuned tasks to share a single model backbone at inference time, and (b) true low-data finetuning, since the approach prescribes hyperparameters, thus obviating the need for separate training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' (2022) suggest that their approach works well with as few as 20 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' bootstrapped to a larger number of positive examples and more than 20 "hard negative" examples using weak models: For our tasks, most paragraphs are not relevant to answering a given question, and it is time-consuming and expensive to collect even a moderate amount of positive examples by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We ensembled multiple weak classifiers such as monoT5, BART-based classifiers, and T0-3B to identify positive examples from thousands of papers, then had experts moderate the examples identified by all the weak classifiers as answering the question, leaving in any negatives from this approach as “hard negatives”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Intuitively, we would expect that this would provide an outsized number of “easy positives” (from the ensemble consensus) and “hard negatives” (false positives from the ensemble consensus, as corrected by experts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Although we would expect this approach to have a difficult time identifying “hard positives”, in practice it created a dataset that worked very well in concert with T-few finetuning, bringing performance on a holdout set on the selection subtask for adherence from 72% before finetuning to 94% accuracy after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Using T0-3B and T0pp (11B parameters) as base models, we finetuned using the T-Few approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=" We found that the models generalized better when we left zero-shot prompts in our finetuning data, which can be interpreted as a form of regularization (in this example, the decoder answer choices are yes and no): Context: {{ paragraph}} Section: {{ section}} Answer yes if the following sentence is about how many participants in the study complied with the study's protocol, had to drop out, or withdrew;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=" answer no if it is about something else, such as the study's design, sampling strategy, or results." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Sentence: {{ sentence}} Answer: Many failures to get the right answer using finetuned classifiers at the selection stage were rooted in false negatives from the classifier (for 67% of failures in the development set with n = 9, the classifier missed the gold standard excerpts), and the generation stage was somewhat robust to “distractors” caused by lower precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We therefore explored variations on this technique to favor recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For instance, we explored classifying sentences (while including the paragraph as context), then, at inference time, classifying a paragraph as positive if any of its sentences were classified as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We also tried finetuning with multiple zero-shot prompt templates, then, at inference time, classifying a paragraph as positive if it was classified as positive via any of the prompt templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' On the adherence subtask, the finetuned classifier significantly outperformed the base monoT5 classifier (paragraph classification F1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='10 for Elicit top-1 approach versus F1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='77 via T-Few finetuning of T0-3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Overall, while this approach led to significant performance improvement, it also had downsides: It requires data and training, and is less interpretable than some of the classification approaches we explored later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In the future, we would consider data-efficient finetuning of classifiers where (a) the task was hard to specify but data was easy to collect or (b) where the inference-time computational advantage of a smaller finetuned classifier was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Task-specific prompting An alternative to finetuning is crafting prompts for specific subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' This reduces the data collection burden of finetuning, requiring just a few examples, at the cost of involving a different sort of expertise in the craft of writing effective prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' With text-davinci-002, we found that we could get considerable performance gains on specific subtasks by prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Naive zero-shot or few-shot instructions performed performed worse than “simulation” approaches where the prompt emulated a genre of text that had certain desirable properties, such as containing correct mathematical 12 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='4 shows an example of a prompt that frames a task as excerpts from a (nonexistent) textbook on evaluating scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5 shows a prompt that simulates a StatsExchange post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Because we wanted to focus on general-purpose decompositions in this case study, and because we expect simulation- based approaches to be less necessary with future instruction-tuned models, we did not evaluate this line of approaches systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Zero-shot perplexity classifier Ideally, we could improve selection performance over the baseline monoT5 classifier with an approach that requires neither substantial data collection nor task-specific prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Inspired by the relative strength of the “generation” stage of the baseline approach, we implemented a zero-shot perplexity-based cross-encoder using text-davinci-002 that performed remarkably well across many questions, outperforming the monoT5 classifier and competitive with the T-Few low-rank finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' In this paragraph classification approach, we used the following prompt with text-davinci-002, echoing the log probs back from the OpenAI API in order to measure perplexity: Answer the question "{{ question}} " based on the excerpt from a research paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Try to answer, but say "The answer to the question is not mentioned in the excerpt" if you don\'t know how to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Include everything that the paper excerpt has to say about the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Make sure everything you say is supported by the excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The excerpt may cite other papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=" answer about the paper you're reading the excerpt from, not the papers that it cites." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Answer in one phrase or sentence: Paper excerpt: {{ paragraph}} Question: {{ question}} Answer: The answer to the question is not mentioned in the excerpt We then scored paragraphs by the inverse perplexity of the tokens X : {(xi=1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' , xt)} in the sequence “The answer to the question is not mentioned in the excerpt”: exp � 1 t t � i=1 log pθ(xi|x 1 else "this"}} excerpt tells us, if anything, about adherence, attrition or compliance: 30 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='5 Adherence subtask prompt, Stats Exchange style We experimented with attempting to calculate a final adherence rate where it was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' One possible approach to such calculations is to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', generate Python code then offload the calculations to a Python interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We also found that we could get stronger in-context mathematical reasoning in a zero-shot setting by simulating a StatsExchange post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A user on Statistics Stack Exchange needs help with this problem: Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention or at what rate they completed the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ignore irrelevant excerpts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If there is not enough information to determine the answer, answer "Unclear".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The intervention is Intracytoplasmic sperm injection (ICSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You must decide which excerpts are relevant to your reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Some may be very relevant, and most will be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Excerpts: Thus, -90% of the couples had an embryo transfer and the viable pregnancy rate was 21% for ejaculated, 22% for epididymal and 19% for testicular spermatozoa (with 25-30% multiple pregnancies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Data forms were sent to all centres that had already taken part in the previous ICSI Task Force survey, as well as to new ICSI centres known from the literature or from national registries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Up to 31 December 1995, 101 centres had submitted their clinical results based on 2 to 2507 cycles (see Table I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Hence, only 17 of the 101 centres that have submitted ICSI results for 1995 are performing a prospective follow-up of the children and only nine as a part of a special project, while another 46 centres are trying to collect information by contacting the infertility specialist, the paediatrician, or the nurses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Before giving your answer describe your reasoning in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Think about how many people participated in the Intracytoplasmic sperm injection (ICSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Explain how to deduce an answer the question "What is the adherence rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" from the relevant excerpts step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' State your final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The best response from Statistics Stack Exchange was the following: The quote "Hence, only 17 of the 101 centres that have submitted ICSI results for 1995 are performing a prospective follow-up of the children and only nine as a part of a special project, while another 46 centres are trying to collect information by contacting the infertility specialist, the paediatrician, or the nurses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" Talks about the number of centres that performed follow-ups but makes no mention of the number of patients who completed, adhered or complied with the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Therefore it is impossible to tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Final answer: Unclear A user on Statistics Stack Exchange needs help with this problem: Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention or at what rate they completed the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ignore irrelevant excerpts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If there is not enough information to determine the answer, answer "Unclear".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The intervention is malaria chemoprevention (SMC) and long-acting artemisinin-based combination therapies (ACTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You must decide which excerpts are relevant to your reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Some may be very relevant, and most will be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Excerpts: A study including approximately 800 children in each group, allowing for a 10% loss to follow-up, would have 90% power to detect a reduction of 20% in the malaria incidence rate between children given a long-acting ACT and a short-acting ACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.A study with 800 children in each group would have greater than 90% power to detect a difference of 25% between children given a long-acting ACT and those given SMC, and very high power to detect larger differences between the SMC group and the short-acting ACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Caregivers were followed up by CHWs between 4 and 7 days after each SMC administration to ask about adverse events and adherence to the SMC regimen using a standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.However, it is unlikely that adherence was in reality this high, as during the qualitative component of this study, some caregivers were found to have tablets remaining that had not been administered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Adherence data were collected but implausibly high values of adherence were reported, as seen elsewhere [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Before giving your answer describe your reasoning in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Think about how many people participated in the malaria chemoprevention (SMC) and long-acting artemisinin-based combination therapies (ACTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Explain how to deduce an answer the question "What is the adherence rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" from the relevant excerpts step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' State your final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The best response from Statistics Stack Exchange was the following: A quote says "However, it is unlikely that adherence was in reality this high, as during the qualitative component of this study, some caregivers were found to have tablets remaining that had not been administered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" This tells us that the follow-up system was not completely accurate, and that some people did not adhere to the SMC regimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' However, we are not given any numbers or percentages, so it is impossible to say what the adherence rate was quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' But this along with "Adherence data were collected but implausibly high values of adherence were reported, as seen elsewhere".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Leads us to believe that authors found the adherence to be high but thought is was implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Final answer: The adherence was stated to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' But authors thought is was implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A user on Statistics Stack Exchange needs help with this problem: Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention or at what rate they completed the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ignore irrelevant excerpts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If there is not enough information to determine the answer, answer "Unclear".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 31 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes The intervention is pulsed electromagnetic field (PEMF) therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You must decide which excerpts are relevant to your reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Some may be very relevant, and most will be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Excerpts: A total of 286 subjects indicated their interest in the study, and they were sent a health questionnaire with a prepaid return envelope to assess their preliminary eligibility to participate in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Twenty-three respondents declined to take part in the trial and 263 respondents returned the questionnaire, 133 of which were excluded from the study due to exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Fourteen patients did not meet the eligibility criteria during the clinical examinations, and eight participants withdrew from the study, thus leaving a sample of 108 participants for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Power calculations indicated that a sample of 110 patients (55 in each treatment), assuming a dropout rate of approximately 10%, would provide an 80% (b ¼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='20) chance of detecting a 40% (a ¼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='05) difference in improvement between the active and sham treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Of the 108 patients who started the treatments, nine dropped out during the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Four patients dropped out during the active treatment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Five patients dropped out during the sham treatment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.The reasons for those dropouts were amplified overall pain and worsened irritable bowel syndrome (n ¼ 1), lumbar herniated disc (n ¼ 1), cervical herniated disc surgery (n ¼ 1), chronic pneumonia (n ¼ 1), and unwilling to continue (n ¼1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Of the enrolled patients, 92% completed the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Mean treatment compliance, measured as attendance at all 168 planned, 8-min treatment sessions, was 134 AE 41 (80%) when using the active devices and 131 AE 44 (78%) when using the sham devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.The treatment adherence was high ($80%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='.Fourth, considering that high dropout rates are a major issue with crossover design [Mills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=', 2009], in this study the dropout rate was low (8%) despite the relatively long duration of the study (28 weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Before giving your answer describe your reasoning in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Think about how many people participated in the pulsed electromagnetic field (PEMF) therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Explain how to deduce an answer the question "What is the adherence rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" from the relevant excerpts step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' State your final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The best response from Statistics Stack Exchange was the following: In context of this study the adherence could mean either the completion or compliance rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' We will consider both individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For the completion rate we know that of the 108 patient 9 dropout during the course of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' So 108-9 = 99 complete the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Hence the completion rate is 99/108 = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' For the compliance rate we are told "Mean treatment compliance, measured as attendance at all 168 planned, 8-min treatment sessions, was 134 AE 41 (80%) when using the active devices and 131 AE 44 (78%) when using the sham devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" and that "The treatment adherence was high ($80%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" So the compliance rate is 80% for the active devices and 78% for the sham devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Final answer: The completion rate is 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The compliance rate is 80% for the active devices and 78% for the sham devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' A user on Statistics Stack Exchange needs help with this problem: Use the excerpts from an academic paper to identify how many participants in an intervention completed the intervention or at what rate they completed the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Ignore irrelevant excerpts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' If there is not enough information to determine the answer, answer "Unclear".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The intervention is {{ intervention}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' You must decide which excerpts are relevant to your reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Some may be very relevant, and most will be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Excerpts: {{ ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='join(excerpts)}} Before giving your answer describe your reasoning in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Think about how many people participated in the {intervention}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Explain how to deduce an answer the question "What is the adherence rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='" from the relevant excerpts step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' State your final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The best response from Statistics Stack Exchange was the following: 32 Iterated Decomposition: Improving Science Q&A by Supervising Reasoning Processes 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='6 Decontextualization few-shot prompt Instructions: Enrich each Passage with the Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Context: Lisa loves to play practical jokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Passage: But sometimes she goes too far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Rewrite: But sometimes she [Lisa] goes too far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' --- Context: The Super Bowl XLI halftime show took place on February 4, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Passage: It was headlined by Prince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Rewrite: It [The Super Bowl XLI halftime show] was headlined by Prince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' --- Context: More than one fifth of the world’s population lives on less than Purchasing Power Parity (PPP) US$1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='25 a day, and there is an emerging international consensus that this share should (and can) be driven close to zero by 2030 (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Passage: Reaching this objective will require enabling the poorest families, who are often the most marginalized within their villages, to shift from insecure and fragile sources of income to more sustainable income-generating activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Rewrite: Reaching this objective [driving the share of the world’s population that lives on less than Purchaing Power Parity (PPP) US$1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content='25 a day from more than one fifth of the to zero by 2030] will require enabling the poorest families, who are often the most marginalized within their villages, to shift from insecure and fragile sources of income to more sustainable income-generating activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' --- Context: We present results from randomized control trials (RCTs) in six countries of a particular approach to foster self-employment activities amongst the very poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Originally designed and implemented by BRAC, a large Bangladeshi NGO that runs several country-wide programs, the ``Graduation" program provides a holistic set of services, including the grant of a productive asset, to the poorest households in a village (referred to by BRAC as the ``ultra-poor").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' The beneficiaries [of the Graduation program, the poorest housholds in a village, or the "ultra-poor"] are identified through a participatory process in a village meeting, followed by a verification visit by the organization’s [the implmenter of the "Graduation" program] staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Selected beneficiaries [among the poorest housholds in a village, or the "ultra-poor"] are then given a productive asset [by the implementer of the "Graduation Program"] that they choose from a list, training and support for the asset they have chosen, as well as general life skills coaching, weekly consumption support for some fixed period, and typically access to savings accounts and health information or services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Passage: These different activities (plus regular interactions with the households over the course of a year) are designed to complement each other in helping households to start a productive self-employment activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' Rewrite: These different activities [training and support for the assest they have chosen and received,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' general life skills coaching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' weekly consumption support,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' and typically access to savings accounts and health information or services] (plus regular interactions with the households over the course of a year) are designed to complement each other in helping households [beneficiaries selected for the "Graduation" program from among the poorest housholds in a village,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' or the "ultra-poor"] to start a productive self-employment activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} +page_content=' --- Context: {{ context}} Passage: {{ passage}} Rewrite: 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AzT4oBgHgl3EQfyP6v/content/2301.01751v1.pdf'} diff --git a/M9E3T4oBgHgl3EQfYwr8/vector_store/index.faiss b/M9E3T4oBgHgl3EQfYwr8/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..bdb265fe792616b57e9d40b7ea102e80ba9dc20d --- /dev/null +++ b/M9E3T4oBgHgl3EQfYwr8/vector_store/index.faiss 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/dev/null +++ b/TNE2T4oBgHgl3EQfCQao/content/tmp_files/2301.03612v1.pdf.txt @@ -0,0 +1,2776 @@ +MNRAS 000, 1–26 (2023) +Preprint 11 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The Cosmological Simulation Code OpenGadget3 – Implementation of +Meshless Finite Mass +Frederick Groth,1★ Ulrich P. Steinwandel,2 Milena Valentini,1 and Klaus Dolag1,3 +1Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany +2Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA +3Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Subsonic turbulence plays a major role in determining properties of the intra cluster medium (ICM). We introduce a new +Meshless Finite Mass (MFM) implementation in OpenGadget3 and apply it to this specific problem. To this end, we present +a set of test cases to validate our implementation of the MFM framework in our code. These include but are not limited to: the +soundwave and Kepler disk as smooth situations to probe the stability, a Rayleigh-Taylor and Kelvin-Helmholtz instability as +popular mixing instabilities, a blob test as more complex example including both mixing and shocks, shock tubes with various +Mach numbers, a Sedov blast wave, different tests including self-gravity such as gravitational freefall, a hydrostatic sphere, the +Zeldovich-pancake, and the nifty cluster as cosmological application. Advantages over SPH include increased mixing and a better +convergence behavior. We demonstrate that the MFM-solver is robust, also in a cosmological context. We show evidence that the +solver preforms extraordinarily well when applied to decaying subsonic turbulence, a problem very difficult to handle for many +methods. MFM captures the expected velocity power spectrum with high accuracy and shows a good convergence behavior. +Using MFM or SPH within OpenGadget3 leads to a comparable decay in turbulent energy due to numerical dissipation. When +studying the energy decay for different initial turbulent energy fractions, we find that MFM performs well down to Mach numbers +M ≈ 0.007. Finally, we show how important the slope limiter and the energy-entropy switch are to control the behavior and the +evolution of the fluids. +Key words: hydrodynamics – methods: numerical – galaxies: clusters: general – turbulence +1 INTRODUCTION +Turbulence plays a key role in a variety of astrophysical systems +at all scales, ranging from stellar structure, star-formation in the +interstellar medium (ISM) all the way up to the ICM. It leads to +enhanced small-scale mixing, and contributes to the global pressure +of a system. While being mostly supersonic in the ISM, turbulence is +mainly subsonic in the ICM (compare, e.g. Schuecker et al. 2004, for +observations on the Coma cluster). A theoretical framework for sub- +sonic turbulence has been provided by Kolmogorov (1941), assuming +isotropy. Simulations are an essential tool to better understand phys- +ical properties of astrophysical turbulence as well as its influence on +local observables such as star formation in the ISM or its contribution +to heating in the ICM. +Historically, there exist different methods to solve the hydrody- +namical equations in co-moving/cosmological context. Hereby, one +has the option to discretize the hydrodynamic equations by mass or +volume. The former leads to the concept of “Lagrangian” (particle +based) codes and the concept of Smoothed Particle Hydrodynam- +ics (SPH), and the more recent Meshless Finite Mass (MFM) and +Meshless Fintie Volume (MFV). The latter gives rise to the con- +★ E-mail: fgroth@usm.lmu.de +cept of “Eulerian” (grid based) codes and the Godunov finite volume +approach. +Popular SPH codes include Gadget in the different versions in- +cluding Gadget-1 (Springel et al. 2001), Gadget-2 (Springel 2005), +and Gadget-4 (Springel et al. 2021), Phantom (Lodato & Price +2010; Price et al. 2018) and gasoline (Wadsley et al. 2004, 2017). +MFM has been implemented in e.g. gizmo (Hopkins 2015), GAN- +DALF (Hubber et al. 2018), Gadget-3 (Steinwandel et al. 2020), and +pkdgrav-3 (Asensio et al. 2022). +Mesh codes exist in two flavors: either as a stationary mesh, pos- +sibly with adaptive mesh refinement, as implemented e.g. in Zeus +(Stone & Norman 1992), TVD (Ryu et al. 1993, 1998), Enzo (Bryan +et al. 1995, 2014), FLASH (Fryxell et al. 2000), RAMSES (Teyssier +2002), athena (Stone et al. 2008), and athena++ (Stone et al. 2020) +or as a moving mesh as in Arepo (Springel 2010; Weinberger et al. +2020) and shadowfax (Vandenbroucke & De Rijcke 2016). The +latter have the advantage of being Pseudo-Lagrangian. While mesh +codes as well as MFM employ a Godunov-method and calculate +fluxes between neighbors (Godunov 1959), SPH directly retrieves +the hydrodynamical fluid vectors from the kernel density estimation +that is obtained by adopting a weighted sum over a certain (typically +non-constant) number of neighbors. +All of them can be used for computations of turbulence, with +earlier calculations primarily carried out in the supersonic regime, +© 2023 The Authors +arXiv:2301.03612v1 [astro-ph.IM] 9 Jan 2023 + +2 +F. Groth et al. +relevant in the ISM for regulating star formation. Many results have +been obtained assuming driven turbulence in which an energy input +at large scales is provided during the whole simulation. In contrast +to driven turbulence, we expect decaying turbulence to be present in +galaxy clusters. Turbulence is injected at large scales for example due +to collapse of large scale structure and subsequent merger activity +(Roettiger & Burns 1999; Subramanian et al. 2006), after which it +energy is transported down to the smaller scales (“turbulent cascade”) +on which it is dissipated (generally below the resolution scale of any +given code). +In the series of papers by Federrath et al. (2008, 2009, 2010), they +have used a stationary grid code to calculate turbulent boxes with +driven turbulence. They found that the choice of the driving scheme +plays an important role in determining properties of the resulting +turbulence, leading to significant differences in the density statis- +tics. Their results suggest a different mixture of driving-mechanisms +for different star forming regions. Overall, they found good agree- +ment with observations as well as other results, independent of the +driving-mechanism employed. More recently, Federrath et al. (2021) +increased the resolution to even resolve the sonic scale, starting from +supersonic turbulence with a resolution of ∼ 100003 cells. +Kitsionas et al. (2009) and Price & Federrath (2010) also compared +the performance of different implementations of SPH and hydro +schemes with a stationary mesh, and find good agreement between +these two methods at high Mach numbers. Mesh codes are more +efficient to obtain volumetric statistics such as the power spectrum, +while SPH recovers the high-density tail better due to automatically +adapting the resolution. +While all these methods work well in the supersonic turbulent +regime, they have problems dealing with subsonic turbulence. Going +to smaller Mach numbers (M) Padoan et al. (2007) showed that SPH +performs sub-optimum when compared to finite volume methods. +Based on this work, Bauer & Springel (2012) studied the capabilities +of SPH for subsonic turbulence at M = 0.3. They found that clas- +sic (vanilla) SPH fails in reproducing the expected velocity power +spectrum as well as the dissipation range. Reasons are mainly the +artificial viscosity scheme used and velocity noise introduced by the +kernel. These results raised the general question of whether SPH can +deal with subsonic turbulence to begin with. +An answer has been provided by Price (2012) who showed that +these limitations are not intrinsic to SPH, but rather a consequence of +some SPH setups adopted to study subsonic turbulence. In contrast to +what previous studies reported, SPH can capture the expected power +spectrum by using more modern formulations of SPH that are able +to reduce artificial viscosity in subsonic regimes. +The role of subsonic turbulence in galaxy clusters has been ana- +lyzed both from observational and theoretical perspectives. Simula- +tions of turbulence in the ICM have been carried out mostly using +grid codes (Vazza et al. 2009, 2018; Mohapatra et al. 2021, 2022; +Iapichino & Niemeyer 2008; Iapichino et al. 2017). Miniati (2014, +2015) found a lack of turbulent energy at small scales depending on +the refinement technique. In addition, they discussed the importance +of microphysics for the evolution of turbulence. A possible improve- +ment for modeling turbulence has been presented by Maier et al. +(2009) combining AMR with large eddy simulations. Simulations +by Dolag et al. (2005b) have shown that also SPH can model turbu- +lence in galaxy clusters when properly reducing artificial viscosity. +In addition to the impact on gas dynamics, turbulence is responsi- +ble for amplifying magnetic fields through a turbulent dynamo. Sim- +ulations by Schekochihin et al. (2001, 2004) and Steinwandel et al. +(2021) have focused on this turbulent dynamo, analyzing its growth. +Other work of Kritsuk et al. (2020) has focused again on turbulent +boxes with stochastic forcing, comparing different hydrodynamical +methods. +More recently, Sayers et al. (2021) have compared simulated clus- +ters to observed ones. Especially, there should be a difference de- +pending on the dynamical state, with more relaxed clusters showing +less turbulence. Simulations, however, do not always find such a +difference. Thus, it is important to accurately capture the turbulent +cascade and the decay in turbulent energy. While the latter would re- +quire including additional microphysics such as viscosity, the former +also depends on the hydro-scheme. +We use Meshless Finite Mass (MFM) as an alternative, newer +method to the aforementioned ones to study subsonic turbulence. +MFM combines ideas of SPH with those of a moving mesh and +thus aims solving several of their individual issues. The develop- +ment of MFM goes back to Godunov SPH (Inutsuka 2002; Cha & +Whitworth 2003), which was still unstable, and to a Meshless Fi- +nite Element Method suggested by Idelsohn et al. (2003), until the +nowadays used version first formulated by Lanson & Vila (2008a,b). +We present a new implementation in the Gadget derivative Open- +Gadget3, originally based on the implementation in the code GAN- +DALF. Several extensions allow its use in cosmological simulations +compared to the implementation in GANDALF that is focused on +star and planet formation. This allows for a stable baseline frame- +work for applications on scales of star and planet formation that we +extend into the cosmological integration framework of OpenGad- +get3, which is a re-base of Gadget-2 with the ability to be compiled +with C++ compilers, and making vast use of templating. It comes +with modules containing state-of-the-art physics and sub-resolution +models, as for instance: self-interacting dark matter (Fischer et al. +2022), MHD (Dolag & Stasyszyn 2009; Stasyszyn et al. 2013), ther- +mal conduction (Arth et al. 2014), cosmic rays (Böss et al. 2022), +star formation and stellar/blackhole feedback according to the Mag- +neticum-model (Springel & Hernquist 2003; Tornatore et al. 2003, +2004, 2007; Hirschmann et al. 2014; Steinborn et al. 2015; Dolag +2015) or with the MUPPI (MUlti Phase Particle Integrator) exten- +sion for non-equilibrium star formation. (Murante et al. 2010, 2014; +Valentini et al. 2017, 2020). +To make use of modern computer architectures, it includes a hybrid +MPI-OpenMP parallelization. In addition, calculations of gravity, +SPH density, hydro-force and thermal conduction, can be carried out +on GPUs. These modules requiring most of the runtime (Ragagnin +et al. 2020) GPU offloading can be useful for some applications, +leading to a speed up by a factor of a few (2-4, depending on the +exact application). The long-term goal is to have a fully publicly +available updated Gadget version for OpenMP and OpenACC. +Before the introduction of this paper the code was solving the +hydrodynamical equations using modern SPH as formulated by +Springel & Hernquist (2002), including modern, time-dependent arti- +ficial viscosity (Beck et al. 2016a) and conduction (Price 2008). With +the new implementation of MFM as a modern meshless method, we +can combine both advantages of this method and efforts previously +made to optimize the pre-existing code base that also involves a treat- +ment in order to evolve strong shocks for which we need the timestep +limiter to be non-local which is ensured by a wakeup scheme (Saitoh +& Makino 2009; Pakmor 2010; Pakmor et al. 2012). OpenGadget3 +closely follows the implementation described by Beck et al. (2016a). +A main goal of this paper is to use Meshless Finite Mass to study +decaying, subsonic turbulence, as present in galaxy clusters. To this +end, we present a new implementation in the cosmological simulation +code OpenGadget3 as an alternative hydro-solver to the currently +implemented SPH. +This paper is structuered as follows. We first describe the code- +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +3 +base of OpenGadget3 including its SPH implementation in Sec. 2. +We continue with a brief overview on MFM and a description of our +MFM implementation in Sec. 3. In Sec. 4, we use a suite of test cases, +each probing specific aspects and properties of the code, to validate +the performance of our MFM implementation. All settings are kept +exactly the same between test cases, independent of the individual +test case, without further tuning. We continue with an analysis of de- +caying subsonic turbulence with our new implementation presented +in Sec. 4.6. In all cases, comparisons between different codes and +methods are provided, including MFM and SPH in OpenGadget3, +MFM in gizmo and a moving and stationary mesh in the publicly +available Arepo version. We analyze the effect of specific numerical +parameters in Sec. 4.7. Our main findings are discussed in Sec. 5. +Additional material such as the hydrostatic square as additional +test case, the formulation of the slope-limiters and a comparison of +the Riemann solvers implemented are presented in App. A, B, and C, +respectively. +2 OPENGADGET3 – NUMERICAL METHOD +Solving the system of differential equations describing the evolution +of the gas, as written in Eqn. (12), requires discretizing them. In the +temporal dimension a sufficiently small timestep Δ𝑡 is introduced. +The spatial discretisation can be obtained using various different +approaches. In OpenGadget3, hydrodynamics is discretized either +using Smoothed Particle Hydrodynamics (SPH) or with the newly +implemented Meshless Finite Mass (MFM). Gravity is solved by a +TreePM method. +2.1 Integrator and Timestepping +For the time integration, we employ a Leapfrog scheme in kick- +drift-kick (KDK) form to achieve second order accuracy (compare, +e.g., Hernquist & Katz 1989) in the implementation following Verlet +(1967); Springel (2005). +Starting from values at timestep number 𝑛, velocities v are updated +in a first half-step kick. It is followed by drifting the positions r, and +another, second half-step kick: +v𝑛+1/2 = v𝑛 + 1 +2a𝑛Δ𝑡 +(1) +r𝑛+1 = r𝑛 + v𝑛+1/2Δ𝑡 +(2) +v𝑛+1 = v𝑛+1/2 + 1 +2a𝑛+1Δ𝑡. +(3) +The acceleration a = ahydro+agrav consists of hydrodynamical accel- +erations ahydro and gravitational accelerations agrav. Following the +operator splitting approach, they are calculated separately. Gravity is +evaluated before the drift, and hydrodnamical accelerations between +the drift and the second half-kick. +OpenGadget3 uses hierarchical timestepping to ensure synchro- +nization, while allowing adaptive timesteps, depending on different +timestep limiters such as a Courant-like timestep criterion +Δ𝑡Courant +𝑖 += 𝐶Courant𝑎ℎ𝑖 +𝑐max +(4) +with maximum signal velocity 𝑐max, scale factor 𝑎, smoothing length +ℎ𝑖, and free parameter 𝐶Courant, as described by Springel (2005). +2.2 Gravity Solver – TreePM +The accurate treatment of gravity is of great importance for cosmo- +logical simulations (Springel 2010). In principle, it can be solved +accurately by a direct summation, which is, however, computation- +ally expensive (O(𝑁2)). Instead, we follow the much more efficient +combined Oct-Tree-Particle Mesh (PM) approach (Xu 1995; Bode +et al. 2000; Springel 2005, 2010; Springel et al. 2021). OpenGadget3 +mainly follows the implementation in Gadget-2, which has been ex- +tensively described by Springel (2005). In the following, we briefly +review the main concept. The potential is split into short-range and +long-range contributions. Short-range forces are calculated following +the oct-tree algorithm, while long-range forces are calculated using +a particle mesh. The idea of a tree algorithm has been proposed +by Appel (1985) and Barnes & Hut (1986). Nodes of an oct-tree +are constructed by splitting the domain into a sequence of cubes. +Force-contributions of nodes satisfying an opening angle criterion +are calculated. For numerical reasons to keep the equation linear +with respect to adding and removing particles from nodes, only the +monopole contributions are taken into account. The implementation +in Gadget has been described by Springel et al. (2001). The total +gravitational acceleration of particle 𝑖 from other nodes/particles 𝑗 +with mass 𝑚 𝑗 at location r𝑖 𝑗 relative to particle 𝑖 and with (gravita- +tional) softening length 𝜖 𝑗 is given by +agrav,𝑖 = 𝐺 +𝑁tot +∑︁ +𝑗 +r𝑖 𝑗 +��� +��� +𝑚𝑗 +𝑟3 +𝑖 𝑗 +if 𝑟𝑖 𝑗 > 𝜖 𝑗 +𝑚𝑗 +𝜖 3 +𝑗 +Corr(𝑟𝑖 𝑗/𝜖) +if 𝑟𝑖 𝑗 ≤ 𝜖 𝑗. +(5) +Corr is a correction term, taking into account the softening. 𝐺 is the +gravitational constant. For the particle mesh (Eastwood & Hockney +1974), all particles are assigned to grid-cells, such that a discrete +Fourier-transformation can be calculated, with the gravitational po- +tential Φ𝑘 in Fourier space at wavenumber 𝑘 being calculated as +−𝑘2Φ𝑘 = 4𝜋𝐺𝜌𝑘. +(6) +Corrections for small-range truncation as well as periodic boundaries +are applied by multiplications in Fourier space. The gravitational +potential in real space is calculated as inverse Fourier-transform, +and is interpolated to the original particle positions to finally obtain +gravitational accelerations. OpenGadget3 uses the more modern +FFTW3 (“Fastest Fourier Transform in the West”) library (Frigo +& Johnson 2005) instead of FFTW2 for the implementation of the +Fourier transform. +2.3 Hydrodynamical Solver – SPH +For Smoothed Particle Hydrodynamics (SPH), the domain is decom- +posed into a finite number of “particles”. The physical quantities +at each point are represented by contributions of close-by (neigh- +boring) particles weighted by a kernel 𝑊𝑖(𝑟𝑖, ℎ𝑖), depending on the +distance 𝑟𝑖 from particle 𝑖, and its smoothing length ℎ𝑖. The ker- +nel has to be continuous, radially symmetric, have compact support +and fulfill the limit limℎ→0 𝑊 = 𝛿, but otherwise can be chosen +arbitrarily. OpenGadget3 offers the choice between different com- +monly used kernels, including a cubic spline (Monaghan & Lattanzio +1985), quintic spline (Morris 1996), or a Wendland C2/C4/C6 kernel +(Wendland 1995; Dehnen & Aly 2012). The effective volume of each +particle is well approximated by 𝑉−1 +𝑖 += 𝑊(𝑟𝑖), such that the density +follows as +𝜌(r𝑖) = +∑︁ +𝑗 ∈Ngb +𝑚 𝑗𝑊 ���r𝑖 − r 𝑗 +�� , ℎ𝑖 +� . +(7) +We allow for adaptive smoothing, automatically increasing resolution +in high-density regions compared to low-density ones. Smoothing +MNRAS 000, 1–26 (2023) + +4 +F. Groth et al. +length and neighbor number are related to the density via: +4𝜋 +3 𝜌𝑖ℎ3 +𝑖 = ¯𝑚𝑁Ngb +(8) +with mean neighbor mass ¯𝑚. As Eqns. (7) and (8) are coupled for +fixed neighbor number, one solves for smoothing length and density +iteratively via finding roots. Quantities other than the density, labeled +with 𝑋, are approximated via +𝑋(r0) ≈ +∑︁ +𝑖∈Ngb +𝑋𝑖 +𝜌𝑖 +𝑊(|r0 − r𝑖| , ℎ)𝑚𝑖. +(9) +Different formulations of the hydrodynamical acceleration can be +derived. In OpenGadget3 the fully conservative formulation for the +hydrodynamical acceleration (Springel & Hernquist 2002) +ahydro,𝑖 = − +∑︁ +𝑗 ∈Ngb +𝑚 𝑗 +� +𝑓𝑖 +𝑃𝑖 +𝜌2 +𝑖 +∇𝑖𝑊𝑖 𝑗 (ℎ𝑖) + 𝑓 𝑗 +𝑃 𝑗 +𝜌2 +𝑗 +∇𝑖𝑊𝑖 𝑗 (ℎ 𝑗) +� +, +(10) +𝑓𝑖 = +� +1 + ℎ𝑖 +3𝜌𝑖 +𝜕𝜌𝑖 +𝜕ℎ𝑖 +�−1 +(11) +is utilized. Instead of calculating gradients of physical quantities, +all spatial derivatives are expressed by gradients of the kernel func- +tion. Traditional SPH has problems dealing with shocks, as well as +reproducing mixing instabilities (Morris 1996; Agertz et al. 2007). +These issues can be resolved by including artificial viscosity and +conductivity. In OpenGadget3, time and spatial dependent artificial +viscosity (Beck et al. 2016a) and artificial conductivity (Price 2008) +are utilized, minimizing their impact in regions where they are not +desired. +3 MESHLESS FINITE MASS +As a second, newly implemented option, the hydrodynamical equa- +tions can be discretized and solved following the Meshless Finite +Mass (MFM) approach. This method conceptually combines SPH +with a moving mesh, calculating fluxes between neighboring cells in +a scheme otherwise similar to SPH, including weighting by a kernel. +Thus, it is combining advantages of both methods. In contrast to +Godunov SPH, the domain associated to a particle is not spherical +due to the kernel weighting, but the particle interfaces in the flux +calculation are subject to the weighting. +3.1 Basic Hydrodynamical Equations +The evolution of any ideal fluid is described by three main equa- +tions. Mass conservation leads to the continuity equation. The second +equation is an equation of motion (Eulers equation), corresponding +to Newton’s second law. Energy conservation is ensured by the first +law of thermodynamics. Within an inertial frame of reference, all +these equations can be combined into +dU +d𝑡 + ∇ · (F − vframe|U) = S +(12) +with outer product | and, for pure hydrodynamics, field vector U = +(𝜌, 𝜌v, 𝜌𝑒), flux F = (𝜌v, 𝜌vv𝑇 + 𝑃1, (𝜌𝑒 + 𝑃) v) and source S = 0. +In total, Eqn. (12) provides 5 constraints for 6 variables: fluid +density 𝜌, energy density 𝑒, pressure 𝑃, and the three components of +the velocity v. The missing constraint is provided by an equation of +state, connecting the pressure to the internal energy density 𝑢. For an +ideal gas it takes the form +𝑃 = (𝛾 − 1) 𝜌𝑢 +(13) +where the adiabatic index 𝛾 amounts to 5/3 if the gas is monoatomic. +3.1.1 Equations in an Expanding Universe +In a cosmological context, the expansion of the universe has to be +taken into account. One possibility is to re-write Eqn. (12) for a +universe with scale factor 𝑎, accounting for these effects, as realized +e.g. in Gadget-1: +𝜕v +𝜕𝑡 + 1 +𝑎 (v · ∇) v + �𝑎 +𝑎 v = − 1 +𝑎𝜌 ∇𝑃 − 1 +𝑎 ∇Φ, +(14) +𝜕𝜌 +𝜕𝑡 + 3 �𝑎 +𝑎 𝜌 + 1 +𝑎 ∇ · (𝜌v) = 0, +(15) +𝜕 +𝜕𝑡 (𝜌𝑢) + 1 +𝑎 �𝑣 · ∇ (𝜌𝑢) = − (𝜌𝑢 + 𝑃) +� 1 +𝑎 ∇ · v + 3 �𝑎 +𝑎 +� +. +(16) +In OpenGadget3 we follow a different approach, and do calcula- +tions using the so called super-co-moving coordinates, as first intro- +duced by Martel & Shapiro (1998). Code units (denoted by subscript +𝑐) are related to physical units (𝑝) via +𝑥𝑐 = 𝑎𝑥𝑝 +(17) +𝜌𝑐 = 𝑎3𝜌𝑝 +(18) +𝑣𝑐 = 𝑎𝑣 𝑝 +(19) +𝑝𝑐 = 𝑎3𝑝 𝑝 +(20) +𝑢𝑐 = 𝑢𝑝, +(21) +such that Eqn. (12) keeps the same form when written in code units. +3.2 MFM Discretization +Mathematically, Eqn. (12) is discretized by multiplying by a partition +function +𝜓𝑖 = +1 +� +𝑗 ∈Ngb 𝑊 𝑗 +𝑊𝑖 +(22) +and integrating over the volume, such that for every particle 𝑖 changes +in the quantities U𝑖 = (𝜌𝑖, 𝜌𝑖𝑣𝑖, 𝑒𝑖), with 𝑒 being the total energy +density, are given by source terms S𝑖 = 0, which vanish for pure +hydrodynamics, and pairwise fluxes F𝑖 𝑗 with the neighbors 𝑗 via +d +d𝑡 (𝑉𝑖U𝑖) + +∑︁ +𝑗 ∈Ngb +� +F𝑖 𝑗 · Aeff +𝑖 𝑗 +� += S𝑖𝑉𝑖. +(23) +Calculating pairwise fluxes automatically ensures mass, momentum +and energy conservation. The effective interface area Aeff +𝑖 𝑗 depends on +the partition function and effective volume 𝑉𝑖, which itself depends +on the integrated partition function, translating to the number density +𝑛𝑖: +𝑉𝑖 = +∫ +𝜓𝑖 = 𝑛−1 +𝑖 , +(24) +Aeff +𝑖 𝑗 = 𝑉𝑖 ˜𝜓 𝑗 − 𝑉𝑗 ˜𝜓𝑖, +(25) +where +˜𝜓𝛼 +𝑗 (x𝑖) = +∑︁ +𝑖∈Ngb +𝐵𝛼𝛽 +𝑖 +(𝑥 𝑗 − 𝑥𝑖)𝛽𝜓 𝑗 (𝑥𝑖) +(26) +B𝑖 = E−1 +𝑖 +(27) +𝐸 𝛼𝛽 +𝑖 += +∑︁ +𝑗 ∈Ngb +(x𝑗 − x𝑖)𝛼(x𝑗 − x𝑖)𝛽𝜓 𝑗 (𝑥𝑖) +(28) +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +5 +with Einstein summation convention over 𝛽 in Eqn. (26). The matrix +B is chosen in order to be second order accurate. A more detailed +derivation has been provided by Gaburov & Nitadori (2011). +Most importantly, the tessellation does not have to be calculated +explicitly, but an SPH-like neighbor search is used, drastically re- +ducing the computational costs compared to a moving mesh. This +has the drawback that the face area is not well defined, but has to +be calculated in an approximate way using the neighbors according +to Eqn. (25). As MFM is typically used with 32 neighbors in 3d +(compare, e.g. Gaburov & Nitadori 2011; Hopkins 2015) this leads +to particles being treated as neighbors that would not be considered +for a moving mesh. A possible improvement would be to only use +the nearest neighbors, constructed in an approximate way (compare +e.g. Błaszczyszyn & Schott 2003), to get closer to what a mesh +reconstruction would do. +In contrast to SPH, for which the mass density is estimated accord- +ing to Eqn. (8), for MFM the number density 𝑛𝑖 is estimated together +with the smoothing length in an iterative process, solving +𝑛(r𝑖) = +∑︁ +𝑗 ∈Ngb +𝑊 ���r𝑖 − r 𝑗 +�� , ℎ𝑖 +� , +(29) +4𝜋 +3 𝑛𝑖ℎ3 +𝑖 = 𝑁Ngb. +(30) +The flux in Eqn. (23) is calculated numerically using a Riemann +solver, where we use an exact Riemann solver, following the im- +plementation by Toro (2009). Alternatively, we implemented the +Riemann-solver that provides an exact solution to the linearized sys- +tem of equations (Roe-solver, Roe 1981), as well as the two most +common flavors of a Harten-Lax-van-Leer solver (HLL) and HLLC +(Toro 2009). For all these, the exact Riemann solver is used as fall- +back in case the faster, approximate solver fails. The effect of the +choice of the solver is discussed in greatre detail in App. C. +By choosing the reference frame corresponding to the rest-frame +of the interface, the scheme becomes Lagrangian. In MFM, also the +boundaries are assumed to deform in a Lagrangian way, eliminating +mass fluxes between neighbors. As the actual deformation does not +exactly correspond to the one assumed during a timestep, second or- +der errors are introduced. An alternative is allowing for mass fluxes +using the Meshless Finite Volume (MFV) method, which, however, +also is only second order accurate. In addition, it has been shown that +MFV can run into problems by draining the mass for particles ac- +celerated into low density environments in cosmological simulations +(Asensio et al. 2022). For this reason, we do not use this scheme here +but focus on the MFM method. An additional advantage over SPH is +that no additional dissipation terms are necessary. +The Riemann solver requires knowledge about velocity, density +and pressure values at the interfaces, summarized in the primitive +fluid vector +W = �� +� +𝜌 +v +𝑝 +�� +� +. +(31) +In principle, values at the cell center can be used directly, following +a zeroth order interpolation. This method can lead to strong jumps, +unphysical oscillations, and numerical errors. To this end, we follow +a two-step approach, as illustrated in Fig. 1, similar to what is usually +done for grid-based methods. In a first step, gradients of the primi- +tive fluid vector are calculated using a second-order accurate matrix +gradient estimator +(∇|W)𝛼 +𝑖 = +∑︁ +𝑗 ∈𝑁 𝑔𝑏 +�W 𝑗 − W𝑖 +� ˜𝜓𝛼 +𝑗 (x𝑖). +(32) +Figure 1. Sketch of extrapolation from central cell values to face values. Using +the central values corresponds to a zeroth order scheme (black solid lines). +It can be extended to be second order by extrapolating using a slope defined +by neighboring cells (blue dashed line), which however can lead to over- +/undershooting at the faces (see left face) or even negative densities/pressures +(see right face). This issue can be solved by limiting the slopes using different +procedures (red dash-dot line). See text for further details. +The position and velocity of the face is estimated via +drface +𝑖 += dr𝑖 𝑗𝑠𝑖, +(33) +vface +𝑖 𝑗 += 𝑠 𝑗v 𝑗 + 𝑠𝑖v𝑖, +(34) +where we set +𝑠𝑖 = +ℎ𝑖 +ℎ𝑖 + ℎ 𝑗 +(35) +to be second order accurate instead of 𝑠𝑖 = 1/2 for a first-order +accurate interpolation. The face-values are extrapolated according to +Wface +𝑖 += W𝑖 + drface +𝑖 +· ∇|W𝑖. +(36) +To avoid over- or undershooting or even unphysical, negative densities +or pressures when strong gradients are present in the fluid, these +gradients are reduced by a factor ∇𝑊𝑖 → 𝛼𝑖∇𝑊𝑖, 0 ≤ 𝛼𝑖 ≤ 1 in the +face interpolation. We implement different options for such a slope- +limiter, including a total variation diminishing (TVD) one (Hess & +Springel 2010), the one from Arepo (Springel 2010) and the one +used in the gizmo code (Hopkins 2015), described further in App. B. +In addition, the pairwise limiter according to the gizmo code can be +used. +3.3 Energy-Entropy Switch +While the Riemann solver outputs total energy changes, the rest of +the code requires internal energies. The total energy change can +straightforwardly be converted into internal energy change via +d𝑈 +d𝑡 = d𝐸tot +d𝑡 +− +� +v + 1 +2dv +� +· dv. +(37) +We introduce the additional term 1 +2d𝑣 in the bracket, which is a +second order correction and improves the accuracy. While this trans- +formation does not conserve total energy to machine-precision, it +increases the precision in the evolution of the internal energy itself. +For very cold flows, the internal energy evolution is still dominated +by numerical errors. This is avoided by assuming purely adiabatic +changes in these rare cases. We follow the idea of the implementation +in the gizmo code, where the switch is only active for specific test +problems such as the Zeldovich pancake. If active, internal energy +𝑈 = 𝑈𝑖 + d𝑈𝑖 +(38) +MNRAS 000, 1–26 (2023) + +zero slopes +-: extrapolated +-.. limited slopes +x6 +F. Groth et al. +is compared to potential and/or kinetic energy +𝐸pot = 𝑚𝑖𝑎grav · 0.5ℎ𝑖, +(39) +𝐸kin = 0.5𝑚𝑖 max +𝑗 ∈Ngb +�v 𝑗 − v𝑖 +�2 . +(40) +If the internal energy is small enough compared to other energy +contributions +𝑈 < 𝛼1𝐸pot + 𝛼2𝐸kin +(41) +in physical units, the new internal energy is instead calculated assum- +ing adiabatic expansion or contraction. The parameters 𝛼1/2 have to +be tuned to only affect the evolution of particles where necessary. +We provide a comparison between different values in Sec. 4.7.2. +3.4 Switching between SPH and MFM in OpenGadget3 +To substitute SPH with MFM, the general code-structure does not +have to be altered. Mainly, the SPH specific force calculation has to +be replaced by the three steps of the MFM calculation, consisting of +gradient calculations, slope-limiting and the actual flux calculation. +As the Riemann solver both requires and outputs physical quantities, +while the rest of the code deals with code units, these units have +to be converted according to Eqn. (17) to (21) just before the flux +calculation. At all places, where results of that calculation, includ- +ing the hydrodynamical acceleration, are used, they first have to be +converted back to physical units. +Also, MFM calculates internal energy changes following the out- +put of the Riemann solver, while in SPH the entropy is evolved. +3.5 Differences to previous implementations of MFM +While the general concept of MFM with respect to the implementa- +tions introduced in gizmo and GANDALF stays the same, there are +several differences compared to these previously made implementa- +tions. Our implementation is based on the one in GANDALF, which +is originally intended to be well suited for star and planet formation. +We expand this implementation by including co-moving integration +and other extensions such as an energy-entropy switch to be used +for cosmological applications. In addition, we change the time in- +tegration scheme from a second-order accurate MUSCL-Hancock +to a second-order accurate Leapfrog KDK, consistent with SPH in +OpenGadget3. +The main difference of OpenGadget3 compared to gizmo is that +fluxes are by default calculated using an iterative, exact Riemann +solver compared to an approximate HLLC Riemann solver used in +gizmo, with an exact Riemann solver only used as fallback. +In addition, there are a few minor differences such as the second- +order correction in Eqn. (37) and making the pairwise limiter La- +grangian, as described in App. B. The convergence of the density +calculation is slightly different between the codes. We follow the +same implementation as for SPH in OpenGadget3, just replacing +the mass density by the number density. Finally, our implementation +employs a hybrid MPI-OpenMP parallelization as done for other +modules of OpenGadget3. +4 TEST CASES +We use several test cases to probe the ability of the different hydro- +methods to accurately follow gas evolution. All of them explore +specific numerical aspects important for cosmological simulations. +We use these tests to compare our new MFM implementation in +OpenGadget3 to SPH in OpenGadget3, MFM in the public gizmo1 +version and the publicly available version of the moving mesh code +Arepo2. +4.1 Settings +We aim for a fair comparison of the different codes throughout the +paper but adopt a general setting for slope limiters, Riemann-solvers +(MFM) as well as the artificial diffusion terms (SPH) that one would +adopt in cosmological simulations. While this leads to overall good +performance of all solvers on almost all test cases, there are a few +test problems (e.g. the square test in Sec A) for which this is not +working ideal and we will discuss this in detail in the remainder of +the paper. If not otherwise mentioned, we assume an ideal gas with +𝛾 = 5/3 and all code operate on adaptive time steps for all tests (i.e. +we never force a small constant time step to improve the accuracy of +the results). +MFM is used with a cubic spline kernel and 32 (24) neighbors in 3d +(2d). The slope limiter from gizmo in combination with their pairwise +limiter is used. Consistent settings are chosen between OpenGad- +get3 and gizmo. For SPH, a Wendland C6 kernel, including bias +correction (Dehnen & Aly 2012), with 295 (64) neighbors in 3d (2d) +is used. The modern, time-dependent artificial viscosity scheme of +Beck et al. (2016a) and artificial conductivity (Price 2008) are in- +cluded. For Arepo we use additional mesh regularization based on +the center of mass, and the “roundness” of the cells. An overview +of all settings is made publicly available3. If not otherwise stated, +the initial conditions (ICs) are created with equal particle masses. +In most cases, particles are arranged in a (perturbed) regular grid in +order to reduce noise introduced by the initial particle distribution. +4.2 Stability +4.2.1 Soundwave +As a first test we adopt a sinusodial soundwave with density 𝜌 = 1 +and small perturbation amplitude Δ𝜌 = 10−4 in a box of length 1 in +𝑥-direction and 0.75 in 𝑦/𝑧 direction. The particles are arranged in a +perturbed hexagonal close packed (hpc) grid with varying resolution. +The number of particles is ranging from 643·0.752 up to 1283·0.752. +In the following, we will define the resolution by the number of +particles per unit-length in 𝑥 direction. We adopt a wavenumber +𝑘 = 2𝜋 and a speed of sound of 𝑐𝑠 = 2/3. For this test there is an +analytic solution 𝜌(𝑥, 𝑡) = 𝜌0 +Δ𝜌 sin(𝑘(𝑥 +𝑐𝑠𝑡)), which makes this +test well suited to perform a convergence analysis. For this purpose, +we measure the L1 error norm 1 +𝑁 +�𝑁tot +𝑖 +|𝜌𝑖 − 𝜌(𝑥, 𝑡)|. All methods +are able to evolve the soundwave, while the accuracy as well as the +precise convergence behavior differ among the codes. We observe +only first order convergence for resolutions > 64 for all methods. +In order to get a more detailed analysis, we split the error between +errors in the position, in the amplitude, and scatter quantified by an +L1-error as shown in Fig. 2. +Deviations from the expected sound speed are related to disper- +sion errors, and will lead to an offset compared to the analytical +solution. This offset error is shown in the upper panel. We observe +for MFM in both implementation the convergence to be between +1 Obtained from https://bitbucket.org/phopkins/gizmo-public/ +src/master/ February 2021 +2 Obtained from https://gitlab.mpcdf.mpg.de/vrs/arepo June 2021 +3 https://github.com/fgroth/hydro_tests +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +7 +Figure 2. Offset-, amplitude- and scatter-errors of the density of a soundwave +at 𝑡 = +2 +𝑐𝑠 calculated with MFM and SPH in OpenGadget3, MFM in gizmo +and a moving mesh in Arepo at different resolutions. The scatter converges +second order for all methods, while other errors show different convergence +behavior. MFM shows between first and second order convergence for all +error-components. +first and second order. Both, our implementation and the one in the +gizmo code, are very similar. For SPH and Arepo, the overall error +is roughly one order of magnitude smaller at the lowest resolution, +but having a convergence even worse than first order. For SPH, this +trend can be explained by low-order errors, which are prominent for +traditional SPH, and still partly left for modern SPH. The error in +the amplitude, shown in the middle panel, is related to numerical +diffusion. As we see also in other tests, the Riemannn solver and the +slope-limiter introduce numerical diffusivity for MFM, which thus +has the largest error. Differences between the different MFM im- +plementations can be explained by different Riemann solvers used. +SPH and Arepo show much lower errors. The convergence behavior, +however, is again better for MFM compared to the other methods. In +both implementations, it is roughly second order, while for the other +methods it appears to be approximately first order. Finally, it is worth +to note that the resulting soundwave does not have perfect sinusodial +shape but shows scatter in the amplitude. This is mainly a result of +the smoothing length/density iteration and the threshold chosen for +the value to be taken as converged. We quantify this error by the +L1 error norm, shown in the bottom panel of Fig. 2. All methods +show roughly second order convergence, while the amplitude of the +error is different. Differences between MFM and SPH in OpenGad- +get3 can be explained by the different kernel used, while other codes +have differences in the iteration and treat parameters for convergence +slightly differently. The large error for Arepo, even at higher resolu- +tion makes the values for the other errors more uncertain. In addition +to the errors already mentioned, the soundwave deforms and steepens +up due to non-linear terms in the evolution. This non-linearity will +lead to an additional, small but constant term in the scatter error in the +bottom panel of Fig. 2. A reduction could be achieved by reducing the +amplitude, which would also make scatter errors be more significant +or the convergence more expensive. The importance of non-linear +contributions increases with lower scatter, it dominates at the highest +resolutions considered, such that the convergence behavior appears +slightly worse for the other errors. +4.2.2 Kepler Disk +The Kepler disk is an important test case for cosmological simula- +tions, allowing to study the ability of the code to conserve angular +momentum and maintain stable orbits over time. Especially, the ef- +fect of viscosity can be analyzed. To this end we initialize a two- +dimensional box sufficiently large to contain all particles. The ICs +are taken from Hopkins (2015) and are initialized with 48240 gas +particles with equal masses, arranged in a grid-like structure and +setup with vanishing pressure of 𝑃 = 10−6. The gas surface density +distribution is given via: +Σ = 0.01 + +���� +���� +(𝑟/0.5)3 +if 𝑟 < 0.5 +1 +if 0.5 ≤ 𝑟 ≤ 2 +(1 + (𝑟 − 2)/0.1)−3 +if 2 < 𝑟. +(42) +For the Arepo run, we adopt a low density mesh with vanishing +pressure at resolution 16 distributed around the disk as well as inside +the central hole of the disk. +We adopt an external potential Φ = −(𝑟2 + 𝜖2)−1/2 with resulting +MNRAS 000, 1–26 (2023) + +convergence +MFM +- 1st order ++ +SPH +102 +2nd order +GIZMO ++ +AREPO +offset error +10-3 +10~ +error +amplitude +10 +L-01 +10 +32 +64 +128 +resolution8 +F. Groth et al. +Figure 3. Evolution of the Kepler disk using different hydro-methods. Surface +density at two times per method: 𝑡 = 12.5 (upper left) and 𝑡 = 120 (lower +right). In general, all methods are able to evolve a stable disk. Initial per- +turbation introduced by the ICs, however, evolve differently for the different +methods. +gravitational acceleration of the form +g = − r +��������� +��������� +� +(𝑟/0.35)2 +(r2)1.5 +− (0.35−𝑟)/0.35 +(r2)1.5 +� +if 𝑟 ≤ 0.35 +� +1 +(r2)1.5 +� +if 0.35 < 𝑟 < 2.1 +� +1+(𝑟−2.1)/0.1 +(r2)1.5 +� +if 2.1 ≤ 𝑟. +(43) +We follow the evolution of the disk until 𝑡 = 120, corresponding +to ≈ 20 orbits at 𝑟 = 1. The resulting density at 𝑡 = 120 and 𝑡 = +12.5 is shown in Fig. 4.2.2. Initially, all methods produce spirals +as a result of perturbations in the ICs. While for more traditional +SPH with Balsara viscosity switch (Balsara 1998) these lead to a +destruction of the disk after only a few orbits, consistent with the +results of Beck et al. (2016a), the modern SPH implementation in +OpenGadget3 with the improved viscosity scheme of Beck et al. +(2016a) drastically increases the stability of the disk. While the inner +and outer region still show some decay, the main part of the disk +is stable for the whole evolution considered. For MFM the disk +remains stable for more than 20 orbits. We observe that the inner +and outer parts of the disk degrade much less compared to SPH. The +initial perturbations are diffused throughout the disk, which shows +slightly larger perturbations in the main part compared to the SPH +calculation. Both, our implementation and the one in gizmo, show +qualitatively similar results. The Arepo run turns out to produce the +most stable disk. Only a slight degeneration at the boundaries can be +observed. Further studies would be needed to analyze whether this +is a numerical effect or due to interaction with the ambient medium +not present in the other calculations. +4.3 Tests for Fluid Mixing instabilities +Mixing occurs in a variety of cosmological situations, most promi- +nently during ram-pressure-stripping. To this end, we analyze the +ability of the different codes and methods to evolve such mixing +instabilities. +4.3.1 Rayleigh-Taylor Instability +One popular fluid-mixing test is the Rayleigh-Taylor instability. It +can be used to explore how well the code can describe unstable, +growing modes. The setup we use is taken from Hopkins (2015). +The calculations are preformed in a two-dimensional periodic box +with side-lengths 1, where the particles at 𝑦 < 0.1 and 𝑦 > 0.9 +are fixed as boundary conditions. A fluid of high density (𝜌 = 2) +is placed on top of a low-density medium (𝜌 = 1) in hydrostatic +equilibrium. For this test-case, we take 𝛾 = 1.4, as for a diatomic +gas, such as molecular hydrogen and apply the constant gravitational +acceleration: +agrav = − 0.5ˆ𝑦. +(44) +To allow the instability to grow, a small velocity perturbation at the +phase boundary is introduced (for more details see Hopkins 2015). +In Fig. 4 we show that all methods are perfectly able to evolve +the instability. A major difference between the different methods is +the presence of asymmetries and secondary instabilities. While these +can be seen clearly for MFM, both in OpenGadget3 and gizmo, and +are also present in the Arepo calculation where they appear more +symmetric, we find that they are absent from the SPH calculation, +due to the over smoothing over the larger kernel and the effectively +lower spatial resolution (e.g. Marin-Gilabert et al. 2022, for a more +detailed discussion of the occurance of secondary insatbilties and +their physical meaning). The results of Arepo indicate the sharpest +boundary and highest density in the tip, followed by MFM. The +boundary particles for Arepo, show still a clear imprint of the initial +grid-like particle distribution. We note that the numerical diffusivity +within modern SPH causes the boundary of the instability to have a +shallower gradient and smears out initial asymmetries. In addition, +the effective spatial resolution is lower by a factor of ≈ 2 compared +to MFM due to the larger neighbor number and thus SPH reaches a +much lower density in the tip of the instability. +4.3.2 Kelvin-Helmholtz Instability +Similar to the Rayleigh-Taylor instability, also the Kelvin-Helmholtz +instability is a famous example for fluid mixing. Again, we use the +setup provided by Hopkins (2015). Two fluids of densities 𝜌1 = 1 and +𝜌2 = 2 in hydrostatic equilibrium are initialized in a 2d periodic box, +with initial velocities v1 = 0.5ˆ𝑥, v2 = −0.5ˆ𝑥 and a small perturbation +following McNally et al. (2012). At time 𝑡 = 2.5 corresponding +to ≈ 1.2𝜏KH in units of the Kelvin-Helmholtz timescale 𝜏KH, the +instability has produced a roll for all methods, as shown in Fig. 5. +Differences are present in the inner structure of the roll. Overall the +qualitative results are very similar to those for the Rayleigh-Taylor +instability. SPH is smoothing the roll, showing no secondary insta- +bilities and evolving more smoothly towards later times. Compared +to that, MFM in both implementations shows a clear separation be- +tween the higher-density roll and the less dense medium, with the +presence of secondary instabilities. A more detailed analysis of the +Kelvin-Helmholtz instability, also using our new MFM implementa- +tion, has been done by Marin-Gilabert et al. (2022). They also show +that the secondary instabilities can be avoided by using a higher +MNRAS 000, 1–26 (2023) + +0 +-1 +MFM +97.5 +GIZMC +120.0 +t= +y0 +-1 +SPH +20.0 +AREPC +120.0 +-1 +0 +1 +-1 +0 +1 +x +xMFM in OpenGadget3 +9 +Figure 4. Rayleigh-Taylor instability at time 𝑡 = 3.6. Comparison between the different hydro-methods. Vertical line marks the initial position of the phase +boundary. Differences are mainly the presence or absence of secondary instabilities. +neighbor number in combination with a higher-order kernel. This +will increase the intrinsic viscosity and prevent mixing in form of +secondary instabilities. Also Arepo shows secondary instabilities, +present especially inside the roll. When present, these perturbations +will finally dominate the evolution over the build-up of the roll for +𝑡 ≳ 3. +4.3.3 The “Blob” test +A more complex problem is the blob test. It is designed to mimic +ram-pressure stripping by an interplay of the evolution of shocks and +fluid-mixing instabilities. We use the setup described by Hopkins +(2015) (compare also Agertz et al. 2007). A cloud of higher density +𝜌cloud = 10𝜌wind is placed into a wind tunnel with supersonic flow +at M = 2.7 and density 𝜌wind = 2.6 · 10−8. Both phases are setup in +pressure equilibrium. +The resulting density in a slice through the cloud at 𝑡 = 𝜏𝐾 𝐻 is +shown in Fig. 6. In front of the cloud, a bow shock forms. At the +MNRAS 000, 1–26 (2023) + +MFM +GIZMO +N +SPH +AREPO +n10 +F. Groth et al. +Figure 5. Build-up of a 2d Kelvin-Helmholtz instability at 𝑡 = 2.5 comparing different methods. Horizontal dashed lines mark the initial position of the phase +boundary. All methods produce the roll, but with differences in their inner structure. +Kelvin-Helmholtz timescale 𝜏KH = 2, the cloud has developed in- +stabilities. These are much more pronounced for MFM and Arepo, +while for SPH the cloud deforms, without showing instabilities. The +precise form of the cloud differs between our MFM implementation, +that in gizmo and the moving mesh code Arepo. Nevertheless, the +cloud mass, defined by the particles obeying 𝜌 > 0.64𝜌cloud,𝑖 and +𝑢 < 0.9𝑢amb,𝑖, is very similar for all methods until 𝜏𝐾 𝐻 , shown in +Fig. 7. As expected, the MFM calculations line up with the calcu- +lations done by Hopkins (2015). The periodic bumps are a result +of the self-interaction of the shock due to the choice of boundary +conditions. +At later times the evolution strongly deviates. While for MFM as +well a moving mesh secondary instabilities build up and lead to a +disruption of the cloud, it is more stable in SPH. Compared to the +more traditional SPH results of Hopkins (2015), however, we find the +blob to decay stronger, as modern SPH with time-dependent artificial +viscosity and conductivity is able to evolve instabilities much better, +thus allowing for more mixing. +MNRAS 000, 1–26 (2023) + +MFM +GIZMO +SPH +AREPOMFM in OpenGadget3 +11 +Figure 6. Blob at 𝑡 = 𝜏KH and 𝑡 = 4𝜏KH as small insertion comparing different hydro-methods. At the earlier time, SPH leads to much less deformation due to +less instabilities building up, while MFM in both implementations as well as Arepo agree qualitatively. At late time, MFM and Arepo are fully mixed, while +SPH still has some structure remaining. +Figure 7. Decay of the cloud fraction surviving for the different methods. In +the background, comparison lines of the results by Hopkins (2015) for MFM +(black, solid) and (traditional) SPH (orange dashed) are shown. MFM and +Arepo agree very well, while SPH shows less mixing. +4.4 Tests for Shock-capturing +4.4.1 Sod Shock-tubes +Another important capability of the code is to capture strong shocks +of (arbitrarily) large Mach number. We begin testing this on a sim- +ple Sod shock-tube based on the setup of Sod (1978). The test is +preformed in a periodic box with two fluids of different density and +pressure (𝜌1 = 1, 𝑃1 = 1; 𝜌2 = 1/8, 𝑃2 = 0.1 for 𝛾 = 1.4) that are +initialized in a glass-like configuration. When the two phases start +interacting, a shock begins to move to the right. In Fig. 8, we show +the resulting structure at 𝑡 = 2.5 for the MFM calculations at dif- +ferent Mach number and compare them to the analytic solution. The +expected profiles are matched very well, for all the Mach numbers +adopted in this work, ranging from a very low M = 1.5 shock to +a strong M = 100 shock. This ability is directly connected to the +accuracy of the Riemann solver. For higher Mach numbers, increas- +ing peaks in velocity and entropy at the shock front are present as a +result of the slope-limiting procedure, which has also been reported +by Hopkins (2015). We note that this can be avoided by using a +TVD-limiter has more disadvantages in other cases. With increasing +Mach number, a sufficiently small timestep becomes more important. +The scatter in velocity for the high M = 100 shock, as well as the +small offset in the position of the shock front converge away with +decreasing timesteps. +The scatter in density present at all Mach numbers is a result of +the choice of the ICs, which are setup in a glass-like configuration +and designed for a higher neighbor number. It does not converge +for low neighbor numbers, as chosen for MFM. The pressure profile +shows the typical bump at the rarefaction fan, as well as the pressure +blip at the contact discontinuity, shown in more detail in Fig. 9 +for the intermediate M = 10 shock. This indicates the presence of +surface tension-like error terms, introduced by the slope limiter. As +MNRAS 000, 1–26 (2023) + +GIZMO +AREPOMFM +SPH +0.8 +GIZMO +AREPO +Hopkins (2015) +0.6 +SPH +MFM +0.4 +0.2 +0.0 +1 +2 +3 +4 +t/TKH12 +F. Groth et al. +Figure 8. Density, pressure, velocity and entropy profile of the shock tube at 𝑡 = 2.5 calculated with our MFM implementation, comparison between different +Mach numbers. MFM is able to reproduce the general structure of the shocks. Artifacts of surface tension introduced by the slope-limiter are visible at higher +Mach numbers. The scatter is a result of the choice of ICs. +Figure 9. Pressure profile of the M = 10 shock tube at 𝑡 = 2.5, comparison between different hydro-methods. The different codes show different amount of +surface tension and also slight differences in the position of the shock front due to different timestepping +discussed in App. A on the example of the hydrostatic square, these +terms are present for SPH and both MFM implementations, but not +for Arepo, manifesting also in the presence or absence of the pressure +blip for the different methods. The shock front is captured equally +well for MFM and SPH, though less smoothed out for MFM due to +the lower neighbor number. Arepo poorly captures the behavior at +the shock front due to several reasons. First, it has troubles in the +mesh reconstruction in this strongly anisotropic region, which leads +to a shift in the position of the shockfront. Second, the public Arepo +version does not include slope-limiters, which leads to the oscillatory +behavior in the shocked region. It could be improved using a static +mesh, which would remove other advantages of this method, however. +Also the inclusion of a slope limiter would improve results. +4.4.2 Sedov-Taylor Blastwave +This very strong, radially symmetric shock has first been introduced +by Sedov (1959). Besides the capability to deal with jumps, Saitoh & +Makino (2009) describe how it can be used to analyze the timestep +limiter and shows the need for the limiting to be non-local, as provided +by the wakeup scheme. The test has become a popular benchmark for +MNRAS 000, 1–26 (2023) + +1.0 +M= 1.5 +M=3.0 +M= 10.0 +M= 100.0 +P 0.5 +0 +40 +P20 +10 +2 +5 +0 +100 +rdid: +50 +A +0 +40 +60 +80 +100 40 +60 +80 +100 40 +60 +80 +100 40 +60 +80 +100 +x +x +x +xMFM +10 +SPH +10 +GIZMO +10 +AREPO +10 +40 +: +P +0 +0 +0 +0 +20 +92 +93 +92 +93 +92 +93 +92 +93 +9 +9 +6 +8 +8 +8 +8 +88 90 +0688 +88 90 +8890 +0 +40 +60 +80 +100 40 +60 +80 +100 40 +60 +80 +100 40 +60 +80 +100 +x +x +x +xMFM in OpenGadget3 +13 +Figure 10. Sedov blast at 𝑡 = 0.02. Comparison between different meth- +ods. The main difference is the height of the peak, which is reduced due to +smoothing of the jump. +Supernova blast wave evolution in recent years (e.g. Kim & Ostriker +2015; Steinwandel et al. 2020). +As ICs, we setup a regular grid with 643 particles and density +𝜌 = 1. While almost all particles exhibit a vanishing pressure 𝑃𝑎 = +10−6, energy of 𝑈 = 10 is distributed equally into the eight central +particles. A shock with very high M𝑖 ≳ 2 · 104 arises, and quickly +moves outwards.The radial density distribution is shown in Fig. 10. +All methods are able to capture the shock, though slightly smooth- +ing it, thus underestimating the height of the density-peak. SPH shows +the strongest smoothing, followed by the two MFM implementations. +Arepo is able to reproduce the height of the peak best. +The position of the peak is similar for all methods, with minor +differences. While Arepo and gizmo’s MFM implementation predict +the peak position correctly, MFM and SPH in OpenGadget3 lag +slightly behind, which results in a more accurate position of the +low-density side of the shock. This position strongly depends on the +precise timestep settings, indicating differences in the timestepping +between the codes. +4.5 Including self-gravity +In cosmological contexts, not only hydrodynamical forces, but also +gravitational accelerations are of great importance. Gravity domi- +nates the evolution on large scales due to its long-range character. It +can lead to collapse of clouds, e.g. in the ISM for star formation, or +balance thermal pressure and lead to hydrostatic equilibrium, such +as in the global structure of galaxies or galaxy clusters. Thus, we +analyze the interplay between hydrodynamical forces and gravity in +the following. +Figure 11. Evolution of the half-mass radius for the gravitational freefall test. +All methods agree at early time, but deviate from the expected solution at +later times when hydrodynamical contributions become more important. +4.5.1 Gravitational Freefall +As a first test including self-gravity, we simulate a collapsing sphere. +The ICS are set up on a regular grid of 203 particles and cut out +a sphere of radius 1, which has a total mass of 𝑀sphere = 1 and +a negligible pressure of 𝑃 = 10−6. For the Arepo run, we fill the +region not occupied by the sphere with low mass, low energy particles +at resolution 8, in order to improve the mesh reconstruction at the +boundary. We follow the evolution of the half-mass radius, to not +be influenced by boundary effects as for the full radius, shown in +Fig. 11. Comparing to the analytic solution +𝑡(𝑟) = arccos +�√︂ 𝑟 +𝑟0 ++ +√︂ 𝑟 +𝑟0 +√︂ +1 − 𝑟 +𝑟0 +� +· 2 +𝜋 +√︄ +3𝜋 +32𝜌0 +, +(45) +all methods agree at early times. At late times, pressure and thus ef- +fects of the hydro-scheme become more relevant, and deviations are +visible. All methods overestimate the radius. MFM lies closest to the +analytic solution with both implementations being indistinguishable. +The moving mesh code Arepo performs worst, which can be ex- +plained by poor treatment of the non-periodic boundary conditions. +In order to construct the grid for the hydro-calculations, the box has +to be treated periodically, which is not the case for all other methods. +Including the low mass cells at the boundary already decreased the +error by a factor of 2. SPH lies in between the other methods except +at very late times, when the error strongly increases. +MNRAS 000, 1–26 (2023) + +MFM +GIZMO +4 +3 +1 +0 +SPH +AREPO +4 +3 +2 +1 +0.2 +0.4 +0.0 +0.2 +0.4 +r0.2 +residuals +0.0 +0.8 +0.7 +0.6 +0.5 +0.4 +R +0.3 +analytical +0.2 +MFM +SPH +0.1 +MFM (GIZMO) +AREPO +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +t14 +F. Groth et al. +4.5.2 Hydrostatic Sphere +In cosmological contexts, e.g. for the ICM, the ability of the code +to preserve hydrostatic equilibrium against gravity is of great impor- +tance. To test this, we calculate a hydrostatic sphere as a second test +including self-gravity. It is also the first test including dark matter as +second, only gravitationally interacting particle type. The ICs have +been created following Viola et al. (2008). 88088 DM particles are +setup following an NFW profile (Navarro et al. 1997), populated +with 95156 gas particles in hydrostatic equilibrium. The correspond- +ing density and internal energy profiles at different times are shown +in Fig. 12. After a short relaxation period, happening on a timescale +approximately corresponding to the dynamical time, we expect the +gas to keep hydrostatic equilibrium. SPH shows the lowest deforma- +tion in density, as the ICs were also designed assuming SPH. MFM +in both implementations, as well as Arepo show a slightly stronger +increase in density, especially in the central region. The convergence +of the profile can best be checked by following the evolution of the in- +ternal energy profile, which is more directly affected by (numerical) +diffusivity. For SPH, this appears to be stable and also shows only +minor changes for Arepo at early time, reaching a stable situation +later. For MFM, in contrast, an impact of the numerical diffusivity +can be observed. Resulting mixing in the central region leads to a de- +crease in internal energy, leading to the observed increase in central +density. This effect is very similar for our MFM implementation and +the one in gizmo. Despite these findings, the effect on the density +profile is quite small for all methods. +4.5.3 Zeldovich pancake +The Zeldovich pancake is the first problem to test our implementation +of co-moving integration. In addition, it is well suited to show effects +of very high M flows, shocks, highly anisotropic particle arrange- +ments, and also very low internal energies. It has been introduced by +Zel’dovich (1970). We start our calculation at 𝑧𝑖 = 100, setting up a +single Fourier mode density perturbation. During the linear growth +until the caustic formation at 𝑧𝑐 = 1, the evolution can be described +by +𝑥 = 𝑥𝑖 − 1 + 𝑧𝑐 +1 + 𝑧 +sin(𝑘𝑥𝑖) +𝑘 +(46) +𝜌 = +𝜌0 +1 − 1+𝑧𝑐 +1+𝑧 cos(𝑘𝑥𝑖) +(47) +vpec = − 𝐻0 +1 + 𝑧𝑐 +√ +1 + 𝑧 sin(𝑘𝑥𝑖) +𝑘 +ˆ𝑥 +(48) +𝑇 = 𝑇𝑖 +� 1 + 𝑧𝑐 +1 + 𝑧 +�2 � 𝜌(𝑥, 𝑧) +𝜌0 +�2/3 +(49) +starting from the unperturbed position 𝑥𝑖. 𝜌𝑐 is the critical density, +𝐻0 = ℎ0 · 100 km s−1 Mpc−1 the Hubble parameter (today) with +ℎ0 = 1, and 𝑇𝑖 = 100 K the initial temperature, such that pressure +forces are negligible. The wavenumber 𝑘 = 2𝜋/(64 ℎ−1Mpc) cor- +responds to the first-order soundwave. We use the ICs provided by +Hopkins (2015), with a resolution of 323 particles. After the lin- +ear growth, an accretion shock forms close to the center. As the +scale factor increases, the background density decreases strongly +and the background temperature decreases adiabatically. This causes +a huge temperature contrast of ≈ 10 orders of magnitude between +the shocked region and the background. Due to the very low internal +energy compared to other energy contributions 𝑈 ≲ 10−3𝐸kin and +𝑈 ≲ 10−2𝐸pot in physical units, which thus is on the same order +as the precision of the Riemann solver, the implementation of the +energy-entropy switch described in Sec. 3.3 is important. Its effect +on the evolution of the Zeldovich pancake is described further in +Sec. 4.7.2. The resulting structure at 𝑧 = 0 is shown in Fig. 13. +Again, we compare the performance of the different hydro-methods. +The energy-entropy switch is included for MFM if 𝑈 < 0.01𝐸pot. +For Arepo, we had to use additional mesh regularization to avoid too +irregular cell shapes in the highly unisotropically compressed shock +region and allow the code to run until the end. All methods agree +with the peculiar velocity profile with only slight differences. Com- +pared to Hopkins (2015) we find that all methods seem to have a too +low viscosity and show particle over- or under-shooting compared to +the predicted velocity profile, as a result of a punch-through of some +particles in the high M shock. The density peak is captured equally +well by almost all methods. Only the Arepo run shows a slightly +higher peak, contrarily to what Hopkins (2015) found. Compared to +the expected profile, all methods over-smooth the central region. Due +to the higher neighbor number and thus larger kernel for SPH, this +smoothing is larger compared to MFM, both in our new implemen- +tation and gizmo. MFM shows a similar performance as the moving +mesh. Most difficult for all methods is to capture the temperature +structure with its very strong contrasts. Both MFM implementations +work very well, as the energy-entropy switch suppresses any numer- +ical noise in the low-energy background and allows a clear jump +between shocked and unshocked region. The jump for SPH is more +strongly smoothed in comparison to the other methods. In addition, +amplified initial (numerical) noise causes a large scatter of several +orders of magnitude in the very cold background. For Arepo, we +find that this behavior is much more drastic, and the background is +dominated entirely by numerical noise. To properly resolve it, some +energy-entropy switch would be required also in Arepo, which does +not seem to be implemented in the public version. +4.5.4 Nifty Cluster +Finally, we apply our newly implemented method on more complex, +cosmological cases. As an example, we re-simulate a cluster from +the MUSIC-2 sample (Prada et al. 2012; Sembolini et al. 2013, 2014; +Biffi et al. 2014), analyzed in detail with different codes by a col- +laboration formed during a nifty workshop (Sembolini et al. 2016), +thus called nifty cluster in the following. The cluster has a mass +𝑀200c = 1015 M⊙ with resolution 𝑚DM = 9.01 · 108ℎ−1M⊙ for +dark matter and 𝑚gas = 1.9 · 108ℎ−1M⊙ for gas particles. The back- +ground cosmology has parameters ΩM = 0.27, Ωb = 0.0469, ΩΛ = +0.73, 𝜎8 = 0.82, 𝑛 = 0.95, ℎ = 0.7 (Komatsu et al. 2011). The pro- +jected surface density at 𝑧 = 0 is shown in Fig. 14, where the cluster +center and virial radius are obtained using Subfind (Springel et al. +2001; Dolag et al. 2009). +We compare MFM to SPH with a different amount of artificial +conductivity, ranging from the usually used amount 𝛼max = 0.25, +𝛼min = 0 (notation following Price 2008) over a run with physical +conductivity at 1/20th of the Spitzer value (Dolag et al. 2004), effec- +tively corresponding to an intermediate amount, to more traditional +SPH without artificial conductivity. The usual amount is chosen to +mimic the behavior of Godunov methods such as MFM, which have +intrinsic numerical diffusivity due to the Riemann solver. For re- +duced artificial conductivity, structures are slightly less “smeared +out”, while the global structure does not change. +A more quantitative analysis can be done using gas radial density, +temperature and entropy profiles shown in Fig. 15. As a comparison, +we provide lines from the nifty paper, obtained using Arepo and +Gadget3-MUSIC as an example of a more traditional SPH code, +which mark the range of solutions obtained. SPH can span the whole +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +15 +Figure 12. Evolution of gas density (top) and internal energy radial profiles (bottom) for the hydrostatic sphere for a few dynamical times until 𝑡 = 10, colored +by the time. Calculated using different hydro-methods. MFM shows a slightly larger numerical diffusivity, but overall still preserves the density profile. +Figure 13. Zeldovich pancake at 𝑧 = 0 for different hydro-methods. As a comparison, a high resolution 1d simulation of Hopkins (2015) is shown. While +velocity and density profiles agree between the methods, strong deviations can be seen for the temperature profile. MFM performs best due to the energy-entropy +switch employed. +MNRAS 000, 1–26 (2023) + +10 +10 +MFM +SPH +GIZMO +AREPO +8 +Q 10-5 +6 +10-6 +i06 +2 +105[ +102 +103 +102 +103 +102 +103 +102 +103 +r +r +r +r2 +MFM +SPH +GIZMO +AREPO +[km/s] +1000 +23 +log(T) [K] +0 +20 +0 +20 +20 +0 +20 +20 +0 +20 +20 +0 +20 +x [h-1Mpc] +x [h-1Mpc] +x [h-1Mpc] +x [h-1Mpc]16 +F. Groth et al. +Figure 14. Projected surface density of the nifty cluster at 𝑧 = 0, comparison between MFM and SPH with usual amount (𝛼cond +max = 0.25), physical (𝜅phys), +corresponding to an intermediate amount, and without artificial conductivity 𝛼cond +max = 0. The overall structure is very similar. Small sub-structures, however, +appear less compact for MFM. +range of possible solutions provided by Sembolini et al. (2016). By +construction traditional SPH without artificial conductivity has no +mixing and thus forms low entropy cores. Subgrid mixing due to the +Riemann solver for MFM and Arepo leads to mixing into the core, +increasing the entropy compared traditional SPH. Thus, the central +density is reduced. By including artificial conductivity in SPH, it can +reach the same profile as MFM, and also lie in between for effectively +intermediate values by using physical conductivity.. +The computational costs for running MFM are comparable to those +of SPH. Each single timestep takes 41 s walltime for SPH, compared +to 31 s for MFM4. Thus, the runtime of MFM per timestep is smaller +by a factor ≈ 1.3. The majority of the time is spent in the gravity cal- +culation. For MFM, the flux calculation and density iteration makes +up only 1 s per timestep, while for SPH, density and acceleration +calculations take 10 s per timestep, explaining the difference in total +4 Run on Supermuc on 8 nodes, each with 4 MPI tasks, and 24 OpenMP +threads. +MNRAS 000, 1–26 (2023) + +2.6 +Ruir +2.8 +0 +Rir +3.0 +[zd -,go d Bo +MFM +SPH(Kphys) +3.2 +3.4 +Ruir +3.6 +0 +Ruir +3.8 +=0.25) +=0) +Rrir +0 +Ruir +Rrir +0 +Rrir +4.0MFM in OpenGadget3 +17 +10 1 +100 +R [h +1Mpc] +1013 +1014 +1015 +1016 +gas [h +1M /(h +3Mpc3)] +MFM +SPH( +cond +max =0.25) +SPH( phys) +SPH( +cond +max =0) +Sembolini et al. (2016) +AREPO +G3-MUSIC +10 1 +100 +R [h +1Mpc] +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +kBT [keV] +10 1 +100 +R [h +1Mpc] +101 +102 +103 +S=T/n2/3 +e [keV h +2cm2] +Figure 15. Gas density (left), temperature (middle) and entropy (right) radial profiles of the nifty cluster at 𝑧 = 0, comparison between different hydro methods, +including our MFM implementation (red plus), SPH in OpenGadget with usual (green), physical, corresponding to an intermediate value, (turquoise) and without +artificial conductivity. As a comparison, the Arepo (black dashed) and G3-MUSIC (traditional) SPH line (red solid) from Sembolini et al. (2016) are shown. +The vertical line marks 𝑅200. Our modern SPH run with sufficiently high artificial conductivity, as well as Arepo and MFM produce higher entropy cores with +lower, less peaked density, while the central entropy is much lower for SPH with lower artificial conductivity. +time per step. The lower neighbor number causes a decrease in com- +putational cost in the density and also the actual flux calculation. This +is only partly compensated by the more expensive Riemann solver. +Thus, the decrease is even more significant for pure hydrodynamical +calculations. An addition, the computational cost could be further +decreased by using a faster approximate Riemann solver instead of +the exact one, which, however, has other disadvantages as discussed +an App. C. +While for pure hydrodynamical problems the number of timesteps +increases for MFM due to the effectively higher spacial resolution, +the timesteps for more complex simulations including gravity are +limited by other criteria than the Courant-criterion, not depending +on the smoothing. Thus, they are similar between the methods and +rather depend on the precise differences in evolution. Overall, MFM +on average even yields a slight decrease in runtime for cosmological +simulations. +The size of the structure holding the gas particle data, MFM has +a much larger requirement by a factor ≈ 5, which could be used by +more efficient use of existing variables in SPH. As also data of other +particle types is saved, the total memory requirement is only larger +by a factor of ≈ 2.7. The difference would decrease even further if +more physics was included. +4.6 Decaying Subsonic Turbulence +In many astrophysical systems, ranging from the atmosphere over +the ISM up to galaxy clusters, turbulence plays a crucial role. In the +ICM, we expect subsonic turbulence with a turbulent energy fraction +of 𝑋 ≈ 0.1 to be excited, for instance after a merger (compare, +e.g. Schuecker et al. 2004; Subramanian et al. 2006). The different +hydro-schemes have problems to capture its full behavior. It has been +shown that traditional SPH is not well suited to calculate sub-sonic +turbulence (Bauer & Springel 2012), but can be improved using +modern SPH with more ideal settings for artificial diffusion terms +(Price 2012). +To test and compare the performance of our MFM implementa- +tion, we setup a 300 kpc cubic box with varying number of particles, +and seed the largest ≈ 70 modes, similar to Bauer & Springel (2012). +Due to the low initial density of 𝜌 ≈ 1.5 · 10−6, gravitational acceler- +ation can be neglected. The initial turbulent energy fraction is varied +between 𝑋𝑖 = 𝐸turb,𝑖/𝐸therm,𝑖 = 0.3 and 𝑋𝑖 = 0.0001. In addition, +the resolution is varied, ranging from 643 up to 5123 particles. We +evolve the turbulence for 1.5 sound-crossing-times. The turbulent +kinetic energy cascades down to smaller scales, forming a turbulent +power spectrum. In order to analyze the velocity power spectrum, +the data are binned to a grid using the code Sph2Grid5. From that, +a power spectrum is calculated. We use a D20 sampling, in order to +conserve energy (Cui et al. 2008). Theoretically, a Kolmogorov slope +𝐸(𝑘) ∼ 𝑘−5/3 would be expected (Kolmogorov 1941). In Fig. 16, +we compare the power spectra of the different methods, normalized +by the expected slope. +While all methods agree at large scales, where the energy was +seeded, they show huge discrepancies at intermediate to small scales. +Arepo shows deviations at small scales, close to the resolution limit, +underestimating the energy present at these scales. SPH starts de- +viating at slightly larger scales, with a less deep dip in the power +spectrum. For MFM the power spectrum shows a dip in energy at +similar scales as the moving mesh code Arepo, but with a much shal- +lower depth than in all other cases, thus being closer to the expected +slope. +In addition, the MFM result converges quickly with resolution, +shown in Fig. 17. As the dip moves towards smaller scales, the overall +spectrum becomes closer to the Kolmogorov one. At the highest +resolution considered, it almost perfectly resembles the expected +Kolmogorov slope over a wide range of scales. +While the power spectrum builds up, energy is not only transported +to smaller scales, but also partly converted into internal energy. We +plot this decay of kinetic, turbulent energy in Fig. 18, comparing +the different hydro-methods. While in a physical situation the decay +would depend on gas microphysics such as its viscosity, here we can +use it to get an insight into the code behavior. The decay is mainly +determined by numerical dissipation. In all cases, the energy shows +a periodic variation, caused by the “ringing” of the initially seeded +5 Developed by J. Donnert, available at https://github.com/jdonnert/ +Sph2Grid +MNRAS 000, 1–26 (2023) + +18 +F. Groth et al. +10 2 +10 1 +k=2 /L [kpc 1] +10 5 +10 4 +10 3 +10 2 +10 1 +100 +101 +102 +k 5/3E(k) [arb. units] +k 128 +SML +k 128 +Nyquist +kbox +kSEED, min +kSEED, max +Kolmogorov +MFM (OpenGadget3) +SPH (OpenGadget3) +Moving Mesh (AREPO) +Static Mesh (AREPO) +Figure 16. Normalized turbulent velocity power spectrum for different methods at 𝑋𝑖 = 0.3. All methods agree at large scales, but show a lack in energy at +intermediate to small scales compared to the expected Kolmogorov-slope 𝑃 ∼ 𝑘−5/3. MFM works best overall reproducing the expected spectrum. +modes. MFM has a decay time of a few sound crossing times. The +decay for SPH depends strongly on the artificial viscosity, varying +from a value similar to that of MFM for SPH with the standard +amount of viscosity, up to even an (unphysical) increase for the cal- +culation without artificial viscosity. The power spectrum, in contrast, +is only weakly influenced by the amount of artificial viscosity. With +increasing resolution, the decay time increases, indicating numerical +dissipation errors converge away. Arepo shows a much slower decay +with 𝑡dec > 20𝑡sc already at lower resolution. A comparison for the +decay at different initial turbulent energy fractions, corresponding to +variations in the Mach number, is shown in Fig. 18 for MFM and +SPH. The variation between 0.3 and 0.0001 for the initial turbulent +energy fraction corresponds to a range of Mach numbers from 0.07 +down to below 0.007. For SPH the decay is independent of the Mach +number, as one would expect, so it is for Arepo. This is true also +for MFM down to 𝑋𝑖 > 0.003, corresponding to M = 0.007. For +even smaller Mach numbers, the turbulent energy increases. At the +same point also the density pdf deviates from the Gaussian shape, +indicating the evolution is dominated by numerical artifacts for such +low Mach numbers. +4.7 Effects of Numerical Parameters +The performance of the numerical methods strongly depend on the +precise parameters used. Effects of neighbor number and kernel have +already been analyzed in detail by various authors (compare, e.g., +Dehnen & Aly 2012; Tricco & Price 2013; Hu et al. 2014). To this +end, we focus on two other parameters that play a major role for MFM, +namely the slope-limiting scheme and the energy-entropy-switch. +4.7.1 Slope-Limiter +The different slope-limiting procedures, which are implemented in +our code, differ not only in how aggressively they limit the slope, but +also in how much numerical diffusivity they introduce. In general, +different limiters are shown to produce different results for specific +test-cases (compare e.g. May & Berger 2013; Hubber et al. 2018). +In the following, we compare the three cases of the limiter from +gizmo that we usually use, the Arepo and, the TVD limiter. The +gizmo and TVD limiters are the most extreme cases of the limiters +implemented, with lowest and highest numerical diffusivity, respec- +tively. The Arepo limiter lies in between. We analyze the effect on the +hydrostatic square (compare also App. A) and the Rayleigh-Taylor +instability (Sec. 4.3.1). The results are shown in Fig. 20. +While for the Rayleigh-Taylor instability the much less diffu- +sive gizmo limiter performs best, evolving a much finer structure, +this causes the strongest deformation of the hydrostatic square. The +Arepo limiter is slightly more diffusive, leading to less strong sec- +ondary instabilities for the Rayleigh-Taylor instability and slightly +less deformation of the square, especially at the edges. The TVD lim- +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +19 +10 2 +10 1 +k=2 /L [kpc 1] +10 5 +10 4 +10 3 +10 2 +10 1 +100 +101 +102 +k 5/3E(k) [arb. units] +k 128 +SML +=k 128 +Nyquist +kbox +kSEED, min +kSEED, max +resolution +643 +1283 +2563 +5123 +Figure 17. Normalized turbulent velocity power spectrum for MFM with different resolutions at 𝑋𝑖 = 0.3. MFM converges fast with resolution towards the +expected Kolmogorov-slope 𝑃 ∼ 𝑘−5/3. +iter has an even higher numerical diffusivity, thus strongly smooths +the Rayleigh-Taylor instability, not only preventing secondary insta- +bilities to form, but also reducing the overall growth of the instabil- +ity. The hydrostatic square, however, is preserved best, due to lower +surface-tension like errors, as it can be observed also e.g. for shocks. +Combining the results, we show that it is not always clear which +slope-limiting procedure would be the overall preferred choice. As +in most cases the gizmo limiter performs best, we chose this as our +reference method. +4.7.2 Energy-Entropy-Switch +To avoid numerical errors to dominate the evolution of the internal +energy, an energy-entropy switch as described in Sec. 3.3 has to be +used in specific problems such as the Zeldovich pancake. Especially, +the numerical noise should be suppressed in the very cold, unshocked +region, while the shock should not be influenced at all. +The resulting structure at 𝑧 = 0, comparing different possibili- +ties for the switch based on potential and kinetic energy estimates +(compare also Eqn. (41)), is shown in Fig. 21. We increase the tuned +values (𝛼1 = 10−2 for the potential energy and 𝛼2 = 3 · 10−3 for +kinetic energy) by a factor 2 and decrease them by a factor ≈ 3. +A more strict switch (larger 𝛼) causes less particles to be treated +with the adiabatic approximation. For the kinetic energy switch, this +difference causes strong variations in the temperature profiles. While +for 𝛼2 = 1 · 10−3 more extended wings form and some scatter in the +low-temperature background close to the peak appears, the increased +value of 6 · 10−3 treats even particles inside the peaked region with +the adiabatic approximation and causes too low temperatures. A +very fine-tuned choice of 𝛼2 is necessary to accurately capture all +particles, both the shocked ones and the low-temperature ones. +Compared to that, a variation of 𝛼 within the switch based on +potential energy influences the temperature profile only weakly. Thus, +it seems to be much more stable and should be the preferred option. +5 DISCUSSION AND CONCLUSIONS +We presented a new MFM implementation into OpenGadget3 as +an alternative hydro-solver to the currently used modern SPH. We +verified its capabilities, both in idealized and more complex, cos- +mological test cases. Tests range from smooth, simple situations, +mixing instabilities, shocks, tests including self-gravity, to the nifty +cluster as cosmological example and decaying, subsonic turbulence. +A comparison has been preformed between MFM and SPH in Open- +Gadget3, the MFM implementation in gizmo and the moving mesh +code Arepo. In addition, two parameters have been analyzed in more +detail. +Overall, we find very good agreement between the MFM imple- +mentation in OpenGadget3 and that in gizmo. Minor differences +are found in the precise appearance, while global properties are in- +MNRAS 000, 1–26 (2023) + +20 +F. Groth et al. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t/tsc +100 +4 × 10 1 +6 × 10 1 +E/E fit +i +hydro method +MFM +SPH ( +visc +max =0) +SPH ( +visc +max =3) +SPH ( +visc +max =10) +AREPO +AREPO(static) +Figure 18. Decay time of turbulent energy for different methods at 𝑋𝑖 = 0.3. +For SPH, the viscosity is varied between 𝛼visc +max = 10 and 𝛼visc +max = 0, where +𝛼visc +max = 3 is the value typically used (notation following Beck et al. 2016a). +Arepo has the highest decay time corresponding to the lowest numerical +dissipation, while MFM and SPH at typical value of viscosity are on a similar +order with a decay time of a few dynamical timescales. +100 +4 × 10 1 +6 × 10 1 +E/E fit +i +SPH: Xi = +0.3 +0.1 +0.01 +0.003 +0.001 +0.0003 +0.0001 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t/tsc +100 +4 × 10 1 +6 × 10 1 +E/E fit +i +MFM +Figure 19. As Fig. 18, but for varying initial turbulent energy fractions 𝑋𝑖, +corresponding to variations in the turbulent Mach number. The decay is +consistent for all 𝑋𝑖 for SPH, and down to 𝑋𝑖 = 0.003 for MFM, when +numerical artifacts lead to an unphysical increase in energy. +distinguishable. Even without further tuning, MFM reproduces the +expected behavior in all test cases considered. The soundwave test +is well suited for a convergence analysis, as an analytical solution +exists. MFM shows a very good convergence behavior between first +and second order for dispersion errors. Diffusion errors as well as +the scatter converge second order. While the convergence is better +than for SPH and a moving mesh, these methods show lower errors, +especially for the dispersion error. +An important advantage of MFM over SPH is the capability to +accurately evolve mixing instabilities without additional artificial +viscosity or conductivity as for SPH. In addition, a lower neighbor +number compared to SPH is sufficient. MFM as a moving mesh even +show secondary instabilities to occur. The blob test as combination +Figure 20. Hydrostatic square (top) and Rayleigh-Taylor instability (bottom), +developed using different slope-limiters, the gizmo limiter we usually use +(left), compared the the same test, but evolved using the Arepo limiter (mid- +dle) and TVD limiter (right). Depending on the test, different slope-limiters +could be preferred. +between mixing and shocks emphasizes the ability of MFM to allow +for more mixing. The rate of the decay of the cloud is similar to that of +a moving mesh simulation and larger than for SPH. Compared to the +more traditional SPH implementation shown by Hopkins (2015), the +modern SPH implementation OpenGadget3 allows for more mixing +and leads to a faster decay of the cloud. As this test is designed to +mimic ram-pressure stripping, we expect this effect to be modeled +more accurately using MFM compared to SPH. This should also lead +to an overall more accurate evolution of galaxies in the environment +of galaxy clusters. To fully understand and follow the evolution of +such gas blob in cosmological contexts more physics such as cooling, +and, depending on the context, star formation, is necessary. Gronke +& Oh (2018, 2020, 2022) have analyzed this test in detail with such +additional physics and found a great importance of the cooling. +In addition, MFM can model shocks for a wide range of Mach +numbers. For the shock tube tests MFM performs especially well +for lower Mach numbers, while effects of surface tension due to +the choice of the slope-limiter are visible at higher Mach numbers. +Nevertheless, it is still able to capture the main features of the shock +including the position of the shock front, the contact discontinuity, +and the rarefaction fan. Different methods lead to differences in +the smoothing of the shock front. The lower neighbor number in +MFM compared to SPH increases the effective spatial resolution by +a factor of ≈ 2. For Arepo, the shock front is dominated by numerical +artifacts due to the absence of a slope-limiter in the publicly available +version as well as difficulties in the mesh reconstruction in such highly +anisotropic region. +The Sedov blast works well for all methods, verifying the capa- +bility of the wakeup scheme as non-local timestep criterion. Main +differences are the smoothing and resulting lower amplitude of the +density peak, revealing an even smaller smoothing for the moving +mesh compared to MFM. The narrower shock front will help e.g. for +shock detection in cosmological simulations (compare, e.g., Pfrom- +mer et al. 2006; Beck et al. 2016b). +In general, MFM is able to preserve hydrostatic equilibrium accu- +rately, as well as preserving stable orbits. The better stability of the +Kepler disk compared to SPH will improve results for simulations +of e.g. isolated galaxies. For this case, a moving mesh leads to even +better results, but requires additional boundary particles. +MNRAS 000, 1–26 (2023) + +M +MMIARmMFM in OpenGadget3 +21 +Figure 21. Effect of the choice of the energy-entropy switch on the Zeldovich +pancake. Comparison between the switch based on kinetic and potential +energy, each with three different 𝛼-values. The switch based on potential +energy is much more stable. +The hydrostatic sphere test showed that MFM coupled to gravity +has a slightly higher numerical diffusivity compared to SPH and a +moving mesh. Thus, on could expect isolated galaxies or also the +core of galaxy clusters to be more compact and cooler in the center. +For the nifty galaxy cluster, however, we saw that there is no +difference between MFM, Arepo and modern SPH in the global +structure. Numerical diffusivity introduced by the Riemann solver +allows mixing of entropy into the core, thus decreasing the central +density compared to traditional SPH, which suppresses any mixing. +Modern SPH mimics the same effect by applying artificial conduc- +tivity, while the precise amount introduced can lead to significant +changes in the structure. As observed galaxy clusters show a wide +range of central entropy profiles (Cavagnolo et al. 2009), both re- +sults are consistent with observations. Especially, we expect a more +complex interplay with cooling, as well as stellar and AGN feed- +back to influence the entropy-evolution of the core (compare, e.g. +Pearce et al. 2000; Borgani et al. 2005; Rasia et al. 2015). These ef- +fects lead to the whole range of possible central profiles, dominating +over effects of the hydro-solver. Thus, further studies including such +processes would be necessary. +In the intra cluster medium, we expect turbulence at low Mach +number to be seeded e.g. by mergers at large scales. It will then decay +and build up a turbulent power spectrum. Such decaying, subsonic +turbulence is a very challenging problem for many hydro-methods. +MFM is able to recover the turbulent power spectrum best compared +to SPH and a moving and stationary mesh. Only a small lack of +energy at intermediate to small scales close to the resolution limit – +similar to where this occurs also for Arepo – is present. This “dip” in +energy moves to smaller scales for higher resolution, overall leading +to fast convergence towards the expected Kolmogorov spectrum. +The decay rate of turbulent energy due to numerical dissipation +is on the same order as for modern SPH, and decreases towards +higher resolution. The results are consistent down to very small initial +turbulent energy fractions 𝑋𝑖 = 0.003, corresponding to small Mach +numbers M = 0.007. For smaller 𝑋𝑖 < 0.003 numerical effects +dominate and and lead to unphysical increase in turbulent energy. +Overall, the results are very promising for the accurate evolution of +turbulence also within galaxy clusters. +An energy-entropy switch is of great importance to accurately +evolve the temperature profile for the Zeldovich pancake. When it +is included, MFM yields the best results, having a clear jump in the +temperature. Comparing different possible values for such a switch, +we found that careful tuning is required. In general, the switch based +on potential energy produces more stable results. +Arepo misses the implementation of such a switch in the public +version, such that the low-temperature region is entirely dominated +by numerical noise. SPH also shows noise in the low-temperature +region, originating from the amplification of noise present in the +ICs, and also much broader wings around the peak. All methods +show some punch-through in the temperature profile, indicating a +too low viscosity. +In addition to comparing different methods, we used two tests to +analyze the impact of the slope-limiter. Depending on the problem, +different slope-limiters can be preferred. While the gizmo limiter +performs best in most test-cases, having a much lower numerical +diffusivity, specific cases such as the hydrostatic square and also +strong shocks work better using a more diffusive TVD-limiter. The +Arepo limiter has an intermediate diffusivity and lies in between the +two other results. +Overall, our implementation of MFM produces accurate results +for the cases considered. It avoids some of the disadvantages of SPH, +while requiring an even smaller computational cost per timestep. The +total number of timesteps and thus the total runtime increases as a +result of the smaller smoothing length and effectively higher spatial +resolution. A faster, approximate Riemann solver can further decrease +the computational costs, but has the drawback of introducing more +numerical diffusivity. Compared to that, a moving mesh requires a +very expensive tessellation to be performed, such that the required +computational costs for many tests are drastically increased. +Overall, MFM is a promising alternative for cosmological simu- +lations. +5.1 Outlook – possible extensions in the future +To make use of the full advantages of OpenGadget3, it will be useful +to couple MFM not only to gravity, but also to include more physical +processes, such as cooling, star formation and stellar feedback, AGN +feedback, physical conductivity and viscosity. For these, we can make +use of already existing implementations in OpenGadget3. +Finally, MFM can be expanded to an MHD method, including +magnetic fields. This will also allow to include the existing imple- +mentation of cosmic rays. +For many of these extensions, coupling can be done in a similar +way as for SPH, while others such as magnetic fields will require +more significant changes including another Riemann solver. +In principle, also a general-relativistic (GR) extension would be +possible, which has been implemented both for SPH (Liptai & Price +2019; Rosswog & Diener 2021) and a moving mesh (Chang & Eti- +enne 2020; Lioutas et al. 2022) and also exits for MFM within the +gizmo code (Lupi 2022). As GR is mainly important in extreme sit- +MNRAS 000, 1–26 (2023) + +2 +U<αiEpot +U<α2Ekin +1 +[km/s] +α1 = +α2 = +0001 +3.10-3 +1.10-3 +X +1·10-2 +3.10-3 +2.10-2 +6.10-3 +log(plpo) +K +log(T) +0 +-25 +0 +25 +-25 +0 +25 +x [h-1Mpc] +x [h-1Mpc]22 +F. Groth et al. +uations such as accretion discs around black holes, this would also +make use of the fact that our MFM implementation is originally +based on GANDALF, which itself was designed to deal with star and +planet formation, and thus for disks. +ACKNOWLEDGEMENTS +FG and KD acknowledge support by the COMPLEX project from the +European Research Council (ERC) under the European Union’s Hori- +zon 2020 research and innovation program grant agreement ERC- +2019-AdG 882679. UPS is supported by a Flatiron Research Fel- +lowship at the Center for Computational Astrophysics (CCA) of the +Flatiron Institute. The Flatiron Institute is supported by the Simons +Foundation. FG, MV and KD acknowledges support by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foundation) un- +der Germany’s Excellence Strategy - EXC-2094 - 390783311. MV +is supported by the Alexander von Humboldt Stiftung and the Carl +Friedrich von Siemens Stiftung. We are especially grateful for the +support by M. Petkova through the Computational Center for Particle +and Astrophysics (C2PAP) under the project pn68va. Some calcu- +lations for the hydrodynamical simulations were carried out at the +Leibniz Supercomputer Center (LRZ) under the project pr86re (Su- +perCast). We thank C. Alig for a turbulence Arepo setup and L. Böss +for providing ICs for the shock-tubes. The analysis was performed +mainly in julia (Bezanson et al. 2014), including the package Gad- +getIO by Böss & Valenzuela (2022). The surface density of the nifty +cluster was calculated using Smac (Dolag et al. 2005a). We thank +the developers of gizmo and Arepo for making the codes publicly +available. +DATA AVAILABILITY +The setup for the different tests are publicly available at https: +//github.com/fgroth/hydro_tests. This includes parameter +and config files for the different codes used, as well as our analysis +routines. If applicable, routines to create ICs are also included. 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In Fig. A1, we compare the resulting density distribu- +tion at time 𝑡 = 10, evolved with the different methods. As the ICs +are set in hydrostatic equilibrium, we would expect no changes to +occur. This ideal state is only achieved using the moving mesh code +Arepo. Theoretically, we would expect the same to be true for MFM, +as shown by Hopkins (2015). They use, however, a strongly ideal- +ized setup compared to ours. Especially, they use a regular grid for +all particles, and increased particle masses within the square. For +our setup, the gradient estimate at the boundary does not conserve +linear gradients. Instead, it is biased by the in-homogeneous parti- +cle distribution due to two separate grids, especially in combination +with the slope-limiter. A more detailed analysis of the effect of the +slope limiter is provided in Sec. 4.7.1, where we have shown that +the amount of surface tension strongly depends on the slope-limiter. +We observe, using both our MFM implementation and gizmo, that +for MFM the edges of the square start to deform, followed by some +numerical instability, which leads to a more asymmetric deforma- +tion. Increasing the resolution by a factor of 4, as shown in Fig. A2, +Figure A1. Density of the hydrostatic square evolved until 𝑡 = 10 using +different methods. The initial location of the high density “square” region is +overplotted as contour. Only Arepo is able to keep the initial square shape, +while other methods lead to deformation of the square. +this instability occurs slower and the square preserves its shape much +better. Also using SPH, the square deforms. As expected, it becomes +more circular, caused by numerical errors, which behave as surface +tension (compare, e.g., Price 2008). For traditional SPH, these errors +should be low-order. We observe, however, that this effect can be +drastically reduced by increasing the resolution, as shown in Fig. A2 +indicating that modern SPH implementations, as used in OpenGad- +get3, reduce low-order errors and improve convergence. Overall, for +this specific test surface tension for SPH, but also for MFM can be +observed. A moving mesh performs best, preserving the situation +perfectly. MFM at later times shows some numerical errors lead- +ing to a more asymmetric deformation, which converge away with +increasing resolution. +APPENDIX B: SLOPE-LIMITERS IN OPENGADGET3 +We implemented seven different slope-limiters and therein variants of +their specific parameters in OpenGadget3. The main concept is de- +scribed in Sec. 4.7.1. In general, we substitute ∇W𝑖,𝑘 → 𝛼𝑖,𝑘∇W𝑖,𝑘 +for each particle 𝑖 and component 𝑘, for the face interpolation, with +𝛼𝑖,𝑘 ∈ [0, 1]. In the following, we briefly describe the implemented +limiters. +The simplest option are to use a zeroth order interpolation setting +𝛼ZERO SLOPES +𝑖,𝑘 += 0 +(B1) +or to include no slope-limiter +𝛼NULL +𝑖,𝑘 += 1. +(B2) +Alternatively, we implemented several more complex limiters. A +commonly used one is a TVD scalar limiter (Hess & Springel 2010), +which is designed to produce good results especially for strong +MNRAS 000, 1–26 (2023) + +IZMO +ST +ARHPOMFM in OpenGadget3 +25 +Figure A2. Hydrostatic Square at 𝑡 = 10. Comparison of MFM and SPH +at two different resolutions, Top: 7168 particles, Bottom: 114688 particles +(increase in resolution by factor 4. Both, MFM and SPH, show convergence +of the shape of the square. +shocks. Compared to the other limiters implemented, it is the most +diffusive one. It sets +𝛼TVD SCALAR +𝑖,𝑘 += +min +𝑗 ∈Ngb max +���� +���� +0 +min +� +1 +d𝑊𝑖 𝑗,𝑘𝑘/d𝑊𝑘 +(B3) +where dW𝑖 𝑗 = W 𝑗 − W𝑖, dW = dr𝑖 𝑗 · ∇|W. +An alternative is the scalar limiter (Balsara 2004; Gaburov & +Nitadori 2011), which looses the TVD behavior but is less diffusive. +It sets +𝛼SCALAR +𝑖,𝑘 += max +������� +������� +0 +min +����� +����� +1 +min +��� +��� +d𝑊𝑘,max +|d𝑟 |max |∇𝑊𝑘 | +d𝑊𝑘,min +|d𝑟 |max |∇𝑊𝑘 | +(B4) +where d𝑊𝑘,min/max = +��𝑊𝑖,𝑘 − min/max𝑗 ∈Ngb 𝑊 𝑗,𝑘 +��, and |d𝑟|max = +max +� +max𝑗 ∈Ngb +��𝑟𝑖 𝑗 +�� , ℎ𝑖 +� +. In contrast to the TVD limiter, only the +global neighbor distribution is considered. Thus, values calculated +from all neighbors individually for the TVD limiter are calculated in +an approximate way. Finally, we implemented the limiters used both +in the Arepo and gizmo code. In the Arepo code (Springel 2010), +the slope is limited using +𝛼Arepo +𝑖,𝑘 += +min +𝑖∈Ngb +���� +���� +d𝑊𝑘,max/d𝑊𝑘 +if d𝑊𝑘 > 0 +d𝑊𝑘,min/d𝑊𝑘 +if d𝑊𝑘 < 0 +1 +d𝑊𝑘 = 0. +(B5) +It lies in between the TVD and scalar limiter, as only the dividend is +approximated from the global neighbor distribution, while the divisor +is still calculated for all neighbors individually. +In gizmo (Hopkins 2015), a general limiter is introduced described +by +𝛼gizmo +𝑖,𝑘 += min +����� +����� +1 +𝛽𝑖 min +� +d𝑊𝑘,max +0.5ℎ𝑖 |∇𝑊𝑘 | +d𝑊𝑘,min +0.5ℎ𝑖 |∇𝑊𝑘 | . +(B6) +The parameter 𝛽 has to be 𝛽𝑖 > 0.5 to ensure second order stability. +A higher number corresponds to a more aggressive, less diffusive +and less stable limiter. We use the suggested value 𝛽 = 2 of Hop- +kins (2015), which is a compromise to reduce numerical diffusivity +while still working for very strong interacting shocks. For 𝛽 = 2, this +limiter is also similar to the scalar limiter with the difference that the +theoretically possible distance between neighbors is defined only by +the smoothing length. In addition, Hopkins (2015) provide a pairwise +limiter, acting on only one specific interaction, instead of all neigh- +bors. For this, it uses already limited slopes for the interpolation. +The pairwise limiter described by Hopkins (2015) limits the already +interpolated face values. The aim is to directly calculate the face +value 𝑊new +𝑖 𝑗,𝑘, starting from the extrapolated value 𝑊face +𝑖 𝑗,𝑘 according to +Eqn. (36), possible already with limited gradients. If 𝑊𝑖,𝑘 = 𝑊 𝑗,𝑘, +the face value is just chosen the same as the cell values 𝑊new +𝑖 𝑗,𝑘 = 𝑊𝑖,𝑘. +Otherwise, the values +𝛿1 = 𝜓1 +��𝑊𝑖,𝑘 − 𝑊𝑗,𝑘 +�� +(B7) +𝛿2 = 𝜓2 +��𝑊𝑖,𝑘 − 𝑊𝑗,𝑘 +�� +(B8) +are calculated. The free parameters 𝜓1/2 are tuned to 𝜓1 = 0.5, +𝜓2 = 0.25. A simple intermediate value used later is given by +¯𝑊𝑖 𝑗,𝑘 = 𝑊𝑖,𝑘 + d𝑟𝑖 𝑗 +d𝑟face +𝑖 +(𝑊 𝑗,𝑘 − 𝑊𝑖,𝑘). +(B9) +The +maximum/minimum +value +is +𝑊𝑘,min/max += +min/max(𝑊𝑖,𝑘, 𝑊𝑗,𝑘). Depending on how the two face val- +ues compare, the new face value is calculated: If 𝑊𝑖,𝑘 < 𝑊 𝑗,𝑘, +then +𝑊new +𝑖 𝑗,𝑘 = max +�������� +�������� +��� +��� +𝑊𝑘,min − 𝛿1 +if SIGN(𝑊𝑘,min − 𝛿1) = 𝑆𝐼𝐺𝑁(𝑊𝑘,min) +𝑊𝑘,min +1+ +𝛿1 +|𝑊𝑘,min| +else +min +� +𝑊face +𝑖 𝑗,𝑘 +¯𝑊𝑖 𝑗,𝑘 + 𝛿2. +(B10) +If 𝑊𝑖,𝑘 ≥ 𝑊 𝑗,𝑘, then +𝑊new +𝑖 𝑗,𝑘 = min +�������� +�������� +��� +��� +𝑊𝑘,max + 𝛿1 +if SIGN(𝑊𝑘,max + 𝛿1) = 𝑆𝐼𝐺𝑁(𝑊𝑘,max) +𝑊𝑘,max +1+ +𝛿1 +|𝑊𝑘,max| +else +max +� +𝑊face +𝑖 𝑗,𝑘 +¯𝑊𝑖 𝑗,𝑘 − 𝛿2. +(B11) +The same limiter is applied for particle 𝑗. Finally, the gizmo code +uses a slightly different pairwise limiter. Depending on the tolerance +𝑡 chosen, the parameters +𝜓1 = +���� +���� +0 +𝑡 = 0 +0.5 +𝑡 = 1 +0.75 +𝑡 = 2 +(B12) +𝜓2 = +���� +���� +0 +𝑡 = 0 +0.4 +𝑡 = 1 +0.375 +𝑡 = 2 +(B13) +MNRAS 000, 1–26 (2023) + +26 +F. Groth et al. +are defined. To calculate ¯𝑊𝑖 𝑗,𝑘, the factor d𝑟𝑖 𝑗/d𝑟face +𝑖 +is approximated +by the first order value 0.5. Except these differences, the limiter is +identical to the already described one. In our implementation, we +apply the limiter in the reference frame of the interface, such that the +velocity is a relative velocity. This makes the limiter Lagrangian and +increases the symmetry between different directions. +APPENDIX C: EFFECT OF THE RIEMANN SOLVER +In OpenGadget3 we use an exact, iterative Riemann solver by de- +fault. This is however, computationally expensive as up to eight it- +erations are used to get close to the exact solution. An alternative is +using approximate Riemann solvers, where we implemented a Roe +solver, the HLL solver, and the HLLC and HLLE solver. +While these are faster by up to 20 per cent for problems dominated +by hydrodynamical calculations such as the shock tube, the effect +becomes less important when using gravity and possibly even more +extensions in cosmological applications. Already for the hydrostatic +sphere, there is no significant difference in runtime. For the HLLC +solver, there is even a slight increase in runtime due to differences in +the precise evolution, making the gravity calculation more expensive. +In addition, the Riemann solver leads to changes in the precise +evolution as it introduces numerical diffusivity, visible in density +and internal energy changes for the hydrostatic sphere, shown in +Fig. C1. As discussed in Sec. 4.5.2, also the exact Riemann solver +leads to some numerical diffusivity. The change in internal energy +is however even stronger for the alternative approximate Riemann +solvers. While the HLL Riemann solver produces results close to the +exact one, it is also the most unstable one, such that a large fraction +of the calculation is actually done using the exact solver. The Roe +Riemann solver shows a slightly stronger change in the hydrostatic +density and internal energy profile, indicating a higher diffusivity, +followed by the HLLC Riemann solver. +While for specific problems these alternative solvers could lead +to faster results, we in general use the most accurate exact Riemann +solver. The increase in runtime is compensated by the gain in accu- +racy. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–26 (2023) + +MFM in OpenGadget3 +27 +Figure C1. Evolution of the density and internal energy profile for the hydrostatic sphere test case, comparing the exact, HLL, Roe and HLLC Riemann solvers. +A main difference is the amount of numerical diffusivity introduced by the different Riemann solvers. +MNRAS 000, 1–26 (2023) + +10 +8 + 10-5 +6 +MFM(Rs=exact) +MFM(Rs=HLL) +MFM(Rs=Roe) +MFM(Rs=HLLC) +10-6/ +106 +4 +2 +105[ +102 +103 +102 +103 +102 +103 +102 +103 +r +r \ No newline at end of file diff --git a/TNE2T4oBgHgl3EQfCQao/content/tmp_files/load_file.txt b/TNE2T4oBgHgl3EQfCQao/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..021518f01485f58cc30382640204df0b3cdf4f37 --- /dev/null +++ b/TNE2T4oBgHgl3EQfCQao/content/tmp_files/load_file.txt @@ -0,0 +1,1893 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf,len=1892 +page_content='MNRAS 000, 1–26 (2023) Preprint 11 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 The Cosmological Simulation Code OpenGadget3 – Implementation of Meshless Finite Mass Frederick Groth,1★ Ulrich P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Steinwandel,2 Milena Valentini,1 and Klaus Dolag1,3 1Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1, 81679 München, Germany 2Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA 3Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Subsonic turbulence plays a major role in determining properties of the intra cluster medium (ICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We introduce a new Meshless Finite Mass (MFM) implementation in OpenGadget3 and apply it to this specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end, we present a set of test cases to validate our implementation of the MFM framework in our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These include but are not limited to: the soundwave and Kepler disk as smooth situations to probe the stability, a Rayleigh-Taylor and Kelvin-Helmholtz instability as popular mixing instabilities, a blob test as more complex example including both mixing and shocks, shock tubes with various Mach numbers, a Sedov blast wave, different tests including self-gravity such as gravitational freefall, a hydrostatic sphere, the Zeldovich-pancake, and the nifty cluster as cosmological application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Advantages over SPH include increased mixing and a better convergence behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We demonstrate that the MFM-solver is robust, also in a cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We show evidence that the solver preforms extraordinarily well when applied to decaying subsonic turbulence, a problem very difficult to handle for many methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM captures the expected velocity power spectrum with high accuracy and shows a good convergence behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Using MFM or SPH within OpenGadget3 leads to a comparable decay in turbulent energy due to numerical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' When studying the energy decay for different initial turbulent energy fractions, we find that MFM performs well down to Mach numbers M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, we show how important the slope limiter and the energy-entropy switch are to control the behavior and the evolution of the fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Key words: hydrodynamics – methods: numerical – galaxies: clusters: general – turbulence 1 INTRODUCTION Turbulence plays a key role in a variety of astrophysical systems at all scales, ranging from stellar structure, star-formation in the interstellar medium (ISM) all the way up to the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It leads to enhanced small-scale mixing, and contributes to the global pressure of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While being mostly supersonic in the ISM, turbulence is mainly subsonic in the ICM (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Schuecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2004, for observations on the Coma cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A theoretical framework for sub- sonic turbulence has been provided by Kolmogorov (1941), assuming isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Simulations are an essential tool to better understand phys- ical properties of astrophysical turbulence as well as its influence on local observables such as star formation in the ISM or its contribution to heating in the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Historically, there exist different methods to solve the hydrody- namical equations in co-moving/cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hereby, one has the option to discretize the hydrodynamic equations by mass or volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The former leads to the concept of “Lagrangian” (particle based) codes and the concept of Smoothed Particle Hydrodynam- ics (SPH), and the more recent Meshless Finite Mass (MFM) and Meshless Fintie Volume (MFV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The latter gives rise to the con- ★ E-mail: fgroth@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='de cept of “Eulerian” (grid based) codes and the Godunov finite volume approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Popular SPH codes include Gadget in the different versions in- cluding Gadget-1 (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2001), Gadget-2 (Springel 2005), and Gadget-4 (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2021), Phantom (Lodato & Price 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2018) and gasoline (Wadsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2004, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM has been implemented in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' gizmo (Hopkins 2015), GAN- DALF (Hubber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2018), Gadget-3 (Steinwandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2020), and pkdgrav-3 (Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Mesh codes exist in two flavors: either as a stationary mesh, pos- sibly with adaptive mesh refinement, as implemented e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' in Zeus (Stone & Norman 1992), TVD (Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 1993, 1998), Enzo (Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 1995, 2014), FLASH (Fryxell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2000), RAMSES (Teyssier 2002), athena (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2008), and athena++ (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2020) or as a moving mesh as in Arepo (Springel 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2020) and shadowfax (Vandenbroucke & De Rijcke 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The latter have the advantage of being Pseudo-Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While mesh codes as well as MFM employ a Godunov-method and calculate fluxes between neighbors (Godunov 1959), SPH directly retrieves the hydrodynamical fluid vectors from the kernel density estimation that is obtained by adopting a weighted sum over a certain (typically non-constant) number of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All of them can be used for computations of turbulence, with earlier calculations primarily carried out in the supersonic regime, © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='03612v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='IM] 9 Jan 2023 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' relevant in the ISM for regulating star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Many results have been obtained assuming driven turbulence in which an energy input at large scales is provided during the whole simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In contrast to driven turbulence, we expect decaying turbulence to be present in galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Turbulence is injected at large scales for example due to collapse of large scale structure and subsequent merger activity (Roettiger & Burns 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2006), after which it energy is transported down to the smaller scales (“turbulent cascade”) on which it is dissipated (generally below the resolution scale of any given code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the series of papers by Federrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2008, 2009, 2010), they have used a stationary grid code to calculate turbulent boxes with driven turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' They found that the choice of the driving scheme plays an important role in determining properties of the resulting turbulence, leading to significant differences in the density statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Their results suggest a different mixture of driving-mechanisms for different star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, they found good agree- ment with observations as well as other results, independent of the driving-mechanism employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' More recently, Federrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2021) increased the resolution to even resolve the sonic scale, starting from supersonic turbulence with a resolution of ∼ 100003 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Kitsionas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2009) and Price & Federrath (2010) also compared the performance of different implementations of SPH and hydro schemes with a stationary mesh, and find good agreement between these two methods at high Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Mesh codes are more efficient to obtain volumetric statistics such as the power spectrum, while SPH recovers the high-density tail better due to automatically adapting the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While all these methods work well in the supersonic turbulent regime, they have problems dealing with subsonic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Going to smaller Mach numbers (M) Padoan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2007) showed that SPH performs sub-optimum when compared to finite volume methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Based on this work, Bauer & Springel (2012) studied the capabilities of SPH for subsonic turbulence at M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' They found that clas- sic (vanilla) SPH fails in reproducing the expected velocity power spectrum as well as the dissipation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Reasons are mainly the artificial viscosity scheme used and velocity noise introduced by the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These results raised the general question of whether SPH can deal with subsonic turbulence to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An answer has been provided by Price (2012) who showed that these limitations are not intrinsic to SPH, but rather a consequence of some SPH setups adopted to study subsonic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In contrast to what previous studies reported, SPH can capture the expected power spectrum by using more modern formulations of SPH that are able to reduce artificial viscosity in subsonic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The role of subsonic turbulence in galaxy clusters has been ana- lyzed both from observational and theoretical perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Simula- tions of turbulence in the ICM have been carried out mostly using grid codes (Vazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2009, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Iapichino & Niemeyer 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Iapichino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Miniati (2014, 2015) found a lack of turbulent energy at small scales depending on the refinement technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, they discussed the importance of microphysics for the evolution of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A possible improve- ment for modeling turbulence has been presented by Maier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2009) combining AMR with large eddy simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Simulations by Dolag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2005b) have shown that also SPH can model turbu- lence in galaxy clusters when properly reducing artificial viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition to the impact on gas dynamics, turbulence is responsi- ble for amplifying magnetic fields through a turbulent dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Sim- ulations by Schekochihin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2001, 2004) and Steinwandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2021) have focused on this turbulent dynamo, analyzing its growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Other work of Kritsuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2020) has focused again on turbulent boxes with stochastic forcing, comparing different hydrodynamical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' More recently, Sayers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2021) have compared simulated clus- ters to observed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Especially, there should be a difference de- pending on the dynamical state, with more relaxed clusters showing less turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Simulations, however, do not always find such a difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, it is important to accurately capture the turbulent cascade and the decay in turbulent energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the latter would re- quire including additional microphysics such as viscosity, the former also depends on the hydro-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use Meshless Finite Mass (MFM) as an alternative, newer method to the aforementioned ones to study subsonic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM combines ideas of SPH with those of a moving mesh and thus aims solving several of their individual issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The develop- ment of MFM goes back to Godunov SPH (Inutsuka 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Cha & Whitworth 2003), which was still unstable, and to a Meshless Fi- nite Element Method suggested by Idelsohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2003), until the nowadays used version first formulated by Lanson & Vila (2008a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We present a new implementation in the Gadget derivative Open- Gadget3, originally based on the implementation in the code GAN- DALF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Several extensions allow its use in cosmological simulations compared to the implementation in GANDALF that is focused on star and planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This allows for a stable baseline frame- work for applications on scales of star and planet formation that we extend into the cosmological integration framework of OpenGad- get3, which is a re-base of Gadget-2 with the ability to be compiled with C++ compilers, and making vast use of templating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It comes with modules containing state-of-the-art physics and sub-resolution models, as for instance: self-interacting dark matter (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022), MHD (Dolag & Stasyszyn 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Stasyszyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2013), ther- mal conduction (Arth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2014), cosmic rays (Böss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022), star formation and stellar/blackhole feedback according to the Mag- neticum-model (Springel & Hernquist 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Tornatore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2003, 2004, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hirschmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Steinborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Dolag 2015) or with the MUPPI (MUlti Phase Particle Integrator) exten- sion for non-equilibrium star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (Murante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2010, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To make use of modern computer architectures, it includes a hybrid MPI-OpenMP parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, calculations of gravity, SPH density, hydro-force and thermal conduction, can be carried out on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These modules requiring most of the runtime (Ragagnin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2020) GPU offloading can be useful for some applications, leading to a speed up by a factor of a few (2-4, depending on the exact application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The long-term goal is to have a fully publicly available updated Gadget version for OpenMP and OpenACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Before the introduction of this paper the code was solving the hydrodynamical equations using modern SPH as formulated by Springel & Hernquist (2002), including modern, time-dependent arti- ficial viscosity (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2016a) and conduction (Price 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' With the new implementation of MFM as a modern meshless method, we can combine both advantages of this method and efforts previously made to optimize the pre-existing code base that also involves a treat- ment in order to evolve strong shocks for which we need the timestep limiter to be non-local which is ensured by a wakeup scheme (Saitoh & Makino 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Pakmor 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 closely follows the implementation described by Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A main goal of this paper is to use Meshless Finite Mass to study decaying, subsonic turbulence, as present in galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end, we present a new implementation in the cosmological simulation code OpenGadget3 as an alternative hydro-solver to the currently implemented SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This paper is structuered as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We first describe the code- MNRAS 000, 1–26 (2023) MFM in OpenGadget3 3 base of OpenGadget3 including its SPH implementation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We continue with a brief overview on MFM and a description of our MFM implementation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4, we use a suite of test cases, each probing specific aspects and properties of the code, to validate the performance of our MFM implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All settings are kept exactly the same between test cases, independent of the individual test case, without further tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We continue with an analysis of de- caying subsonic turbulence with our new implementation presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In all cases, comparisons between different codes and methods are provided, including MFM and SPH in OpenGadget3, MFM in gizmo and a moving and stationary mesh in the publicly available Arepo version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We analyze the effect of specific numerical parameters in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Our main findings are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Additional material such as the hydrostatic square as additional test case, the formulation of the slope-limiters and a comparison of the Riemann solvers implemented are presented in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A, B, and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2 OPENGADGET3 – NUMERICAL METHOD Solving the system of differential equations describing the evolution of the gas, as written in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (12), requires discretizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the temporal dimension a sufficiently small timestep Δ𝑡 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The spatial discretisation can be obtained using various different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In OpenGadget3, hydrodynamics is discretized either using Smoothed Particle Hydrodynamics (SPH) or with the newly implemented Meshless Finite Mass (MFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gravity is solved by a TreePM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Integrator and Timestepping For the time integration, we employ a Leapfrog scheme in kick- drift-kick (KDK) form to achieve second order accuracy (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', Hernquist & Katz 1989) in the implementation following Verlet (1967);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Springel (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Starting from values at timestep number 𝑛, velocities v are updated in a first half-step kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It is followed by drifting the positions r, and another, second half-step kick: v𝑛+1/2 = v𝑛 + 1 2a𝑛Δ𝑡 (1) r𝑛+1 = r𝑛 + v𝑛+1/2Δ𝑡 (2) v𝑛+1 = v𝑛+1/2 + 1 2a𝑛+1Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (3) The acceleration a = ahydro+agrav consists of hydrodynamical accel- erations ahydro and gravitational accelerations agrav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Following the operator splitting approach, they are calculated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gravity is evaluated before the drift, and hydrodnamical accelerations between the drift and the second half-kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 uses hierarchical timestepping to ensure synchro- nization, while allowing adaptive timesteps, depending on different timestep limiters such as a Courant-like timestep criterion Δ𝑡Courant 𝑖 = 𝐶Courant𝑎ℎ𝑖 𝑐max (4) with maximum signal velocity 𝑐max, scale factor 𝑎, smoothing length ℎ𝑖, and free parameter 𝐶Courant, as described by Springel (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Gravity Solver – TreePM The accurate treatment of gravity is of great importance for cosmo- logical simulations (Springel 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In principle, it can be solved accurately by a direct summation, which is, however, computation- ally expensive (O(𝑁2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Instead, we follow the much more efficient combined Oct-Tree-Particle Mesh (PM) approach (Xu 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Bode et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Springel 2005, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 mainly follows the implementation in Gadget-2, which has been ex- tensively described by Springel (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the following, we briefly review the main concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The potential is split into short-range and long-range contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Short-range forces are calculated following the oct-tree algorithm, while long-range forces are calculated using a particle mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The idea of a tree algorithm has been proposed by Appel (1985) and Barnes & Hut (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Nodes of an oct-tree are constructed by splitting the domain into a sequence of cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Force-contributions of nodes satisfying an opening angle criterion are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For numerical reasons to keep the equation linear with respect to adding and removing particles from nodes, only the monopole contributions are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The implementation in Gadget has been described by Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The total gravitational acceleration of particle 𝑖 from other nodes/particles 𝑗 with mass 𝑚 𝑗 at location r𝑖 𝑗 relative to particle 𝑖 and with (gravita- tional) softening length 𝜖 𝑗 is given by agrav,𝑖 = 𝐺 𝑁tot ∑︁ 𝑗 r𝑖 𝑗 ��� ��� 𝑚𝑗 𝑟3 𝑖 𝑗 if 𝑟𝑖 𝑗 > 𝜖 𝑗 𝑚𝑗 𝜖 3 𝑗 Corr(𝑟𝑖 𝑗/𝜖) if 𝑟𝑖 𝑗 ≤ 𝜖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (5) Corr is a correction term, taking into account the softening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 𝐺 is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the particle mesh (Eastwood & Hockney 1974), all particles are assigned to grid-cells, such that a discrete Fourier-transformation can be calculated, with the gravitational po- tential Φ𝑘 in Fourier space at wavenumber 𝑘 being calculated as −𝑘2Φ𝑘 = 4𝜋𝐺𝜌𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (6) Corrections for small-range truncation as well as periodic boundaries are applied by multiplications in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The gravitational potential in real space is calculated as inverse Fourier-transform, and is interpolated to the original particle positions to finally obtain gravitational accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 uses the more modern FFTW3 (“Fastest Fourier Transform in the West”) library (Frigo & Johnson 2005) instead of FFTW2 for the implementation of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 Hydrodynamical Solver – SPH For Smoothed Particle Hydrodynamics (SPH), the domain is decom- posed into a finite number of “particles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The physical quantities at each point are represented by contributions of close-by (neigh- boring) particles weighted by a kernel 𝑊𝑖(𝑟𝑖, ℎ𝑖), depending on the distance 𝑟𝑖 from particle 𝑖, and its smoothing length ℎ𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The ker- nel has to be continuous, radially symmetric, have compact support and fulfill the limit limℎ→0 𝑊 = 𝛿, but otherwise can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 offers the choice between different com- monly used kernels, including a cubic spline (Monaghan & Lattanzio 1985), quintic spline (Morris 1996), or a Wendland C2/C4/C6 kernel (Wendland 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Dehnen & Aly 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The effective volume of each particle is well approximated by 𝑉−1 𝑖 = 𝑊(𝑟𝑖), such that the density follows as 𝜌(r𝑖) = ∑︁ 𝑗 ∈Ngb 𝑚 𝑗𝑊 ���r𝑖 − r 𝑗 �� , ℎ𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (7) We allow for adaptive smoothing, automatically increasing resolution in high-density regions compared to low-density ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Smoothing MNRAS 000, 1–26 (2023) 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' length and neighbor number are related to the density via: 4𝜋 3 𝜌𝑖ℎ3 𝑖 = ¯𝑚𝑁Ngb (8) with mean neighbor mass ¯𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (7) and (8) are coupled for fixed neighbor number, one solves for smoothing length and density iteratively via finding roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Quantities other than the density, labeled with 𝑋, are approximated via 𝑋(r0) ≈ ∑︁ 𝑖∈Ngb 𝑋𝑖 𝜌𝑖 𝑊(|r0 − r𝑖| , ℎ)𝑚𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (9) Different formulations of the hydrodynamical acceleration can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In OpenGadget3 the fully conservative formulation for the hydrodynamical acceleration (Springel & Hernquist 2002) ahydro,𝑖 = − ∑︁ 𝑗 ∈Ngb 𝑚 𝑗 � 𝑓𝑖 𝑃𝑖 𝜌2 𝑖 ∇𝑖𝑊𝑖 𝑗 (ℎ𝑖) + 𝑓 𝑗 𝑃 𝑗 𝜌2 𝑗 ∇𝑖𝑊𝑖 𝑗 (ℎ 𝑗) � , (10) 𝑓𝑖 = � 1 + ℎ𝑖 3𝜌𝑖 𝜕𝜌𝑖 𝜕ℎ𝑖 �−1 (11) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Instead of calculating gradients of physical quantities, all spatial derivatives are expressed by gradients of the kernel func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Traditional SPH has problems dealing with shocks, as well as reproducing mixing instabilities (Morris 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Agertz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These issues can be resolved by including artificial viscosity and conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In OpenGadget3, time and spatial dependent artificial viscosity (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2016a) and artificial conductivity (Price 2008) are utilized, minimizing their impact in regions where they are not desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3 MESHLESS FINITE MASS As a second, newly implemented option, the hydrodynamical equa- tions can be discretized and solved following the Meshless Finite Mass (MFM) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This method conceptually combines SPH with a moving mesh, calculating fluxes between neighboring cells in a scheme otherwise similar to SPH, including weighting by a kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, it is combining advantages of both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In contrast to Godunov SPH, the domain associated to a particle is not spherical due to the kernel weighting, but the particle interfaces in the flux calculation are subject to the weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Basic Hydrodynamical Equations The evolution of any ideal fluid is described by three main equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Mass conservation leads to the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The second equation is an equation of motion (Eulers equation), corresponding to Newton’s second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Energy conservation is ensured by the first law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Within an inertial frame of reference, all these equations can be combined into dU d𝑡 + ∇ · (F − vframe|U) = S (12) with outer product | and, for pure hydrodynamics, field vector U = (𝜌, 𝜌v, 𝜌𝑒), flux F = (𝜌v, 𝜌vv𝑇 + 𝑃1, (𝜌𝑒 + 𝑃) v) and source S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In total, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (12) provides 5 constraints for 6 variables: fluid density 𝜌, energy density 𝑒, pressure 𝑃, and the three components of the velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The missing constraint is provided by an equation of state, connecting the pressure to the internal energy density 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For an ideal gas it takes the form 𝑃 = (𝛾 − 1) 𝜌𝑢 (13) where the adiabatic index 𝛾 amounts to 5/3 if the gas is monoatomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Equations in an Expanding Universe In a cosmological context, the expansion of the universe has to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' One possibility is to re-write Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (12) for a universe with scale factor 𝑎, accounting for these effects, as realized e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' in Gadget-1: 𝜕v 𝜕𝑡 + 1 𝑎 (v · ∇) v + �𝑎 𝑎 v = − 1 𝑎𝜌 ∇𝑃 − 1 𝑎 ∇Φ, (14) 𝜕𝜌 𝜕𝑡 + 3 �𝑎 𝑎 𝜌 + 1 𝑎 ∇ · (𝜌v) = 0, (15) 𝜕 𝜕𝑡 (𝜌𝑢) + 1 𝑎 �𝑣 · ∇ (𝜌𝑢) = − (𝜌𝑢 + 𝑃) � 1 𝑎 ∇ · v + 3 �𝑎 𝑎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (16) In OpenGadget3 we follow a different approach, and do calcula- tions using the so called super-co-moving coordinates, as first intro- duced by Martel & Shapiro (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Code units (denoted by subscript 𝑐) are related to physical units (𝑝) via 𝑥𝑐 = 𝑎𝑥𝑝 (17) 𝜌𝑐 = 𝑎3𝜌𝑝 (18) 𝑣𝑐 = 𝑎𝑣 𝑝 (19) 𝑝𝑐 = 𝑎3𝑝 𝑝 (20) 𝑢𝑐 = 𝑢𝑝, (21) such that Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (12) keeps the same form when written in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 MFM Discretization Mathematically, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (12) is discretized by multiplying by a partition function 𝜓𝑖 = 1 � 𝑗 ∈Ngb 𝑊 𝑗 𝑊𝑖 (22) and integrating over the volume, such that for every particle 𝑖 changes in the quantities U𝑖 = (𝜌𝑖, 𝜌𝑖𝑣𝑖, 𝑒𝑖), with 𝑒 being the total energy density, are given by source terms S𝑖 = 0, which vanish for pure hydrodynamics, and pairwise fluxes F𝑖 𝑗 with the neighbors 𝑗 via d d𝑡 (𝑉𝑖U𝑖) + ∑︁ 𝑗 ∈Ngb � F𝑖 𝑗 · Aeff 𝑖 𝑗 � = S𝑖𝑉𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (23) Calculating pairwise fluxes automatically ensures mass, momentum and energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The effective interface area Aeff 𝑖 𝑗 depends on the partition function and effective volume 𝑉𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' which itself depends on the integrated partition function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' translating to the number density 𝑛𝑖: 𝑉𝑖 = ∫ 𝜓𝑖 = 𝑛−1 𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (24) Aeff 𝑖 𝑗 = 𝑉𝑖 ˜𝜓 𝑗 − 𝑉𝑗 ˜𝜓𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (25) where ˜𝜓𝛼 𝑗 (x𝑖) = ∑︁ 𝑖∈Ngb 𝐵𝛼𝛽 𝑖 (𝑥 𝑗 − 𝑥𝑖)𝛽𝜓 𝑗 (𝑥𝑖) (26) B𝑖 = E−1 𝑖 (27) 𝐸 𝛼𝛽 𝑖 = ∑︁ 𝑗 ∈Ngb (x𝑗 − x𝑖)𝛼(x𝑗 − x𝑖)𝛽𝜓 𝑗 (𝑥𝑖) (28) MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 1–26 (2023) MFM in OpenGadget3 5 with Einstein summation convention over 𝛽 in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The matrix B is chosen in order to be second order accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A more detailed derivation has been provided by Gaburov & Nitadori (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Most importantly, the tessellation does not have to be calculated explicitly, but an SPH-like neighbor search is used, drastically re- ducing the computational costs compared to a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This has the drawback that the face area is not well defined, but has to be calculated in an approximate way using the neighbors according to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As MFM is typically used with 32 neighbors in 3d (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gaburov & Nitadori 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hopkins 2015) this leads to particles being treated as neighbors that would not be considered for a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A possible improvement would be to only use the nearest neighbors, constructed in an approximate way (compare e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Błaszczyszyn & Schott 2003), to get closer to what a mesh reconstruction would do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In contrast to SPH, for which the mass density is estimated accord- ing to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (8), for MFM the number density 𝑛𝑖 is estimated together with the smoothing length in an iterative process, solving 𝑛(r𝑖) = ∑︁ 𝑗 ∈Ngb 𝑊 ���r𝑖 − r 𝑗 �� , ℎ𝑖 � , (29) 4𝜋 3 𝑛𝑖ℎ3 𝑖 = 𝑁Ngb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (30) The flux in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (23) is calculated numerically using a Riemann solver, where we use an exact Riemann solver, following the im- plementation by Toro (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Alternatively, we implemented the Riemann-solver that provides an exact solution to the linearized sys- tem of equations (Roe-solver, Roe 1981), as well as the two most common flavors of a Harten-Lax-van-Leer solver (HLL) and HLLC (Toro 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For all these, the exact Riemann solver is used as fall- back in case the faster, approximate solver fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The effect of the choice of the solver is discussed in greatre detail in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' By choosing the reference frame corresponding to the rest-frame of the interface, the scheme becomes Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In MFM, also the boundaries are assumed to deform in a Lagrangian way, eliminating mass fluxes between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As the actual deformation does not exactly correspond to the one assumed during a timestep, second or- der errors are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An alternative is allowing for mass fluxes using the Meshless Finite Volume (MFV) method, which, however, also is only second order accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, it has been shown that MFV can run into problems by draining the mass for particles ac- celerated into low density environments in cosmological simulations (Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this reason, we do not use this scheme here but focus on the MFM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An additional advantage over SPH is that no additional dissipation terms are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Riemann solver requires knowledge about velocity, density and pressure values at the interfaces, summarized in the primitive fluid vector W = �� � 𝜌 v 𝑝 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (31) In principle, values at the cell center can be used directly, following a zeroth order interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This method can lead to strong jumps, unphysical oscillations, and numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end, we follow a two-step approach, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 1, similar to what is usually done for grid-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In a first step, gradients of the primi- tive fluid vector are calculated using a second-order accurate matrix gradient estimator (∇|W)𝛼 𝑖 = ∑︁ 𝑗 ∈𝑁 𝑔𝑏 �W 𝑗 − W𝑖 � ˜𝜓𝛼 𝑗 (x𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (32) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Sketch of extrapolation from central cell values to face values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Using the central values corresponds to a zeroth order scheme (black solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It can be extended to be second order by extrapolating using a slope defined by neighboring cells (blue dashed line), which however can lead to over- /undershooting at the faces (see left face) or even negative densities/pressures (see right face).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This issue can be solved by limiting the slopes using different procedures (red dash-dot line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' See text for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The position and velocity of the face is estimated via drface 𝑖 = dr𝑖 𝑗𝑠𝑖, (33) vface 𝑖 𝑗 = 𝑠 𝑗v 𝑗 + 𝑠𝑖v𝑖, (34) where we set 𝑠𝑖 = ℎ𝑖 ℎ𝑖 + ℎ 𝑗 (35) to be second order accurate instead of 𝑠𝑖 = 1/2 for a first-order accurate interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The face-values are extrapolated according to Wface 𝑖 = W𝑖 + drface 𝑖 ∇|W𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (36) To avoid over- or undershooting or even unphysical, negative densities or pressures when strong gradients are present in the fluid, these gradients are reduced by a factor ∇𝑊𝑖 → 𝛼𝑖∇𝑊𝑖, 0 ≤ 𝛼𝑖 ≤ 1 in the face interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We implement different options for such a slope- limiter, including a total variation diminishing (TVD) one (Hess & Springel 2010), the one from Arepo (Springel 2010) and the one used in the gizmo code (Hopkins 2015), described further in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, the pairwise limiter according to the gizmo code can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 Energy-Entropy Switch While the Riemann solver outputs total energy changes, the rest of the code requires internal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The total energy change can straightforwardly be converted into internal energy change via d𝑈 d𝑡 = d𝐸tot d𝑡 − � v + 1 2dv � dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (37) We introduce the additional term 1 2d𝑣 in the bracket, which is a second order correction and improves the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While this trans- formation does not conserve total energy to machine-precision, it increases the precision in the evolution of the internal energy itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For very cold flows, the internal energy evolution is still dominated by numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This is avoided by assuming purely adiabatic changes in these rare cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We follow the idea of the implementation in the gizmo code, where the switch is only active for specific test problems such as the Zeldovich pancake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' If active, internal energy 𝑈 = 𝑈𝑖 + d𝑈𝑖 (38) MNRAS 000, 1–26 (2023) zero slopes : extrapolated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='. limited slopes x6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' is compared to potential and/or kinetic energy 𝐸pot = 𝑚𝑖𝑎grav · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ℎ𝑖, (39) 𝐸kin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5𝑚𝑖 max 𝑗 ∈Ngb �v 𝑗 − v𝑖 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (40) If the internal energy is small enough compared to other energy contributions 𝑈 < 𝛼1𝐸pot + 𝛼2𝐸kin (41) in physical units, the new internal energy is instead calculated assum- ing adiabatic expansion or contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The parameters 𝛼1/2 have to be tuned to only affect the evolution of particles where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We provide a comparison between different values in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 Switching between SPH and MFM in OpenGadget3 To substitute SPH with MFM, the general code-structure does not have to be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Mainly, the SPH specific force calculation has to be replaced by the three steps of the MFM calculation, consisting of gradient calculations, slope-limiting and the actual flux calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As the Riemann solver both requires and outputs physical quantities, while the rest of the code deals with code units, these units have to be converted according to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (17) to (21) just before the flux calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At all places, where results of that calculation, includ- ing the hydrodynamical acceleration, are used, they first have to be converted back to physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Also, MFM calculates internal energy changes following the out- put of the Riemann solver, while in SPH the entropy is evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 Differences to previous implementations of MFM While the general concept of MFM with respect to the implementa- tions introduced in gizmo and GANDALF stays the same, there are several differences compared to these previously made implementa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Our implementation is based on the one in GANDALF, which is originally intended to be well suited for star and planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We expand this implementation by including co-moving integration and other extensions such as an energy-entropy switch to be used for cosmological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, we change the time in- tegration scheme from a second-order accurate MUSCL-Hancock to a second-order accurate Leapfrog KDK, consistent with SPH in OpenGadget3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The main difference of OpenGadget3 compared to gizmo is that fluxes are by default calculated using an iterative, exact Riemann solver compared to an approximate HLLC Riemann solver used in gizmo, with an exact Riemann solver only used as fallback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, there are a few minor differences such as the second- order correction in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (37) and making the pairwise limiter La- grangian, as described in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The convergence of the density calculation is slightly different between the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We follow the same implementation as for SPH in OpenGadget3, just replacing the mass density by the number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, our implementation employs a hybrid MPI-OpenMP parallelization as done for other modules of OpenGadget3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4 TEST CASES We use several test cases to probe the ability of the different hydro- methods to accurately follow gas evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All of them explore specific numerical aspects important for cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use these tests to compare our new MFM implementation in OpenGadget3 to SPH in OpenGadget3, MFM in the public gizmo1 version and the publicly available version of the moving mesh code Arepo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Settings We aim for a fair comparison of the different codes throughout the paper but adopt a general setting for slope limiters, Riemann-solvers (MFM) as well as the artificial diffusion terms (SPH) that one would adopt in cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While this leads to overall good performance of all solvers on almost all test cases, there are a few test problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' the square test in Sec A) for which this is not working ideal and we will discuss this in detail in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' If not otherwise mentioned, we assume an ideal gas with 𝛾 = 5/3 and all code operate on adaptive time steps for all tests (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' we never force a small constant time step to improve the accuracy of the results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM is used with a cubic spline kernel and 32 (24) neighbors in 3d (2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The slope limiter from gizmo in combination with their pairwise limiter is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Consistent settings are chosen between OpenGad- get3 and gizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH, a Wendland C6 kernel, including bias correction (Dehnen & Aly 2012), with 295 (64) neighbors in 3d (2d) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The modern, time-dependent artificial viscosity scheme of Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016a) and artificial conductivity (Price 2008) are in- cluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For Arepo we use additional mesh regularization based on the center of mass, and the “roundness” of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An overview of all settings is made publicly available3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' If not otherwise stated, the initial conditions (ICs) are created with equal particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In most cases, particles are arranged in a (perturbed) regular grid in order to reduce noise introduced by the initial particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Stability 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Soundwave As a first test we adopt a sinusodial soundwave with density 𝜌 = 1 and small perturbation amplitude Δ𝜌 = 10−4 in a box of length 1 in 𝑥-direction and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='75 in 𝑦/𝑧 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The particles are arranged in a perturbed hexagonal close packed (hpc) grid with varying resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The number of particles is ranging from 643·0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='752 up to 1283·0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the following, we will define the resolution by the number of particles per unit-length in 𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We adopt a wavenumber 𝑘 = 2𝜋 and a speed of sound of 𝑐𝑠 = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this test there is an analytic solution 𝜌(𝑥, 𝑡) = 𝜌0 +Δ𝜌 sin(𝑘(𝑥 +𝑐𝑠𝑡)), which makes this test well suited to perform a convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this purpose, we measure the L1 error norm 1 𝑁 �𝑁tot 𝑖 |𝜌𝑖 − 𝜌(𝑥, 𝑡)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods are able to evolve the soundwave, while the accuracy as well as the precise convergence behavior differ among the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We observe only first order convergence for resolutions > 64 for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In order to get a more detailed analysis, we split the error between errors in the position, in the amplitude, and scatter quantified by an L1-error as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Deviations from the expected sound speed are related to disper- sion errors, and will lead to an offset compared to the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This offset error is shown in the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We observe for MFM in both implementation the convergence to be between 1 Obtained from https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='org/phopkins/gizmo-public/ src/master/ February 2021 2 Obtained from https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='mpcdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='de/vrs/arepo June 2021 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='com/fgroth/hydro_tests MNRAS 000, 1–26 (2023) MFM in OpenGadget3 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Offset-, amplitude- and scatter-errors of the density of a soundwave at 𝑡 = 2 𝑐𝑠 calculated with MFM and SPH in OpenGadget3, MFM in gizmo and a moving mesh in Arepo at different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The scatter converges second order for all methods, while other errors show different convergence behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM shows between first and second order convergence for all error-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' first and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Both, our implementation and the one in the gizmo code, are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH and Arepo, the overall error is roughly one order of magnitude smaller at the lowest resolution, but having a convergence even worse than first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH, this trend can be explained by low-order errors, which are prominent for traditional SPH, and still partly left for modern SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The error in the amplitude, shown in the middle panel, is related to numerical diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As we see also in other tests, the Riemannn solver and the slope-limiter introduce numerical diffusivity for MFM, which thus has the largest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Differences between the different MFM im- plementations can be explained by different Riemann solvers used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH and Arepo show much lower errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The convergence behavior, however, is again better for MFM compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In both implementations, it is roughly second order, while for the other methods it appears to be approximately first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, it is worth to note that the resulting soundwave does not have perfect sinusodial shape but shows scatter in the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This is mainly a result of the smoothing length/density iteration and the threshold chosen for the value to be taken as converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We quantify this error by the L1 error norm, shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods show roughly second order convergence, while the amplitude of the error is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Differences between MFM and SPH in OpenGad- get3 can be explained by the different kernel used, while other codes have differences in the iteration and treat parameters for convergence slightly differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The large error for Arepo, even at higher resolu- tion makes the values for the other errors more uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition to the errors already mentioned, the soundwave deforms and steepens up due to non-linear terms in the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This non-linearity will lead to an additional, small but constant term in the scatter error in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A reduction could be achieved by reducing the amplitude, which would also make scatter errors be more significant or the convergence more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The importance of non-linear contributions increases with lower scatter, it dominates at the highest resolutions considered, such that the convergence behavior appears slightly worse for the other errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Kepler Disk The Kepler disk is an important test case for cosmological simula- tions, allowing to study the ability of the code to conserve angular momentum and maintain stable orbits over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Especially, the ef- fect of viscosity can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end we initialize a two- dimensional box sufficiently large to contain all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The ICs are taken from Hopkins (2015) and are initialized with 48240 gas particles with equal masses, arranged in a grid-like structure and setup with vanishing pressure of 𝑃 = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The gas surface density distribution is given via: Σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='01 + ���� ���� (𝑟/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5)3 if 𝑟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 1 if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 ≤ 𝑟 ≤ 2 (1 + (𝑟 − 2)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1)−3 if 2 < 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (42) For the Arepo run, we adopt a low density mesh with vanishing pressure at resolution 16 distributed around the disk as well as inside the central hole of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We adopt an external potential Φ = −(𝑟2 + 𝜖2)−1/2 with resulting MNRAS 000, 1–26 (2023) convergence MFM 1st order + SPH 102 2nd order GIZMO + AREPO offset error 10-3 10~ error amplitude 10 L-01 10 32 64 128 resolution8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Evolution of the Kepler disk using different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Surface density at two times per method: 𝑡 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 (upper left) and 𝑡 = 120 (lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In general, all methods are able to evolve a stable disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Initial per- turbation introduced by the ICs, however, evolve differently for the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' gravitational acceleration of the form g = − r ��������� ��������� � (𝑟/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='35)2 (r2)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 − (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='35−𝑟)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='35 (r2)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 � if 𝑟 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='35 � 1 (r2)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 � if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='35 < 𝑟 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 � 1+(𝑟−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 (r2)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 � if 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 ≤ 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (43) We follow the evolution of the disk until 𝑡 = 120, corresponding to ≈ 20 orbits at 𝑟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The resulting density at 𝑡 = 120 and 𝑡 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Initially, all methods produce spirals as a result of perturbations in the ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for more traditional SPH with Balsara viscosity switch (Balsara 1998) these lead to a destruction of the disk after only a few orbits, consistent with the results of Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016a), the modern SPH implementation in OpenGadget3 with the improved viscosity scheme of Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016a) drastically increases the stability of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the inner and outer region still show some decay, the main part of the disk is stable for the whole evolution considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For MFM the disk remains stable for more than 20 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We observe that the inner and outer parts of the disk degrade much less compared to SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The initial perturbations are diffused throughout the disk, which shows slightly larger perturbations in the main part compared to the SPH calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Both, our implementation and the one in gizmo, show qualitatively similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Arepo run turns out to produce the most stable disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Only a slight degeneration at the boundaries can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Further studies would be needed to analyze whether this is a numerical effect or due to interaction with the ambient medium not present in the other calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 Tests for Fluid Mixing instabilities Mixing occurs in a variety of cosmological situations, most promi- nently during ram-pressure-stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end, we analyze the ability of the different codes and methods to evolve such mixing instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Rayleigh-Taylor Instability One popular fluid-mixing test is the Rayleigh-Taylor instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It can be used to explore how well the code can describe unstable, growing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The setup we use is taken from Hopkins (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The calculations are preformed in a two-dimensional periodic box with side-lengths 1, where the particles at 𝑦 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 and 𝑦 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='9 are fixed as boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A fluid of high density (𝜌 = 2) is placed on top of a low-density medium (𝜌 = 1) in hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this test-case, we take 𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4, as for a diatomic gas, such as molecular hydrogen and apply the constant gravitational acceleration: agrav = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ˆ𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (44) To allow the instability to grow, a small velocity perturbation at the phase boundary is introduced (for more details see Hopkins 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4 we show that all methods are perfectly able to evolve the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A major difference between the different methods is the presence of asymmetries and secondary instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While these can be seen clearly for MFM, both in OpenGadget3 and gizmo, and are also present in the Arepo calculation where they appear more symmetric, we find that they are absent from the SPH calculation, due to the over smoothing over the larger kernel and the effectively lower spatial resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Marin-Gilabert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022, for a more detailed discussion of the occurance of secondary insatbilties and their physical meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The results of Arepo indicate the sharpest boundary and highest density in the tip, followed by MFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The boundary particles for Arepo, show still a clear imprint of the initial grid-like particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We note that the numerical diffusivity within modern SPH causes the boundary of the instability to have a shallower gradient and smears out initial asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, the effective spatial resolution is lower by a factor of ≈ 2 compared to MFM due to the larger neighbor number and thus SPH reaches a much lower density in the tip of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Kelvin-Helmholtz Instability Similar to the Rayleigh-Taylor instability, also the Kelvin-Helmholtz instability is a famous example for fluid mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Again, we use the setup provided by Hopkins (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Two fluids of densities 𝜌1 = 1 and 𝜌2 = 2 in hydrostatic equilibrium are initialized in a 2d periodic box, with initial velocities v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ˆ𝑥, v2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ˆ𝑥 and a small perturbation following McNally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At time 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 corresponding to ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2𝜏KH in units of the Kelvin-Helmholtz timescale 𝜏KH, the instability has produced a roll for all methods, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Differences are present in the inner structure of the roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall the qualitative results are very similar to those for the Rayleigh-Taylor instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH is smoothing the roll, showing no secondary insta- bilities and evolving more smoothly towards later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to that, MFM in both implementations shows a clear separation be- tween the higher-density roll and the less dense medium, with the presence of secondary instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A more detailed analysis of the Kelvin-Helmholtz instability, also using our new MFM implementa- tion, has been done by Marin-Gilabert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' They also show that the secondary instabilities can be avoided by using a higher MNRAS 000, 1–26 (2023) 0 1 MFM 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 GIZMC 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 t= y0 1 SPH 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 AREPC 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 1 0 1 1 0 1 x xMFM in OpenGadget3 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Rayleigh-Taylor instability at time 𝑡 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparison between the different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Vertical line marks the initial position of the phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Differences are mainly the presence or absence of secondary instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' neighbor number in combination with a higher-order kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This will increase the intrinsic viscosity and prevent mixing in form of secondary instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Also Arepo shows secondary instabilities, present especially inside the roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' When present, these perturbations will finally dominate the evolution over the build-up of the roll for 𝑡 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 The “Blob” test A more complex problem is the blob test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It is designed to mimic ram-pressure stripping by an interplay of the evolution of shocks and fluid-mixing instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use the setup described by Hopkins (2015) (compare also Agertz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A cloud of higher density 𝜌cloud = 10𝜌wind is placed into a wind tunnel with supersonic flow at M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7 and density 𝜌wind = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 · 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Both phases are setup in pressure equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The resulting density in a slice through the cloud at 𝑡 = 𝜏𝐾 𝐻 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In front of the cloud, a bow shock forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At the MNRAS 000, 1–26 (2023) MFM GIZMO N SPH AREPO n10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Build-up of a 2d Kelvin-Helmholtz instability at 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 comparing different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Horizontal dashed lines mark the initial position of the phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods produce the roll, but with differences in their inner structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Kelvin-Helmholtz timescale 𝜏KH = 2, the cloud has developed in- stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These are much more pronounced for MFM and Arepo, while for SPH the cloud deforms, without showing instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The precise form of the cloud differs between our MFM implementation, that in gizmo and the moving mesh code Arepo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Nevertheless, the cloud mass, defined by the particles obeying 𝜌 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='64𝜌cloud,𝑖 and 𝑢 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='9𝑢amb,𝑖, is very similar for all methods until 𝜏𝐾 𝐻 , shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As expected, the MFM calculations line up with the calcu- lations done by Hopkins (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The periodic bumps are a result of the self-interaction of the shock due to the choice of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At later times the evolution strongly deviates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for MFM as well a moving mesh secondary instabilities build up and lead to a disruption of the cloud, it is more stable in SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to the more traditional SPH results of Hopkins (2015), however, we find the blob to decay stronger, as modern SPH with time-dependent artificial viscosity and conductivity is able to evolve instabilities much better, thus allowing for more mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) MFM GIZMO SPH AREPOMFM in OpenGadget3 11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Blob at 𝑡 = 𝜏KH and 𝑡 = 4𝜏KH as small insertion comparing different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At the earlier time, SPH leads to much less deformation due to less instabilities building up, while MFM in both implementations as well as Arepo agree qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At late time, MFM and Arepo are fully mixed, while SPH still has some structure remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Decay of the cloud fraction surviving for the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the background, comparison lines of the results by Hopkins (2015) for MFM (black, solid) and (traditional) SPH (orange dashed) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM and Arepo agree very well, while SPH shows less mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 Tests for Shock-capturing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Sod Shock-tubes Another important capability of the code is to capture strong shocks of (arbitrarily) large Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We begin testing this on a sim- ple Sod shock-tube based on the setup of Sod (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The test is preformed in a periodic box with two fluids of different density and pressure (𝜌1 = 1, 𝑃1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 𝜌2 = 1/8, 𝑃2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 for 𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4) that are initialized in a glass-like configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' When the two phases start interacting, a shock begins to move to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 8, we show the resulting structure at 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 for the MFM calculations at dif- ferent Mach number and compare them to the analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The expected profiles are matched very well, for all the Mach numbers adopted in this work, ranging from a very low M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 shock to a strong M = 100 shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This ability is directly connected to the accuracy of the Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For higher Mach numbers, increas- ing peaks in velocity and entropy at the shock front are present as a result of the slope-limiting procedure, which has also been reported by Hopkins (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We note that this can be avoided by using a TVD-limiter has more disadvantages in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' With increasing Mach number, a sufficiently small timestep becomes more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The scatter in velocity for the high M = 100 shock, as well as the small offset in the position of the shock front converge away with decreasing timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The scatter in density present at all Mach numbers is a result of the choice of the ICs, which are setup in a glass-like configuration and designed for a higher neighbor number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It does not converge for low neighbor numbers, as chosen for MFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The pressure profile shows the typical bump at the rarefaction fan, as well as the pressure blip at the contact discontinuity, shown in more detail in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 9 for the intermediate M = 10 shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This indicates the presence of surface tension-like error terms, introduced by the slope limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As MNRAS 000, 1–26 (2023) GIZMO AREPOMFM SPH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 GIZMO AREPO Hopkins (2015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 SPH MFM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 1 2 3 4 t/TKH12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Density, pressure, velocity and entropy profile of the shock tube at 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 calculated with our MFM implementation, comparison between different Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM is able to reproduce the general structure of the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Artifacts of surface tension introduced by the slope-limiter are visible at higher Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The scatter is a result of the choice of ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Pressure profile of the M = 10 shock tube at 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5, comparison between different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The different codes show different amount of surface tension and also slight differences in the position of the shock front due to different timestepping discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A on the example of the hydrostatic square, these terms are present for SPH and both MFM implementations, but not for Arepo, manifesting also in the presence or absence of the pressure blip for the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The shock front is captured equally well for MFM and SPH, though less smoothed out for MFM due to the lower neighbor number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo poorly captures the behavior at the shock front due to several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' First, it has troubles in the mesh reconstruction in this strongly anisotropic region, which leads to a shift in the position of the shockfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Second, the public Arepo version does not include slope-limiters, which leads to the oscillatory behavior in the shocked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It could be improved using a static mesh, which would remove other advantages of this method, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Also the inclusion of a slope limiter would improve results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Sedov-Taylor Blastwave This very strong, radially symmetric shock has first been introduced by Sedov (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Besides the capability to deal with jumps, Saitoh & Makino (2009) describe how it can be used to analyze the timestep limiter and shows the need for the limiting to be non-local, as provided by the wakeup scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The test has become a popular benchmark for MNRAS 000, 1–26 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 M= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 M=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 M= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 M= 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 0 40 P20 10 2 5 0 100 rdid: 50 A 0 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 x x x xMFM 10 SPH 10 GIZMO 10 AREPO 10 40 : P 0 0 0 0 20 92 93 92 93 92 93 92 93 9 9 6 8 8 8 8 88 90 0688 88 90 8890 0 40 60 80 100 40 60 80 100 40 60 80 100 40 60 80 100 x x x xMFM in OpenGadget3 13 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Sedov blast at 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparison between different meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The main difference is the height of the peak, which is reduced due to smoothing of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Supernova blast wave evolution in recent years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Kim & Ostriker 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Steinwandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As ICs, we setup a regular grid with 643 particles and density 𝜌 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While almost all particles exhibit a vanishing pressure 𝑃𝑎 = 10−6, energy of 𝑈 = 10 is distributed equally into the eight central particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A shock with very high M𝑖 ≳ 2 · 104 arises, and quickly moves outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='The radial density distribution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods are able to capture the shock, though slightly smooth- ing it, thus underestimating the height of the density-peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH shows the strongest smoothing, followed by the two MFM implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo is able to reproduce the height of the peak best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The position of the peak is similar for all methods, with minor differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While Arepo and gizmo’s MFM implementation predict the peak position correctly, MFM and SPH in OpenGadget3 lag slightly behind, which results in a more accurate position of the low-density side of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This position strongly depends on the precise timestep settings, indicating differences in the timestepping between the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 Including self-gravity In cosmological contexts, not only hydrodynamical forces, but also gravitational accelerations are of great importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gravity domi- nates the evolution on large scales due to its long-range character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It can lead to collapse of clouds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' in the ISM for star formation, or balance thermal pressure and lead to hydrostatic equilibrium, such as in the global structure of galaxies or galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, we analyze the interplay between hydrodynamical forces and gravity in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Evolution of the half-mass radius for the gravitational freefall test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods agree at early time, but deviate from the expected solution at later times when hydrodynamical contributions become more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Gravitational Freefall As a first test including self-gravity, we simulate a collapsing sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The ICS are set up on a regular grid of 203 particles and cut out a sphere of radius 1, which has a total mass of 𝑀sphere = 1 and a negligible pressure of 𝑃 = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the Arepo run, we fill the region not occupied by the sphere with low mass, low energy particles at resolution 8, in order to improve the mesh reconstruction at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We follow the evolution of the half-mass radius, to not be influenced by boundary effects as for the full radius, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparing to the analytic solution 𝑡(𝑟) = arccos �√︂ 𝑟 𝑟0 + √︂ 𝑟 𝑟0 √︂ 1 − 𝑟 𝑟0 � 2 𝜋 √︄ 3𝜋 32𝜌0 , (45) all methods agree at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At late times, pressure and thus ef- fects of the hydro-scheme become more relevant, and deviations are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods overestimate the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM lies closest to the analytic solution with both implementations being indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The moving mesh code Arepo performs worst, which can be ex- plained by poor treatment of the non-periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In order to construct the grid for the hydro-calculations, the box has to be treated periodically, which is not the case for all other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Including the low mass cells at the boundary already decreased the error by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH lies in between the other methods except at very late times, when the error strongly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) MFM GIZMO 4 3 1 0 SPH AREPO 4 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 analytical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 MFM SPH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 MFM (GIZMO) AREPO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 t14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Hydrostatic Sphere In cosmological contexts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' for the ICM, the ability of the code to preserve hydrostatic equilibrium against gravity is of great impor- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To test this, we calculate a hydrostatic sphere as a second test including self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It is also the first test including dark matter as second, only gravitationally interacting particle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The ICs have been created following Viola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 88088 DM particles are setup following an NFW profile (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 1997), populated with 95156 gas particles in hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The correspond- ing density and internal energy profiles at different times are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' After a short relaxation period, happening on a timescale approximately corresponding to the dynamical time, we expect the gas to keep hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH shows the lowest deforma- tion in density, as the ICs were also designed assuming SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM in both implementations, as well as Arepo show a slightly stronger increase in density, especially in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The convergence of the profile can best be checked by following the evolution of the in- ternal energy profile, which is more directly affected by (numerical) diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH, this appears to be stable and also shows only minor changes for Arepo at early time, reaching a stable situation later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For MFM, in contrast, an impact of the numerical diffusivity can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Resulting mixing in the central region leads to a de- crease in internal energy, leading to the observed increase in central density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This effect is very similar for our MFM implementation and the one in gizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Despite these findings, the effect on the density profile is quite small for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 Zeldovich pancake The Zeldovich pancake is the first problem to test our implementation of co-moving integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, it is well suited to show effects of very high M flows, shocks, highly anisotropic particle arrange- ments, and also very low internal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It has been introduced by Zel’dovich (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We start our calculation at 𝑧𝑖 = 100, setting up a single Fourier mode density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' During the linear growth until the caustic formation at 𝑧𝑐 = 1, the evolution can be described by 𝑥 = 𝑥𝑖 − 1 + 𝑧𝑐 1 + 𝑧 sin(𝑘𝑥𝑖) 𝑘 (46) 𝜌 = 𝜌0 1 − 1+𝑧𝑐 1+𝑧 cos(𝑘𝑥𝑖) (47) vpec = − 𝐻0 1 + 𝑧𝑐 √ 1 + 𝑧 sin(𝑘𝑥𝑖) 𝑘 ˆ𝑥 (48) 𝑇 = 𝑇𝑖 � 1 + 𝑧𝑐 1 + 𝑧 �2 � 𝜌(𝑥, 𝑧) 𝜌0 �2/3 (49) starting from the unperturbed position 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 𝜌𝑐 is the critical density, 𝐻0 = ℎ0 · 100 km s−1 Mpc−1 the Hubble parameter (today) with ℎ0 = 1, and 𝑇𝑖 = 100 K the initial temperature, such that pressure forces are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The wavenumber 𝑘 = 2𝜋/(64 ℎ−1Mpc) cor- responds to the first-order soundwave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use the ICs provided by Hopkins (2015), with a resolution of 323 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' After the lin- ear growth, an accretion shock forms close to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As the scale factor increases, the background density decreases strongly and the background temperature decreases adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This causes a huge temperature contrast of ≈ 10 orders of magnitude between the shocked region and the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Due to the very low internal energy compared to other energy contributions 𝑈 ≲ 10−3𝐸kin and 𝑈 ≲ 10−2𝐸pot in physical units, which thus is on the same order as the precision of the Riemann solver, the implementation of the energy-entropy switch described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Its effect on the evolution of the Zeldovich pancake is described further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The resulting structure at 𝑧 = 0 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Again, we compare the performance of the different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The energy-entropy switch is included for MFM if 𝑈 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='01𝐸pot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For Arepo, we had to use additional mesh regularization to avoid too irregular cell shapes in the highly unisotropically compressed shock region and allow the code to run until the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods agree with the peculiar velocity profile with only slight differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Com- pared to Hopkins (2015) we find that all methods seem to have a too low viscosity and show particle over- or under-shooting compared to the predicted velocity profile, as a result of a punch-through of some particles in the high M shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The density peak is captured equally well by almost all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Only the Arepo run shows a slightly higher peak, contrarily to what Hopkins (2015) found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to the expected profile, all methods over-smooth the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Due to the higher neighbor number and thus larger kernel for SPH, this smoothing is larger compared to MFM, both in our new implemen- tation and gizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM shows a similar performance as the moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Most difficult for all methods is to capture the temperature structure with its very strong contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Both MFM implementations work very well, as the energy-entropy switch suppresses any numer- ical noise in the low-energy background and allows a clear jump between shocked and unshocked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The jump for SPH is more strongly smoothed in comparison to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, amplified initial (numerical) noise causes a large scatter of several orders of magnitude in the very cold background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For Arepo, we find that this behavior is much more drastic, and the background is dominated entirely by numerical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To properly resolve it, some energy-entropy switch would be required also in Arepo, which does not seem to be implemented in the public version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 Nifty Cluster Finally, we apply our newly implemented method on more complex, cosmological cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As an example, we re-simulate a cluster from the MUSIC-2 sample (Prada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Sembolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2013, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Biffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2014), analyzed in detail with different codes by a col- laboration formed during a nifty workshop (Sembolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2016), thus called nifty cluster in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The cluster has a mass 𝑀200c = 1015 M⊙ with resolution 𝑚DM = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='01 · 108ℎ−1M⊙ for dark matter and 𝑚gas = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='9 · 108ℎ−1M⊙ for gas particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The back- ground cosmology has parameters ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='27, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0469, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='73, 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='82, 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='95, ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7 (Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The pro- jected surface density at 𝑧 = 0 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 14, where the cluster center and virial radius are obtained using Subfind (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Dolag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We compare MFM to SPH with a different amount of artificial conductivity, ranging from the usually used amount 𝛼max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='25, 𝛼min = 0 (notation following Price 2008) over a run with physical conductivity at 1/20th of the Spitzer value (Dolag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2004), effec- tively corresponding to an intermediate amount, to more traditional SPH without artificial conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The usual amount is chosen to mimic the behavior of Godunov methods such as MFM, which have intrinsic numerical diffusivity due to the Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For re- duced artificial conductivity, structures are slightly less “smeared out”, while the global structure does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A more quantitative analysis can be done using gas radial density, temperature and entropy profiles shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As a comparison, we provide lines from the nifty paper, obtained using Arepo and Gadget3-MUSIC as an example of a more traditional SPH code, which mark the range of solutions obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH can span the whole MNRAS 000, 1–26 (2023) MFM in OpenGadget3 15 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Evolution of gas density (top) and internal energy radial profiles (bottom) for the hydrostatic sphere for a few dynamical times until 𝑡 = 10, colored by the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Calculated using different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM shows a slightly larger numerical diffusivity, but overall still preserves the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Zeldovich pancake at 𝑧 = 0 for different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As a comparison, a high resolution 1d simulation of Hopkins (2015) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While velocity and density profiles agree between the methods, strong deviations can be seen for the temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM performs best due to the energy-entropy switch employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) 10 10 MFM SPH GIZMO AREPO 8 Q 10-5 6 10-6 i06 2 105[ 102 103 102 103 102 103 102 103 r r r r2 MFM SPH GIZMO AREPO [km/s] 1000 23 log(T) [K] 0 20 0 20 20 0 20 20 0 20 20 0 20 x [h-1Mpc] x [h-1Mpc] x [h-1Mpc] x [h-1Mpc]16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Projected surface density of the nifty cluster at 𝑧 = 0, comparison between MFM and SPH with usual amount (𝛼cond max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='25), physical (𝜅phys), corresponding to an intermediate amount, and without artificial conductivity 𝛼cond max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The overall structure is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Small sub-structures, however, appear less compact for MFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' range of possible solutions provided by Sembolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' By construction traditional SPH without artificial conductivity has no mixing and thus forms low entropy cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Subgrid mixing due to the Riemann solver for MFM and Arepo leads to mixing into the core, increasing the entropy compared traditional SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, the central density is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' By including artificial conductivity in SPH, it can reach the same profile as MFM, and also lie in between for effectively intermediate values by using physical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='. The computational costs for running MFM are comparable to those of SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Each single timestep takes 41 s walltime for SPH, compared to 31 s for MFM4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, the runtime of MFM per timestep is smaller by a factor ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The majority of the time is spent in the gravity cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For MFM, the flux calculation and density iteration makes up only 1 s per timestep, while for SPH, density and acceleration calculations take 10 s per timestep, explaining the difference in total 4 Run on Supermuc on 8 nodes, each with 4 MPI tasks, and 24 OpenMP threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 Ruir 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 0 Rir 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 [zd -,go d Bo MFM SPH(Kphys) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 Ruir 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 0 Ruir 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='25) =0) Rrir 0 Ruir Rrir 0 Rrir 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0MFM in OpenGadget3 17 10 1 100 R [h 1Mpc] 1013 1014 1015 1016 gas [h 1M /(h 3Mpc3)] MFM SPH( cond max =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='25) SPH( phys) SPH( cond max =0) Sembolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016) AREPO G3-MUSIC 10 1 100 R [h 1Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 kBT [keV] 10 1 100 R [h 1Mpc] 101 102 103 S=T/n2/3 e [keV h 2cm2] Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gas density (left), temperature (middle) and entropy (right) radial profiles of the nifty cluster at 𝑧 = 0, comparison between different hydro methods, including our MFM implementation (red plus), SPH in OpenGadget with usual (green), physical, corresponding to an intermediate value, (turquoise) and without artificial conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As a comparison, the Arepo (black dashed) and G3-MUSIC (traditional) SPH line (red solid) from Sembolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (2016) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The vertical line marks 𝑅200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Our modern SPH run with sufficiently high artificial conductivity, as well as Arepo and MFM produce higher entropy cores with lower, less peaked density, while the central entropy is much lower for SPH with lower artificial conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' time per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The lower neighbor number causes a decrease in com- putational cost in the density and also the actual flux calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This is only partly compensated by the more expensive Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, the decrease is even more significant for pure hydrodynamical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An addition, the computational cost could be further decreased by using a faster approximate Riemann solver instead of the exact one, which, however, has other disadvantages as discussed an App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for pure hydrodynamical problems the number of timesteps increases for MFM due to the effectively higher spacial resolution, the timesteps for more complex simulations including gravity are limited by other criteria than the Courant-criterion, not depending on the smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, they are similar between the methods and rather depend on the precise differences in evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, MFM on average even yields a slight decrease in runtime for cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The size of the structure holding the gas particle data, MFM has a much larger requirement by a factor ≈ 5, which could be used by more efficient use of existing variables in SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As also data of other particle types is saved, the total memory requirement is only larger by a factor of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The difference would decrease even further if more physics was included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 Decaying Subsonic Turbulence In many astrophysical systems, ranging from the atmosphere over the ISM up to galaxy clusters, turbulence plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the ICM, we expect subsonic turbulence with a turbulent energy fraction of 𝑋 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 to be excited, for instance after a merger (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Schuecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The different hydro-schemes have problems to capture its full behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It has been shown that traditional SPH is not well suited to calculate sub-sonic turbulence (Bauer & Springel 2012), but can be improved using modern SPH with more ideal settings for artificial diffusion terms (Price 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To test and compare the performance of our MFM implementa- tion, we setup a 300 kpc cubic box with varying number of particles, and seed the largest ≈ 70 modes, similar to Bauer & Springel (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Due to the low initial density of 𝜌 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 · 10−6, gravitational acceler- ation can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The initial turbulent energy fraction is varied between 𝑋𝑖 = 𝐸turb,𝑖/𝐸therm,𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 and 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, the resolution is varied, ranging from 643 up to 5123 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We evolve the turbulence for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 sound-crossing-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The turbulent kinetic energy cascades down to smaller scales, forming a turbulent power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In order to analyze the velocity power spectrum, the data are binned to a grid using the code Sph2Grid5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' From that, a power spectrum is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use a D20 sampling, in order to conserve energy (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Theoretically, a Kolmogorov slope 𝐸(𝑘) ∼ 𝑘−5/3 would be expected (Kolmogorov 1941).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 16, we compare the power spectra of the different methods, normalized by the expected slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While all methods agree at large scales, where the energy was seeded, they show huge discrepancies at intermediate to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo shows deviations at small scales, close to the resolution limit, underestimating the energy present at these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH starts de- viating at slightly larger scales, with a less deep dip in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For MFM the power spectrum shows a dip in energy at similar scales as the moving mesh code Arepo, but with a much shal- lower depth than in all other cases, thus being closer to the expected slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, the MFM result converges quickly with resolution, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As the dip moves towards smaller scales, the overall spectrum becomes closer to the Kolmogorov one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At the highest resolution considered, it almost perfectly resembles the expected Kolmogorov slope over a wide range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the power spectrum builds up, energy is not only transported to smaller scales, but also partly converted into internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We plot this decay of kinetic, turbulent energy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 18, comparing the different hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While in a physical situation the decay would depend on gas microphysics such as its viscosity, here we can use it to get an insight into the code behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The decay is mainly determined by numerical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In all cases, the energy shows a periodic variation, caused by the “ringing” of the initially seeded 5 Developed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Donnert, available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='com/jdonnert/ Sph2Grid MNRAS 000, 1–26 (2023) 18 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 10 2 10 1 k=2 /L [kpc 1] 10 5 10 4 10 3 10 2 10 1 100 101 102 k 5/3E(k) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' units] k 128 SML k 128 Nyquist kbox kSEED, min kSEED, max Kolmogorov MFM (OpenGadget3) SPH (OpenGadget3) Moving Mesh (AREPO) Static Mesh (AREPO) Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Normalized turbulent velocity power spectrum for different methods at 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods agree at large scales, but show a lack in energy at intermediate to small scales compared to the expected Kolmogorov-slope 𝑃 ∼ 𝑘−5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM works best overall reproducing the expected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM has a decay time of a few sound crossing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The decay for SPH depends strongly on the artificial viscosity, varying from a value similar to that of MFM for SPH with the standard amount of viscosity, up to even an (unphysical) increase for the cal- culation without artificial viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The power spectrum, in contrast, is only weakly influenced by the amount of artificial viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' With increasing resolution, the decay time increases, indicating numerical dissipation errors converge away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo shows a much slower decay with 𝑡dec > 20𝑡sc already at lower resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A comparison for the decay at different initial turbulent energy fractions, corresponding to variations in the Mach number, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 18 for MFM and SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The variation between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0001 for the initial turbulent energy fraction corresponds to a range of Mach numbers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='07 down to below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH the decay is independent of the Mach number, as one would expect, so it is for Arepo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This is true also for MFM down to 𝑋𝑖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='003, corresponding to M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For even smaller Mach numbers, the turbulent energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' At the same point also the density pdf deviates from the Gaussian shape, indicating the evolution is dominated by numerical artifacts for such low Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7 Effects of Numerical Parameters The performance of the numerical methods strongly depend on the precise parameters used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Effects of neighbor number and kernel have already been analyzed in detail by various authors (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', Dehnen & Aly 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Tricco & Price 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To this end, we focus on two other parameters that play a major role for MFM, namely the slope-limiting scheme and the energy-entropy-switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Slope-Limiter The different slope-limiting procedures, which are implemented in our code, differ not only in how aggressively they limit the slope, but also in how much numerical diffusivity they introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In general, different limiters are shown to produce different results for specific test-cases (compare e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' May & Berger 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hubber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the following, we compare the three cases of the limiter from gizmo that we usually use, the Arepo and, the TVD limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The gizmo and TVD limiters are the most extreme cases of the limiters implemented, with lowest and highest numerical diffusivity, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Arepo limiter lies in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We analyze the effect on the hydrostatic square (compare also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A) and the Rayleigh-Taylor instability (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for the Rayleigh-Taylor instability the much less diffu- sive gizmo limiter performs best, evolving a much finer structure, this causes the strongest deformation of the hydrostatic square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Arepo limiter is slightly more diffusive, leading to less strong sec- ondary instabilities for the Rayleigh-Taylor instability and slightly less deformation of the square, especially at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The TVD lim- MNRAS 000, 1–26 (2023) MFM in OpenGadget3 19 10 2 10 1 k=2 /L [kpc 1] 10 5 10 4 10 3 10 2 10 1 100 101 102 k 5/3E(k) [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' units] k 128 SML =k 128 Nyquist kbox kSEED, min kSEED, max resolution 643 1283 2563 5123 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Normalized turbulent velocity power spectrum for MFM with different resolutions at 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM converges fast with resolution towards the expected Kolmogorov-slope 𝑃 ∼ 𝑘−5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' iter has an even higher numerical diffusivity, thus strongly smooths the Rayleigh-Taylor instability, not only preventing secondary insta- bilities to form, but also reducing the overall growth of the instabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The hydrostatic square, however, is preserved best, due to lower surface-tension like errors, as it can be observed also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' for shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Combining the results, we show that it is not always clear which slope-limiting procedure would be the overall preferred choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As in most cases the gizmo limiter performs best, we chose this as our reference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 Energy-Entropy-Switch To avoid numerical errors to dominate the evolution of the internal energy, an energy-entropy switch as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 has to be used in specific problems such as the Zeldovich pancake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Especially, the numerical noise should be suppressed in the very cold, unshocked region, while the shock should not be influenced at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The resulting structure at 𝑧 = 0, comparing different possibili- ties for the switch based on potential and kinetic energy estimates (compare also Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (41)), is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We increase the tuned values (𝛼1 = 10−2 for the potential energy and 𝛼2 = 3 · 10−3 for kinetic energy) by a factor 2 and decrease them by a factor ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A more strict switch (larger 𝛼) causes less particles to be treated with the adiabatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the kinetic energy switch, this difference causes strong variations in the temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for 𝛼2 = 1 · 10−3 more extended wings form and some scatter in the low-temperature background close to the peak appears, the increased value of 6 · 10−3 treats even particles inside the peaked region with the adiabatic approximation and causes too low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A very fine-tuned choice of 𝛼2 is necessary to accurately capture all particles, both the shocked ones and the low-temperature ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to that, a variation of 𝛼 within the switch based on potential energy influences the temperature profile only weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, it seems to be much more stable and should be the preferred option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 5 DISCUSSION AND CONCLUSIONS We presented a new MFM implementation into OpenGadget3 as an alternative hydro-solver to the currently used modern SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We verified its capabilities, both in idealized and more complex, cos- mological test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Tests range from smooth, simple situations, mixing instabilities, shocks, tests including self-gravity, to the nifty cluster as cosmological example and decaying, subsonic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A comparison has been preformed between MFM and SPH in Open- Gadget3, the MFM implementation in gizmo and the moving mesh code Arepo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, two parameters have been analyzed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, we find very good agreement between the MFM imple- mentation in OpenGadget3 and that in gizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Minor differences are found in the precise appearance, while global properties are in- MNRAS 000, 1–26 (2023) 20 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 t/tsc 100 4 × 10 1 6 × 10 1 E/E fit i hydro method MFM SPH ( visc max =0) SPH ( visc max =3) SPH ( visc max =10) AREPO AREPO(static) Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Decay time of turbulent energy for different methods at 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For SPH, the viscosity is varied between 𝛼visc max = 10 and 𝛼visc max = 0, where 𝛼visc max = 3 is the value typically used (notation following Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo has the highest decay time corresponding to the lowest numerical dissipation, while MFM and SPH at typical value of viscosity are on a similar order with a decay time of a few dynamical timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 100 4 × 10 1 6 × 10 1 E/E fit i SPH: Xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 t/tsc 100 4 × 10 1 6 × 10 1 E/E fit i MFM Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 18, but for varying initial turbulent energy fractions 𝑋𝑖, corresponding to variations in the turbulent Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The decay is consistent for all 𝑋𝑖 for SPH, and down to 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='003 for MFM, when numerical artifacts lead to an unphysical increase in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Even without further tuning, MFM reproduces the expected behavior in all test cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The soundwave test is well suited for a convergence analysis, as an analytical solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM shows a very good convergence behavior between first and second order for dispersion errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Diffusion errors as well as the scatter converge second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the convergence is better than for SPH and a moving mesh, these methods show lower errors, especially for the dispersion error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An important advantage of MFM over SPH is the capability to accurately evolve mixing instabilities without additional artificial viscosity or conductivity as for SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, a lower neighbor number compared to SPH is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM as a moving mesh even show secondary instabilities to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The blob test as combination Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hydrostatic square (top) and Rayleigh-Taylor instability (bottom), developed using different slope-limiters, the gizmo limiter we usually use (left), compared the the same test, but evolved using the Arepo limiter (mid- dle) and TVD limiter (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Depending on the test, different slope-limiters could be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' between mixing and shocks emphasizes the ability of MFM to allow for more mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The rate of the decay of the cloud is similar to that of a moving mesh simulation and larger than for SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to the more traditional SPH implementation shown by Hopkins (2015), the modern SPH implementation OpenGadget3 allows for more mixing and leads to a faster decay of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As this test is designed to mimic ram-pressure stripping, we expect this effect to be modeled more accurately using MFM compared to SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This should also lead to an overall more accurate evolution of galaxies in the environment of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To fully understand and follow the evolution of such gas blob in cosmological contexts more physics such as cooling, and, depending on the context, star formation, is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gronke & Oh (2018, 2020, 2022) have analyzed this test in detail with such additional physics and found a great importance of the cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, MFM can model shocks for a wide range of Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the shock tube tests MFM performs especially well for lower Mach numbers, while effects of surface tension due to the choice of the slope-limiter are visible at higher Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Nevertheless, it is still able to capture the main features of the shock including the position of the shock front, the contact discontinuity, and the rarefaction fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Different methods lead to differences in the smoothing of the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The lower neighbor number in MFM compared to SPH increases the effective spatial resolution by a factor of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For Arepo, the shock front is dominated by numerical artifacts due to the absence of a slope-limiter in the publicly available version as well as difficulties in the mesh reconstruction in such highly anisotropic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Sedov blast works well for all methods, verifying the capa- bility of the wakeup scheme as non-local timestep criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Main differences are the smoothing and resulting lower amplitude of the density peak, revealing an even smaller smoothing for the moving mesh compared to MFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The narrower shock front will help e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' for shock detection in cosmological simulations (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', Pfrom- mer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In general, MFM is able to preserve hydrostatic equilibrium accu- rately, as well as preserving stable orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The better stability of the Kepler disk compared to SPH will improve results for simulations of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this case, a moving mesh leads to even better results, but requires additional boundary particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) M MMIARmMFM in OpenGadget3 21 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Effect of the choice of the energy-entropy switch on the Zeldovich pancake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparison between the switch based on kinetic and potential energy, each with three different 𝛼-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The switch based on potential energy is much more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The hydrostatic sphere test showed that MFM coupled to gravity has a slightly higher numerical diffusivity compared to SPH and a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, on could expect isolated galaxies or also the core of galaxy clusters to be more compact and cooler in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the nifty galaxy cluster, however, we saw that there is no difference between MFM, Arepo and modern SPH in the global structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Numerical diffusivity introduced by the Riemann solver allows mixing of entropy into the core, thus decreasing the central density compared to traditional SPH, which suppresses any mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Modern SPH mimics the same effect by applying artificial conduc- tivity, while the precise amount introduced can lead to significant changes in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As observed galaxy clusters show a wide range of central entropy profiles (Cavagnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2009), both re- sults are consistent with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Especially, we expect a more complex interplay with cooling, as well as stellar and AGN feed- back to influence the entropy-evolution of the core (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Borgani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Rasia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' These ef- fects lead to the whole range of possible central profiles, dominating over effects of the hydro-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, further studies including such processes would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the intra cluster medium, we expect turbulence at low Mach number to be seeded e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' by mergers at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It will then decay and build up a turbulent power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Such decaying, subsonic turbulence is a very challenging problem for many hydro-methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM is able to recover the turbulent power spectrum best compared to SPH and a moving and stationary mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Only a small lack of energy at intermediate to small scales close to the resolution limit – similar to where this occurs also for Arepo – is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This “dip” in energy moves to smaller scales for higher resolution, overall leading to fast convergence towards the expected Kolmogorov spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The decay rate of turbulent energy due to numerical dissipation is on the same order as for modern SPH, and decreases towards higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The results are consistent down to very small initial turbulent energy fractions 𝑋𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='003, corresponding to small Mach numbers M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For smaller 𝑋𝑖 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='003 numerical effects dominate and and lead to unphysical increase in turbulent energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, the results are very promising for the accurate evolution of turbulence also within galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An energy-entropy switch is of great importance to accurately evolve the temperature profile for the Zeldovich pancake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' When it is included, MFM yields the best results, having a clear jump in the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparing different possible values for such a switch, we found that careful tuning is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In general, the switch based on potential energy produces more stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Arepo misses the implementation of such a switch in the public version, such that the low-temperature region is entirely dominated by numerical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' SPH also shows noise in the low-temperature region, originating from the amplification of noise present in the ICs, and also much broader wings around the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' All methods show some punch-through in the temperature profile, indicating a too low viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition to comparing different methods, we used two tests to analyze the impact of the slope-limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Depending on the problem, different slope-limiters can be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the gizmo limiter performs best in most test-cases, having a much lower numerical diffusivity, specific cases such as the hydrostatic square and also strong shocks work better using a more diffusive TVD-limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Arepo limiter has an intermediate diffusivity and lies in between the two other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, our implementation of MFM produces accurate results for the cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It avoids some of the disadvantages of SPH, while requiring an even smaller computational cost per timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The total number of timesteps and thus the total runtime increases as a result of the smaller smoothing length and effectively higher spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A faster, approximate Riemann solver can further decrease the computational costs, but has the drawback of introducing more numerical diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to that, a moving mesh requires a very expensive tessellation to be performed, such that the required computational costs for many tests are drastically increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, MFM is a promising alternative for cosmological simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1 Outlook – possible extensions in the future To make use of the full advantages of OpenGadget3, it will be useful to couple MFM not only to gravity, but also to include more physical processes, such as cooling, star formation and stellar feedback, AGN feedback, physical conductivity and viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For these, we can make use of already existing implementations in OpenGadget3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, MFM can be expanded to an MHD method, including magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This will also allow to include the existing imple- mentation of cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For many of these extensions, coupling can be done in a similar way as for SPH, while others such as magnetic fields will require more significant changes including another Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In principle, also a general-relativistic (GR) extension would be possible, which has been implemented both for SPH (Liptai & Price 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Rosswog & Diener 2021) and a moving mesh (Chang & Eti- enne 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Lioutas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2022) and also exits for MFM within the gizmo code (Lupi 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As GR is mainly important in extreme sit- MNRAS 000, 1–26 (2023) 2 U<αiEpot U<α2Ekin 1 [km/s] α1 = α2 = 0001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='10-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='10-3 X 1·10-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='10-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='10-2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='10-3 log(plpo) K log(T) 0 25 0 25 25 0 25 x [h-1Mpc] x [h-1Mpc]22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' uations such as accretion discs around black holes, this would also make use of the fact that our MFM implementation is originally based on GANDALF, which itself was designed to deal with star and planet formation, and thus for disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' ACKNOWLEDGEMENTS FG and KD acknowledge support by the COMPLEX project from the European Research Council (ERC) under the European Union’s Hori- zon 2020 research and innovation program grant agreement ERC- 2019-AdG 882679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' UPS is supported by a Flatiron Research Fel- lowship at the Center for Computational Astrophysics (CCA) of the Flatiron Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Flatiron Institute is supported by the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' FG, MV and KD acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un- der Germany’s Excellence Strategy - EXC-2094 - 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MV is supported by the Alexander von Humboldt Stiftung and the Carl Friedrich von Siemens Stiftung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We are especially grateful for the support by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Petkova through the Computational Center for Particle and Astrophysics (C2PAP) under the project pn68va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Some calcu- lations for the hydrodynamical simulations were carried out at the Leibniz Supercomputer Center (LRZ) under the project pr86re (Su- perCast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Alig for a turbulence Arepo setup and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Böss for providing ICs for the shock-tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The analysis was performed mainly in julia (Bezanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2014), including the package Gad- getIO by Böss & Valenzuela (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The surface density of the nifty cluster was calculated using Smac (Dolag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 2005a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We thank the developers of gizmo and Arepo for making the codes publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' DATA AVAILABILITY The setup for the different tests are publicly available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='com/fgroth/hydro_tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This includes parameter and config files for the different codes used, as well as our analysis routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' If applicable, routines to create ICs are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Other data will be shared upon reasonable request to the corresponding au- thor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' OpenGadget3 is a non-public developer version of Gadget-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It is available upon reasonable request from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Dolag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' REFERENCES Agertz O.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', 2020, The Astrophysical Journal Supplement Series, 248, 32 Wendland H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', 1995, Advances in Computational Mathematics, 4, 389 Xu G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', 1995, The Astrophysical Journal Supplement Series, 98, 355 Zel’dovich Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', 1970, Astronomy and Astrophysics, 5, 84 APPENDIX A: HYDROSTATIC SQUARE The Hydrostatic Square test is well suited to study the stability of edges related to numerical surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Similar tests have been performed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' by Hess & Springel (2010) and Hopkins (2013, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We set up a two-dimensional box of size 𝐿 = 1 with peri- odic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It is filled with 7168 gas particles with equal masses, arranged in two regular grids, one grid for the am- bient medium (𝜌𝑎=1, 𝑃𝑎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5) and one for the square with side- length 𝐿/2 with increased density 𝜌𝑠 = 4 in hydrostatic equilibrium (𝑃𝑠 = 𝑃𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A1, we compare the resulting density distribu- tion at time 𝑡 = 10, evolved with the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As the ICs are set in hydrostatic equilibrium, we would expect no changes to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This ideal state is only achieved using the moving mesh code Arepo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Theoretically, we would expect the same to be true for MFM, as shown by Hopkins (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' They use, however, a strongly ideal- ized setup compared to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Especially, they use a regular grid for all particles, and increased particle masses within the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For our setup, the gradient estimate at the boundary does not conserve linear gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Instead, it is biased by the in-homogeneous parti- cle distribution due to two separate grids, especially in combination with the slope-limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A more detailed analysis of the effect of the slope limiter is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1, where we have shown that the amount of surface tension strongly depends on the slope-limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We observe, using both our MFM implementation and gizmo, that for MFM the edges of the square start to deform, followed by some numerical instability, which leads to a more asymmetric deforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Increasing the resolution by a factor of 4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A2, Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Density of the hydrostatic square evolved until 𝑡 = 10 using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The initial location of the high density “square” region is overplotted as contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Only Arepo is able to keep the initial square shape, while other methods lead to deformation of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' this instability occurs slower and the square preserves its shape much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Also using SPH, the square deforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As expected, it becomes more circular, caused by numerical errors, which behave as surface tension (compare, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=', Price 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For traditional SPH, these errors should be low-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We observe, however, that this effect can be drastically reduced by increasing the resolution, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A2 indicating that modern SPH implementations, as used in OpenGad- get3, reduce low-order errors and improve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Overall, for this specific test surface tension for SPH, but also for MFM can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A moving mesh performs best, preserving the situation perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MFM at later times shows some numerical errors lead- ing to a more asymmetric deformation, which converge away with increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' APPENDIX B: SLOPE-LIMITERS IN OPENGADGET3 We implemented seven different slope-limiters and therein variants of their specific parameters in OpenGadget3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The main concept is de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In general, we substitute ∇W𝑖,𝑘 → 𝛼𝑖,𝑘∇W𝑖,𝑘 for each particle 𝑖 and component 𝑘, for the face interpolation, with 𝛼𝑖,𝑘 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the following, we briefly describe the implemented limiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The simplest option are to use a zeroth order interpolation setting 𝛼ZERO SLOPES 𝑖,𝑘 = 0 (B1) or to include no slope-limiter 𝛼NULL 𝑖,𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B2) Alternatively, we implemented several more complex limiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A commonly used one is a TVD scalar limiter (Hess & Springel 2010), which is designed to produce good results especially for strong MNRAS 000, 1–26 (2023) IZMO ST ARHPOMFM in OpenGadget3 25 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Hydrostatic Square at 𝑡 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Comparison of MFM and SPH at two different resolutions, Top: 7168 particles, Bottom: 114688 particles (increase in resolution by factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Both, MFM and SPH, show convergence of the shape of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Compared to the other limiters implemented, it is the most diffusive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It sets 𝛼TVD SCALAR 𝑖,𝑘 = min 𝑗 ∈Ngb max ���� ���� 0 min � 1 d𝑊𝑖 𝑗,𝑘𝑘/d𝑊𝑘 (B3) where dW𝑖 𝑗 = W 𝑗 − W𝑖, dW = dr𝑖 𝑗 · ∇|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An alternative is the scalar limiter (Balsara 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Gaburov & Nitadori 2011), which looses the TVD behavior but is less diffusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' It sets 𝛼SCALAR 𝑖,𝑘 = max ������� ������� 0 min ����� ����� 1 min ��� ��� d𝑊𝑘,max |d𝑟 |max |∇𝑊𝑘 | d𝑊𝑘,min |d𝑟 |max |∇𝑊𝑘 | (B4) where d𝑊𝑘,min/max = ��𝑊𝑖,𝑘 − min/max𝑗 ∈Ngb 𝑊 𝑗,𝑘 ��, and |d𝑟|max = max � max𝑗 ∈Ngb ��𝑟𝑖 𝑗 �� , ℎ𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In contrast to the TVD limiter, only the global neighbor distribution is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Thus, values calculated from all neighbors individually for the TVD limiter are calculated in an approximate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, we implemented the limiters used both in the Arepo and gizmo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In the Arepo code (Springel 2010), the slope is limited using 𝛼Arepo 𝑖,𝑘 = min 𝑖∈Ngb ���� ���� d𝑊𝑘,max/d𝑊𝑘 if d𝑊𝑘 > 0 d𝑊𝑘,min/d𝑊𝑘 if d𝑊𝑘 < 0 1 d𝑊𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B5) It lies in between the TVD and scalar limiter, as only the dividend is approximated from the global neighbor distribution, while the divisor is still calculated for all neighbors individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In gizmo (Hopkins 2015), a general limiter is introduced described by 𝛼gizmo 𝑖,𝑘 = min ����� ����� 1 𝛽𝑖 min � d𝑊𝑘,max 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ℎ𝑖 |∇𝑊𝑘 | d𝑊𝑘,min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5ℎ𝑖 |∇𝑊𝑘 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B6) The parameter 𝛽 has to be 𝛽𝑖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 to ensure second order stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A higher number corresponds to a more aggressive, less diffusive and less stable limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' We use the suggested value 𝛽 = 2 of Hop- kins (2015), which is a compromise to reduce numerical diffusivity while still working for very strong interacting shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For 𝛽 = 2, this limiter is also similar to the scalar limiter with the difference that the theoretically possible distance between neighbors is defined only by the smoothing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, Hopkins (2015) provide a pairwise limiter, acting on only one specific interaction, instead of all neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For this, it uses already limited slopes for the interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The pairwise limiter described by Hopkins (2015) limits the already interpolated face values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The aim is to directly calculate the face value 𝑊new 𝑖 𝑗,𝑘, starting from the extrapolated value 𝑊face 𝑖 𝑗,𝑘 according to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (36), possible already with limited gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' If 𝑊𝑖,𝑘 = 𝑊 𝑗,𝑘, the face value is just chosen the same as the cell values 𝑊new 𝑖 𝑗,𝑘 = 𝑊𝑖,𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Otherwise, the values 𝛿1 = 𝜓1 ��𝑊𝑖,𝑘 − 𝑊𝑗,𝑘 �� (B7) 𝛿2 = 𝜓2 ��𝑊𝑖,𝑘 − 𝑊𝑗,𝑘 �� (B8) are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The free parameters 𝜓1/2 are tuned to 𝜓1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5, 𝜓2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A simple intermediate value used later is given by ¯𝑊𝑖 𝑗,𝑘 = 𝑊𝑖,𝑘 + d𝑟𝑖 𝑗 d𝑟face 𝑖 (𝑊 𝑗,𝑘 − 𝑊𝑖,𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B9) The maximum/minimum value is 𝑊𝑘,min/max = min/max(𝑊𝑖,𝑘, 𝑊𝑗,𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Depending on how the two face val- ues compare, the new face value is calculated: If 𝑊𝑖,𝑘 < 𝑊 𝑗,𝑘, then 𝑊new 𝑖 𝑗,𝑘 = max �������� �������� ��� ��� 𝑊𝑘,min − 𝛿1 if SIGN(𝑊𝑘,min − 𝛿1) = 𝑆𝐼𝐺𝑁(𝑊𝑘,min) 𝑊𝑘,min 1+ 𝛿1 |𝑊𝑘,min| else min � 𝑊face 𝑖 𝑗,𝑘 ¯𝑊𝑖 𝑗,𝑘 + 𝛿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B10) If 𝑊𝑖,𝑘 ≥ 𝑊 𝑗,𝑘, then 𝑊new 𝑖 𝑗,𝑘 = min �������� �������� ��� ��� 𝑊𝑘,max + 𝛿1 if SIGN(𝑊𝑘,max + 𝛿1) = 𝑆𝐼𝐺𝑁(𝑊𝑘,max) 𝑊𝑘,max 1+ 𝛿1 |𝑊𝑘,max| else max � 𝑊face 𝑖 𝑗,𝑘 ¯𝑊𝑖 𝑗,𝑘 − 𝛿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' (B11) The same limiter is applied for particle 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Finally, the gizmo code uses a slightly different pairwise limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Depending on the tolerance 𝑡 chosen, the parameters 𝜓1 = ���� ���� 0 𝑡 = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5 𝑡 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='75 𝑡 = 2 (B12) 𝜓2 = ���� ���� 0 𝑡 = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='4 𝑡 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='375 𝑡 = 2 (B13) MNRAS 000, 1–26 (2023) 26 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Groth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' To calculate ¯𝑊𝑖 𝑗,𝑘, the factor d𝑟𝑖 𝑗/d𝑟face 𝑖 is approximated by the first order value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Except these differences, the limiter is identical to the already described one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In our implementation, we apply the limiter in the reference frame of the interface, such that the velocity is a relative velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This makes the limiter Lagrangian and increases the symmetry between different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' APPENDIX C: EFFECT OF THE RIEMANN SOLVER In OpenGadget3 we use an exact, iterative Riemann solver by de- fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This is however, computationally expensive as up to eight it- erations are used to get close to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' An alternative is using approximate Riemann solvers, where we implemented a Roe solver, the HLL solver, and the HLLC and HLLE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While these are faster by up to 20 per cent for problems dominated by hydrodynamical calculations such as the shock tube, the effect becomes less important when using gravity and possibly even more extensions in cosmological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Already for the hydrostatic sphere, there is no significant difference in runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' For the HLLC solver, there is even a slight increase in runtime due to differences in the precise evolution, making the gravity calculation more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' In addition, the Riemann solver leads to changes in the precise evolution as it introduces numerical diffusivity, visible in density and internal energy changes for the hydrostatic sphere, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content='2, also the exact Riemann solver leads to some numerical diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The change in internal energy is however even stronger for the alternative approximate Riemann solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While the HLL Riemann solver produces results close to the exact one, it is also the most unstable one, such that a large fraction of the calculation is actually done using the exact solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The Roe Riemann solver shows a slightly stronger change in the hydrostatic density and internal energy profile, indicating a higher diffusivity, followed by the HLLC Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' While for specific problems these alternative solvers could lead to faster results, we in general use the most accurate exact Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' The increase in runtime is compensated by the gain in accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) MFM in OpenGadget3 27 Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' Evolution of the density and internal energy profile for the hydrostatic sphere test case, comparing the exact, HLL, Roe and HLLC Riemann solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' A main difference is the amount of numerical diffusivity introduced by the different Riemann solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} +page_content=' MNRAS 000, 1–26 (2023) 10 8 10-5 6 MFM(Rs=exact) MFM(Rs=HLL) MFM(Rs=Roe) MFM(Rs=HLLC) 10-6/ 106 4 2 105[ 102 103 102 103 102 103 102 103 r r' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE2T4oBgHgl3EQfCQao/content/2301.03612v1.pdf'} diff --git a/TtE1T4oBgHgl3EQfugUk/content/tmp_files/2301.03388v1.pdf.txt b/TtE1T4oBgHgl3EQfugUk/content/tmp_files/2301.03388v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a10e0e2db8210d880662312afda62994f2f5a515 --- /dev/null +++ b/TtE1T4oBgHgl3EQfugUk/content/tmp_files/2301.03388v1.pdf.txt @@ -0,0 +1,720 @@ +Runaway signals: +Exaggerated displays of commitment may result +from second-order signaling +Julien Lie-Panis*a,b,c and Jean-Louis Dessallesb +a Institut Jean Nicod, Departement d'etudes cognitives, Ecole normale superieure, +Universite PSL, EHESS, CNRS, 75005 Paris, France +b LTCI, Telecom Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France +c Universite de Paris, EURIP Graduate School for Interdisciplinary Research, 75004 +Paris, France +January 9, 2023 +Abstract +To demonstrate their commitment, for instance during wartime, mem­ +bers of a group will sometimes all engage in a ruinous display. +Such +widespread, high-cost signals are hard to reconcile with standard models +of signaling. For signals to be stable, they must honestly inform their +audience; and to be honest, their costs need deter least commited individ­ +uals. To account for the existence of uniform high-cost signals , we design +a simple game theory model. In our model, senders can engage in second­ +order signaling. They can pay a cost to express outrage at a non-sender. +In doing so, they draw attention to their own signal, and benefit from its +increased visibility. +Outrage is therefore +a +self-serving +behavior +performed at the expense of a target. +Using our model and a +simulation, we show that outrage can stabilize widespread signals and +can lead signal costs to run away. Second-order signaling such as outrage +may explain why groups sometimes demand displays of commitment +from all their members, and why these displays can entail extreme costs. +Keywords: costly signaling; commitment displays; ritual; game theory +* Corresponding author; Email: j liep©protonmail. com; ORCID: 0000-0001-7273-7704 +1 + +11 +Widespread, high-cost displays +Membership in human groups often involves ritual behaviors which appear ar- +bitrary and wasteful to the non-initiated, ranging from the embarrassment of +hazing and the time-constraints of religious practice to the emotional and phys- +ical scarring of certain rites or recruitment devices (Atran & Henrich, 2010; +Cimino, 2011; Densley, 2012; Sosis et al., 2007; Whitehouse & Lanman, 2014). +These behaviors have been explained as displays of prosocial commitment (Bul- +bulia & Sosis, 2011; Gambetta, 2009; Irons, 2001; Sosis, 2003). In accordance +with this explanation, individuals who expend more time and energy in ritual +activities are on average more generous towards other group members (Ruffle +& Sosis, 2006; Soler, 2012; Xygalatas et al., 2013), and are perceived as such +(Power, 2017; Purzycki & Arakchaa, 2013). +Yet, ritual displays differ from the way signals are traditionally understood in +a crucial manner; they involve most, if not all, of the members of a social group +(Gelfand et al., 2020). Widespread costly displays run counter to theoretical +expectations. When individuals all invest in the same signal ( e.g. an initiation +rite), the signal is dishonest (Gintis et al., 2001). If onlookers are unable to +distinguish between participants, the ritual is uninformative; in theory, it should +be abandoned. When individuals invest in different levels of signaling ( e.g. in a +lower-ordeal or higher-ordeal ritual, Xygalatas et al., 2013), the overall signal is +honest, but net costly for the least committed (Dessalles, 2014). If individuals +are unable to distinguish themselves from the bottom of the pack, they are +better off opting out of the display entirely. +Our proposal is that not sending a signal can sometimes expose to more +serious consequences than mere missed social opportunities. In certain contexts, +non-senders will be exploited by senders, who may chastise them to make their +own signal more visible. Widespread displays could then emerge out of a single +motivation: advertising one's prosocial commitment, by any means necessary. +r------------------, +sender +outrage +l +-- infers ~ +~-----~ +receiver +target +Figure 1: Outrage as a second-order signal. A sender can express outrage at a target who +does not invest in the signal. When outrage is honest, receivers can infer that the sender has +invested in the signal, even without having observed the sender's behavior directly. Outrage +makes the sender's signal more visible. As a side-effect, the target is harmed. +More specifically, we argue that widespread costly displays can be propped +up by moral outrage. Outrage can be a credible signal of moral behavior. To +1 + +十targetinfer the moral quality of our partners, we sometimes use their propensity to +express outrage (Jordan et al., 2017). Conversely, to advertise our investment +in desirable behavior, we sometimes express outrage against those who unam­ +biguously display undesirable behavior (Jordan & Rand, 2019); or even against +those whose morality is merely ambiguous (Jordan & Kteily, 2022). +In the context of commitment displays, outrage can be thought of as a +second-order signal - a signal about (the absence of) a signal (see Figure 1). +When we publicly comment on others' perceived lack of investment in a display, +we indirectly broadcast our own investment. In doing so, we increase others' +incentive to display, and lay the groundwork for widespread signaling. To em­ +phasize our own observance, we may for instance draw attention to those who +secretly eat during a fast, and whose transgression may have otherwise gone +unnoticed. +In this paper, we formally explore this hypothesis. We introduce a model, +which we dub the 'signal runaway game', in which individuals may engage in +first- and second-order signaling. Using our model and a computer simulation, +we show that widespread costly displays may emerge endogenously, out of the +motivation to advertise a socially desirable quality. We show that outrage can +enable a step-by-step runaway process, leading individuals to gradually adopt +costlier displays of commitment. Below, we outline the main elements of our +model and simulation, and the main steps leading to our results (for a full +characterization, see the Supplementary Information). +2 +The signal runaway game +2.1 Baseline model +Commitment displays can be studied using the multi-player model introduced by +Gintis, Smith and Bowles (2001), which we adapt. This type of model inevitably +leads to a separating equilibrium in which only high-quality individuals pay the +cost to send the signal. +We consider a large population where individuals are characterized by an un­ +observable quality q, which may take any value between 0 and 1, the minimum +and maximum possible qualities. Individuals alternate between two roles, that +of Signaler and Receiver. Signalers may pay cost c1 ( q) to send, depending on +their quality q. Signaling is cheaper for high quality individuals: c1 is a strictly +decreasing continuous function of individual quality q which takes positive val­ +ues. In the present context, individuals of higher quality can be thought of as +individuals who are more committed to the group and/or its moral values, and +whose commitment translates into an increased ability or willingness to invest +in ritual signaling ( e.g. because they expect to stay in the community for longer, +and extract more social benefits from said community; Brusse, 2020). +Receivers choose a Signaler to follow. A signaling equilibrium occurs when +they condition their choice on the signal; i.e. when Receivers pay to monitor +others' signals, and follow a sender at random (rather than any individual). +Receivers who monitor observe Signalers' behavior with probability p1 < 1. +Each time Signalers are chosen by a Receiver, they gain s. +Competition for followers leads to a separating equilibrium in which indi­ +viduals send the signal when their quality is higher than a certain threshold {j, +2 + +and do not send when it is lower. Let 1r(q) = P(q > q) be the fraction of indi- +viduals who send the signal. On average, Receivers observe a fraction p 1 x 1r(q) +of senders, and choose one to follow. Signalers either do not send, and obtain +nothing; or send, and are observed with probability p 1 . On average, a Signaler +recruits ---1!1_(, = -( +1 ') followers, earning s for each follower. q is the quality at +J)1 7r q +7r q +which benefit and cost of signaling are equal, i.e. verifies: +(1) +For signaling to be stable, it must be honest. We obtain an evolutionar- +ily stable strategy (ESS; Maynard Smith & Price, 1973) as long as Receivers +benefit from following higher quality Signalers (q > q) rather than lower qual- +ity signalers (q ::S q), and that benefit exceeds the cost of monitoring. When +monitoring is cheap, it is sufficient that the signal be prohibitively costly for +individuals of minimum quality q = 0, i.e. that we have: c1 (0) > 7ri'o) = s. In +contrast, widespread signaling (q = 0) is always uninformative, and can never +be stable. +2.2 +Outrage may sustain widespread costly signaling +The signal runaway game occurs when we introduce outrage into the previ- +ous baseline model. Signalers who send the signal may now pay c2 to express +outrage. Individuals who do not send cannot subsequently express outrage in +our model; by assumption, outrage is a reliable indicator of signaling - +a re- +liable second-order signal. We assume outrage increases the visibility of one's +first-order displays. A sender who expresses outrage is observed with increased +probability P2 (P1 < P2 < 1). +Outrage is aimed in priority at non-senders in our model. When a Signaler +expresses outrage, a target is selected at random among those individuals the +Signaler observes opting out of the signal. That target is harmed, and loses +h. A specific case occurs when the entire population sends the signal, and such +targets are absent. In this case, we assume that outraged individuals may target +ambiguous senders, i.e. individuals they do not observe sending the signal. +Signalers now compete to attract followers and evade others' outrage. Simi- +larly to before, let us consider the case where Receivers condition on the signal, +and Signalers send and express outrage when their quality exceeds a threshold +q > 0. As before, non-senders do not gain any followers, and miss out on aver- +age benefit 7r(ii). In addition, they risk becoming a target for the fraction 1r( q) +of outraged senders, with probability p 1 . Outraged senders target one of the +p1 x (1 - 1r( q)) percent of individuals they observe opting out of the signal. Di- +viding, we deduce that non-senders lose on average: 1:S ) be the fraction of indi- +of senders. and choose one to follow. Signalers either do not send. and obtain +nothing; or send, and are observed with probability pi. On average, a Signaler +recruits p + -( followers, earning s for each follower. Q is the quality at +which benefit and cost of signaling are equal, i.e. verifies: +(1) +C1(Q) : +元() +For signaling to be stable, it must be honest. We obtain an evolutionar- +ily stable strategy (ESS; Maynard Smith & Price, 1973) as long as Receivers +ity signalers (q ≤ q), and that +t benefit exceeds the cost of monitoring. When +monitoring is cheap, it is sufficient that the signal be prohibitively costly for +contrast, widespread signaling (q = O) is always uninformative, and can never +be stable. +2.2 Outrage may sustain widespread costly signaling +ous baseline model. Signalers who send the signal may now pay C2 to express +a re +liable second-order signal. We assume outrage increases the visibility of one's +first-order displays. A sender who expresses outrage is observed with increased +probability p2 (p1 < p2 < 1). +Outrage is aimed in priority at non-senders in our model. When a Signaler +expresses outrage, a target is selected at random among those individuals the +Signaler observes opting out of the signal. That target is harmed, and loses +h. A specific case occurs when the entire population sends the signal, and such +targets are absent. In this case, we assume that outraged individuals may target +ambiguous senders, i.e. individuals they do not observe sending the signal. +Signalers now compete to attract followers and evade others' outrage. Simi- +larly to before, let us consider the case where Receivers + condition on the signal +and Signalers send and express outrage when their quality exceeds a threshold +q > O. As before, non-senders do not gain any followers, and miss out on aver +of outraged senders, with probability pi. Outraged senders target one of the +pi x (1 - π(Q)) percent of individuals they observe opting out of the signal. Di- +) × h. q is the quality +at which total benefit and cost of signaling are equal, and now verifies: +π(α)h +(2) +Ci(q) ＋ C2 = +1- π(Q) +π(Q) +Outrage perturbs the typical signaling equilibrium, by increasing the incen- +tive to signal. Sending the first- and second-order signal allows individuals to +more individuals are pushed to send (the minimum bar q decreases). +3Without outrage +·····--------- +···--- · -- : -,-:. . +------------------------------------· i +Equilibrium: +Lowest quality +(q=O) +Ben fit 1r(q) +Cost c1(q) +Highest quality +(q=1) +Do not send the signal +Send the signal +With outrage +(i) Equilibrium +when h small: +(ii) Equilibrium +when h large: +·····-···---- +Lowest quality +(q=O) +Do not send +V +' +. +--}.:_------~::,/ - +·--··· +Send the signal +Send the signal +B +fl +8 +tr( p2 ; targets lose h. Similarly to +before, individuals are pushed to increase their investment in the signal (they +are prevented from decreasing their investment to O for the same reasons as +before). We expect full escalation to the new signal level when: +~ +i 60 +,,_ ________ _____, +0 +20 +40 +60 +Impact of being followed (s) +(a) Average level of signaling +1- +Average investment in first-order signaling I +_ +Averageprobab 1l1tyof outrage +(b) Step-by-step runaway +(5) +Figure 4: Average investment in the signal after many rounds (left), and step-by-step runaway +(right), for four evenly spaced levels of signaling. When harm hand benefit of being followed +s are sufficiently high, agents learn to invest in the highest level of first-order signaling, and +in high levels of second-order signaling (high probability of expressing outrage). +Outrage may lead a population to adopt exaggerated displays. We relaunch +our simulation with several evenly spaced levels of signaling (proportional costs). +Agents may now express outrage at non-senders and lower-level senders (whom +they still observe directly and indirectly). When h and s are sufficiently large, +outrage enables a step-by-step runaway process: individuals gradually learn to +invest in the highest level of signaling (see Figure 4). This is in accordance with +equation (5); when levels are evenly spaced, the marginal cost of signaling one +level above is constant from one level to the next, and signal escalation may +continue indefinitely. In reality, we expect marginal costs to increase at each +step to infinity, as individuals are forced to miss out on increasingly important +opportunities. The process will necessarily come to a halt. Eventually, high +quality individuals will not benefit from creating a costlier display ( and adver- +tising it at the expense of others), and low quality individuals will prefer not to +increase their investment, even if this means appearing relatively uncommitted. +3 +Discussion +This paper offers a proof of concept for the existence of widespread costly dis- +plays. Our model is agnostic about any function the emerging behavior may +serve at the level of the collective ( e.g. encouraging group cohesion or coopera- +tion; Atran & Henrich, 2010; Bulbulia & Sosis, 2011; Cimino, 2011; Durkheim, +6 + +40 +60 +801.0 +0.75 +0.50 +0.25 +Average investment in first-order signaling +Averageprobabilityof outrage +0.02008; Gambetta, 2009; Irons, 2001; Whitehouse & Lanman, 2014; Xygalatas +et al., 2013). Widespread signals are explained at the individual level. Outrage +benefits senders, by making their signal easier to spot. We show that, under +certain conditions, outrage is sufficient to generate widespread signaling, and +escalating costs. +We consider signals which take discrete values. Our model applies for dis- +plays of commitment which categorize individuals ( e.g. into participants of a +high-ordeal ritual, of a low-ordeal ritual, and non-participants; Xygalatas et al., +2013), not when evaluations are based on a more continuous metric ( e.g. time +given to community work). This is a feature of the model, and not a bug. +Though continuously-valued signals may emerge and remain stable (Grafen, +1990), outrage requires clear-cut comparisons. In some cases, committed indi- +viduals could design discrete displays precisely for that purpose. +We assume however that outrage is honest, in our model and simulation. +Outrage is generally believed to be honest when hypocrites suffer sufficient re- +taliatory costs; yet, retaliation against hypocrites is subject to much variation +(Sommers & Jordan, 2022). Further research should investigate the conditions +under which outrage is more likely to be honest, and/or treated as such by +onlookers; ensuring that it can function as a second-order signal. +Our model may help explain mandatory displays of commitment, such as +rites of passage (see also: Cimino, 2011; Densley, 2012; Gambetta, 2009; Iannac- +cone, 1992). Outrage can create a positive feedback loop, and sustain uniform, +and therefore uninformative, displays. The resulting behavior is a specific type +of norm. In general, norms can emerge from a variety of positive feedback loops, +such as those created by social punishment or benchmark effects (Young, 2015). +In our case, uniform displays arise endogenously, from the motivation to adver- +tise one's prosocial commitment to group members, via first- and second-order +signaling (we do not need to assume non-senders are punished). +Our model may also help explain exaggerated displays of commitment, e.g. +during wartime (see also: Sosis et al., 2007; Whitehouse, 2018). Times of crisis +tend to favor expression of commitment over others (Hahl et al., 2018), and may +provide the initial push enabling signal runaway. In extreme cases, the system is +expected to stop at extreme levels of signaling and outrage, pushing individuals +to ever greater lengths to avoid appearing uncommitted. A similar logic may be +at play with witch hunts or other collective crazes which follow a self-fulfilling +pattern (Lotto, 1994). +The present model is kept minimal. It needs to be completed to explain +why many widespread signals remain stable without reaching extreme values, +or why they may deescalate. Depending on the context, individuals may look +for commitment to other groups or values. Signals and non-signals can change +meaning ( e.g. pacifism instead of cowardice, or closed-mindedness instead of +dedication to the group). +Methods +Static analysis. To explore the conditions under which outrage may evolve, +and lead to widespread signaling, we characterize all evolutionarily stable strat- +egy (ESS) of the signal runaway game (for all details, see Supplementary Infor- +mation). +7 + +2008; Gambetta, 2009; Irons, 2001; Whitehouse & Lanman, 2014; Xygalatas +et al., 2013). Widespread signals are explained at the individual level. Outrage +benefits senders, by making their signal easier to spot. We show that, under +certain conditions, outrage is suficient to generate widespread signaling, and +escalating costs. +We consider signals which take discrete values. Our model applies for dis- +plays of commitment which categorize individuals (e.g. into participants of a +high-ordeal ritual, of a low-ordeal ritual, and non-participants; Xygalatas et al., +2013), not when evaluations are based on a more continuous metric (e.g. time +given to community work). This is a feature of the model, and not a bug. +Though continuously-valued signals may emerge and remain stable (Grafen, +1990), outrage requires clear-cut comparisons. In some cases, committed indi- +We assume however that outrage is honest, in our model and simulation. +Outrage is generally believed to be honest when hypocrites suffer sufficient re +taliatory costs; yet, retaliation against hypocrites is subject to much variation +(Sommers & Jordan, 2022). Further research should investigate the conditions +under which outrage is more likely to be honest, and/or treated as such by +onlookers; ensuring that it can function as a second-order signal. +Our model may help explain mandatory displays of commitment, such as +rites of passage (see also: Cimino, 2011; Densley, 2012; Gambetta, 2009; Iannac- +and therefore uninformative, displays. The resulting behavior is a specific type +of norm. In general, norms can emerge from a variety of positive feedback loops, +In our case, uniform displays arise endogenously, from the motivation to adver- +tise one's prosocial commitment to group members, via first- and second-order + signaling (we do not need to assume non-senders are punished). +Our model may also help explain exaggerated displays of commitment, e.g. +during wartime (see also: Sosis et al., 2007; Whitehouse, 2018). Times of crisis +tend to favor expression of commitment over others (Hahl et al., 2018), and may +provide the initial push enabling signal runaway. In extreme cases, the system is +expected to stop at extreme levels of signaling and outrage, pushing individuals +to ever greater lengths to avoid appearing uncommitted. A similar logic may be +at play with witch hunts or other collective crazes which follow a self-fulfilling +pattern (Lotto, 1994). +The present model is kept minimal. It needs to be completed to explain +why many widespread signals remain stable without reaching extreme values, +for commitment to other groups or values. Signals and non-signals can change +meaning (e.g. pacifism instead of cowardice, or closed-mindedness instead of +dedication to the group). +Methods +Static analysis. To explore the conditions under which outrage may evolve, +and lead to widespread signaling, we characterize all evolutionarily stable strat- +egy (ESS) of the signal runaway game (for all details, see Supplementary Infor- +mation). +7Evolutionary simulations. To explore the conditions under which outrage +may lead to widespread signaling and/or exaggerated signaling costs, and the +evolution of strategies in a more realistic setting, we implement the model into +an agent-based simulation (with one or several available signal levels). In the +simulation, agents are characterized by a fixed quality, and three flexible fea- +tures. They interact locally, based on their feature values at a given point in +time. They learn optimal feature values by exploring the feature space, based +on the outcome of these interactions. +The simulation is written in Python and based on the Evolife platform (for +all details, see Supplementary Information). All programs are open source and +available from the companion website, along with instructions for installation, +figures, and chosen parameter values. +Acknowledgements +We thank A. Sijilmassi for feedback on a early version of the manuscript. This +research was supported by funding from the EURIP Graduate School for Inter- +disciplinary Reseach, and from the Agence Nationale pour la Recherche (ANR- +17-EURE-0017, ANR-10-IDEX-0001-02). +References +Atran, S., & Henrich, J. (2010). The Evolution of Religion: How Cognitive By- +Products, Adaptive Learning Heuristics, Ritual Displays, and Group +Competition Generate Deep Commitments to Prosocial Religions. Bi- +ological Theory, 5(1), 18-30. https://doi.org/10.1162/BIOT_a_00018 +Brusse, C. (2020). Signaling theories of religion: Models and explanation. Re- +ligion, Brain &J Behavior, 10(3), 272-291. https://doi.org/10.1080/ +2153599X.2019.1678514 +Bulbulia, J., & Sosis, R. (2011). 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Discerning devotion: Testing the signaling theory of reli- +gion. Evolution and Human Behavior, 38(1), 82-91. https://doi.org/ +10.1016/j.evolhumbehav.2016.07.003 +Purzycki, B. G., & Arakchaa, T. (2013). Ritual Behavior and Trust in the Tyva +Republic. Current Anthropology, 54(3), 381-388. https://doi.org/10. +1086/670526 +Ruffle, B. J., & Sosis, R. (2006). Cooperation and the in-group-out-group bias: +A field test on Israeli kibbutz members and city residents. Journal of +Economic Behavior &f Organization, 60(2), 147-163. https://doi.org/ +10.1016/j.jebo.2004.07.007 +Soler, M. (2012). Costly signaling, ritual and cooperation: Evidence from Can- +domble, an Afro-Brazilian religion. Evolution and Human Behavior, +33(4), 346-356. https://doi.org/10.1016/j.evolhumbehav.2011.11.004 +Sommers, R., & .Jordan, J. (2022). When does moral engagement risk triggering +a hypocrisy penalty? https://doi.org/10.31234/osf.io/w23ec +9 + +Gelfand, M. J., Caluori, N., Jackson, J. C., & Taylor, M. K. (2020). The cultural +evolutionary trade-off of ritualistic synchrony. Philosophical Transac. +tions of the Royal Society B: Biological Sciences, 375(1805), 20190432. +https://doi.org/10.1098/rstb.2019.0432 +Gintis, H., Smith, E. A., & Bowles, S. (2001). Costly Signaling and Cooperation. +Journal of Theoretical Biology, 213(1), 103-119. https: / /doi.org/10. +1006/jt bi.2001.2406 +Grafen, A. (1990). Biological signals as handicaps. Journal of Theoretical Biol +8-88008(90)861-200910101/81010//:s+* *9-1g()1“60 +Hahl, O., Kim, M., & Zuckerman Sivan, E. W. (2018). The Authentic Appeal +of the Lying Demagogue: Proclaiming the Deeper Truth about Political +Illegitimacy. American Sociological Review, 88(1), 1-33. https: / /doi. +0rg/10.1177/0003122417749632 +Iannaccone, L. R. (1992). Sacrifice and Stigma: Reducing Free-riding in Cults +Communes, and Other Collectives. Journal of Political Economy, 100(2), +88192/980101/81010//:sd4 *162-12 +Irons, W. (2001). Religion as a hard-to-fake sign of commitment. In R. M. Nesse +(Ed.), Evolution and the Capacity for Commitment. +Jordan, J. J., & Kteily, N. S. (2022). People punish moral transgressions for rep +utational gain, even when they personally question whether punishment +is merited. Available at PsyArXiv. +Jordan, J. J., & Rand, D. G. (2019). Signaling when no one is watching: A +reputation heuristics account of outrage and punishment in one-shot +anonymous interactions. Journal of Personality and Social Psychology, +No Pagination Specified-No Pagination Specified. https:/ /doi.org/10. +1037/pspi0000186 +Jordan, J. J., Sommers, R., Bloom, P., & Rand, D. G. (2ol7). Why Do We Hate +Science. 28(3), 356-368. https://doi.0rg/10.1177/095679761668577 +Lotto, D. (1994). On Witches and Witch Hunts: Ritual and Satanic Cult Abuse. +The Journal of Psychohistory; New York, 21(4), 373-396. +246(5427), 15-18. https:/ /doi.0rg/10.1038/246015a0 +Power, E. A. (2017). Discerning devotion: Testing the signaling theory of reli- +gion. Evolution and Human Behavior, 38(1), 82-91. https:/ /doi.org/ +10.1016/j.evolhumbehav.2016.07.003 +Purzycki, B. G., & Arakchaa, T. (2013). Ritual Behavior and Trust in the Tyva +Republic. Current Anthropology, 54 (3), 381-388. https: / / doi.org/10. +1086/670526 +Rufle, B. J., & Sosis, R. (2006). Cooperation and the in-group-out-group bias +Economic Behavior 3 Organization, 60(2), 147-163. https: / /doi.org/ +10.1016/j.jebo.2004.07.007 +Soler, M. (2012). Costly signaling, ritual and cooperation: Evidence from Can- +domblé, an Afro-Brazilian religion. Evolution and Human Behavior +33(4), 346-356. https:/ /doi.org/10.1016/j.evolhumbehav.2011.11.004 +Sommers, R., & Jordan, J. (2022). When does moral engagement risk triggering +a hypocrisy penalty? https:/ /doi.org/10.31234/osf.io/w23ec +9Sosis, R. (2003). Why aren't we all hutterites?: Costly signaling theory and +religious behavior. Human Nature, 14 (2), 91-127. https://doi.org/10. +1007/s12110-003-1000-6 +Sosis, R., Kress, H. C., & Boster, J. S. (2007). Scars for war: Evaluating alter- +native signaling explanations for cross-cultural variance in ritual costs. +Evolution and Human Behavior, 28(4), 234-247. https://doi.org/10. +1016/j.evolhumbehav.2007.02.007 +Whitehouse, H. (2018). Dying for the group: Towards a general theory of ex- +treme self-sacrifice. Behavioral and Brain Sciences, 41. https://doi.org/ +10.1017 /S0140525X18000249 +Whitehouse, H., & Lanman, J. A. (2014). The Ties That Bind Us: Ritual, +Fusion, and Identification. Current Anthropology, 55(6), 674-695. https: +//doi.org/10.1086/678698 +Xygalatas, D., Mitkidis, P., Fischer, R., Reddish, P., Skewes, J., Geertz, A. W., +Roepstorff, A., & Bulbulia, J. (2013). Extreme Rituals Promote Proso- +ciality. Psychological Science, 24 (8), 1602-1605. https: / / doi. org/ 10. +1177/0956797612472910 +Young, H. P. (2015). The Evolution of Social Norms. Annual Review of Eco- +nomics, 'l(l), 359-387. https: / / doi.org/ 10.1146 / annurev-economics- +080614-115322 +10 + +Sosis, R. (2003). Why aren't we all hutterites?: Costly signaling theory and +religious behavior. Human Nature, 14(2), 91-127. https:/ /doi.org/10. +1007/s12110-003-1000-6 +Sosis, R., Kress, H. C., & Boster, J. S. (2007). Scars for war: Evaluating alter- +native signaling explanations for cross-cultural variance in ritual costs. +Evolution and Human Behavior, 28(4), 234-247. https: / /doi.org /10. +1016/j.evolhumbehav.2007.02.007 +e e e () +treme self-sacrifice. Behavioral and Brain Sciences, 41. https:/ /doi.org/ +10.1017/S0140525X18000249 +Fusion, and Identification. Current Anthropology, 55(6), 674-695. https: +//doi.0rg/10.1086/678698 +Xygalatas, D., Mitkidis, P., Fischer, R., Reddish, P., Skewes, J., Geertz, A. W. +Roepstorff, A., & Bulbulia, J. (20l3). Extreme Rituals Promote Proso- +ciality. Psychological Science, 24(8), 1602-1605. https://doi.0rg/ 10. +1177/0956797612472910 +Young, H. P. (2015). The Evolution of Social Norms. Annual Review of Eco- +nomics, 7(1), 359-387. https: / /doi.org/10.1146/annurev-economics +080614-115322 +10 \ No newline at end of file diff --git a/TtE1T4oBgHgl3EQfugUk/content/tmp_files/load_file.txt b/TtE1T4oBgHgl3EQfugUk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c99d652d8c6e9c113f93e0c7f4ca303cc49ccc0 --- /dev/null +++ b/TtE1T4oBgHgl3EQfugUk/content/tmp_files/load_file.txt @@ -0,0 +1,900 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf,len=899 +page_content='Runaway signals: Exaggerated displays of commitment may result from second-order signaling Julien Lie-Panis*a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='c and Jean-Louis Dessallesb a Institut Jean Nicod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" Departement d'etudes cognitives," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Ecole normale superieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Universite PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' EHESS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' 75004 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' France January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' 2023 Abstract To demonstrate their commitment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' for instance during wartime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' mem\xad bers of a group will sometimes all engage in a ruinous display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Such widespread, high-cost signals are hard to reconcile with standard models of signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' For signals to be stable, they must honestly inform their audience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' and to be honest, their costs need deter least commited individ\xad uals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' To account for the existence of uniform high-cost signals , we design a simple game theory model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In our model, senders can engage in second\xad order signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' They can pay a cost to express outrage at a non-sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In doing so, they draw attention to their own signal, and benefit from its increased visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Outrage is therefore a self-serving behavior performed at the expense of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Using our model and a simulation, we show that outrage can stabilize widespread signals and can lead signal costs to run away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Second-order signaling such as outrage may explain why groups sometimes demand displays of commitment from all their members, and why these displays can entail extreme costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Keywords: costly signaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' commitment displays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' ritual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' game theory Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Email: j liep©protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' ORCID: 0000-0001-7273-7704 1 11 Widespread, high-cost displays Membership in human groups often involves ritual behaviors which appear ar- bitrary and wasteful to the non-initiated, ranging from the embarrassment of hazing and the time-constraints of religious practice to the emotional and phys- ical scarring of certain rites or recruitment devices (Atran & Henrich, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Cimino, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Densley, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Sosis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Whitehouse & Lanman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' These behaviors have been explained as displays of prosocial commitment (Bul- bulia & Sosis, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Gambetta, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Irons, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Sosis, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In accordance with this explanation, individuals who expend more time and energy in ritual activities are on average more generous towards other group members (Ruffle & Sosis, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Soler, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Xygalatas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2013), and are perceived as such (Power, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Purzycki & Arakchaa, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Yet, ritual displays differ from the way signals are traditionally understood in a crucial manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' they involve most, if not all, of the members of a social group (Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Widespread costly displays run counter to theoretical expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' When individuals all invest in the same signal ( e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' an initiation rite), the signal is dishonest (Gintis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' If onlookers are unable to distinguish between participants, the ritual is uninformative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' in theory, it should be abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' When individuals invest in different levels of signaling ( e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' in a lower-ordeal or higher-ordeal ritual, Xygalatas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2013), the overall signal is honest, but net costly for the least committed (Dessalles, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' If individuals are unable to distinguish themselves from the bottom of the pack, they are better off opting out of the display entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Our proposal is that not sending a signal can sometimes expose to more serious consequences than mere missed social opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In certain contexts, non-senders will be exploited by senders, who may chastise them to make their own signal more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" Widespread displays could then emerge out of a single motivation: advertising one's prosocial commitment, by any means necessary." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' r------------------, sender outrage l -- infers ~ ~-----~ receiver target Figure 1: Outrage as a second-order signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' A sender can express outrage at a target who does not invest in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" When outrage is honest, receivers can infer that the sender has invested in the signal, even without having observed the sender's behavior directly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" Outrage makes the sender's signal more visible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' As a side-effect, the target is harmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' More specifically, we argue that widespread costly displays can be propped up by moral outrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Outrage can be a credible signal of moral behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' To 1 十targetinfer the moral quality of our partners, we sometimes use their propensity to express outrage (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Conversely, to advertise our investment in desirable behavior, we sometimes express outrage against those who unam\xad biguously display undesirable behavior (Jordan & Rand, 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' or even against those whose morality is merely ambiguous (Jordan & Kteily, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In the context of commitment displays, outrage can be thought of as a second-order signal - a signal about (the absence of) a signal (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" When we publicly comment on others' perceived lack of investment in a display, we indirectly broadcast our own investment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" In doing so, we increase others' incentive to display, and lay the groundwork for widespread signaling." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' To em\xad phasize our own observance, we may for instance draw attention to those who secretly eat during a fast, and whose transgression may have otherwise gone unnoticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In this paper, we formally explore this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" We introduce a model, which we dub the 'signal runaway game', in which individuals may engage in first- and second-order signaling." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Using our model and a computer simulation, we show that widespread costly displays may emerge endogenously, out of the motivation to advertise a socially desirable quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' We show that outrage can enable a step-by-step runaway process, leading individuals to gradually adopt costlier displays of commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Below, we outline the main elements of our model and simulation, and the main steps leading to our results (for a full characterization, see the Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' 2 The signal runaway game 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='1 Baseline model Commitment displays can be studied using the multi-player model introduced by Gintis, Smith and Bowles (2001), which we adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' This type of model inevitably leads to a separating equilibrium in which only high-quality individuals pay the cost to send the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' We consider a large population where individuals are characterized by an un\xad observable quality q, which may take any value between 0 and 1, the minimum and maximum possible qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Individuals alternate between two roles, that of Signaler and Receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Signalers may pay cost c1 ( q) to send, depending on their quality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Signaling is cheaper for high quality individuals: c1 is a strictly decreasing continuous function of individual quality q which takes positive val\xad ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In the present context, individuals of higher quality can be thought of as individuals who are more committed to the group and/or its moral values, and whose commitment translates into an increased ability or willingness to invest in ritual signaling ( e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' because they expect to stay in the community for longer, and extract more social benefits from said community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Brusse, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Receivers choose a Signaler to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' A signaling equilibrium occurs when they condition their choice on the signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" when Receivers pay to monitor others' signals, and follow a sender at random (rather than any individual)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" Receivers who monitor observe Signalers' behavior with probability p1 < 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Each time Signalers are chosen by a Receiver, they gain s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Competition for followers leads to a separating equilibrium in which indi\xad viduals send the signal when their quality is higher than a certain threshold {j, 2 and do not send when it is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Let 1r(q) = P(q > q) be the fraction of indi- viduals who send the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' On average, Receivers observe a fraction p 1 x 1r(q) of senders, and choose one to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Signalers either do not send, and obtain nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' or send, and are observed with probability p 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' On average, a Signaler recruits ---1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content="1_(, = -( 1 ') followers, earning s for each follower." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' q is the quality at J)1 7r q 7r q which benefit and cost of signaling are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' verifies: (1) For signaling to be stable, it must be honest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' We obtain an evolutionar- ily stable strategy (ESS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Maynard Smith & Price, 1973) as long as Receivers benefit from following higher quality Signalers (q > q) rather than lower qual- ity signalers (q ::S q), and that benefit exceeds the cost of monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' When monitoring is cheap, it is sufficient that the signal be prohibitively costly for individuals of minimum quality q = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" that we have: c1 (0) > 7ri'o) = s." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In contrast, widespread signaling (q = 0) is always uninformative, and can never be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='2 Outrage may sustain widespread costly signaling The signal runaway game occurs when we introduce outrage into the previ- ous baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Signalers who send the signal may now pay c2 to express outrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Individuals who do not send cannot subsequently express outrage in our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' by assumption, outrage is a reliable indicator of signaling - a re- liable second-order signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" We assume outrage increases the visibility of one's first-order displays." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' A sender who expresses outrage is observed with increased probability P2 (P1 < P2 < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Outrage is aimed in priority at non-senders in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' When a Signaler expresses outrage, a target is selected at random among those individuals the Signaler observes opting out of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' That target is harmed, and loses h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' A specific case occurs when the entire population sends the signal, and such targets are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In this case, we assume that outraged individuals may target ambiguous senders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' individuals they do not observe sending the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=" Signalers now compete to attract followers and evade others' outrage." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Simi- larly to before, let us consider the case where Receivers condition on the signal, and Signalers send and express outrage when their quality exceeds a threshold q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' As before, non-senders do not gain any followers, and miss out on aver- age benefit 7r(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' In addition, they risk becoming a target for the fraction 1r( q) of outraged senders, with probability p 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Outraged senders target one of the p1 x (1 - 1r( q)) percent of individuals they observe opting out of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE1T4oBgHgl3EQfugUk/content/2301.03388v1.pdf'} +page_content=' Di- viding, we deduce that non-senders lose on average: 1:S d. The commutator of diagrams in the circle at (s, d) +with H0 generates the diagrams in the crosses at (s ± 1, d) +and (s, d ± 1), indicated by the solid arrow tip. +The dia- +grams in crosses are to be canceled with the contribution from +� +Qj−1 +k +, Hint +� +(Fig. 3). +solid arrow in the plane indicates the commutator with +H0 in Fig. 2, 3. The vertical dotted arrow in Fig. 3 in- +dicates the commutator with Hint. The diagrams at the +crosses are to be canceled. +We give an example of commutator of land with H0 +and Hint: +� +I , H0 +� += I I +2 − I I −2 I +I + +− I +I and +� +I , Hint +� += I − I . We note that the commutator of +connected diagrams and H also generates a disconnected +diagram with a hole or a disconnecting gap. The details +of the commutation relation of diagrams with H are given +in [49]. We can construct Qj +k recursively by calculating +the cancellation at the crosses in Fig. 2, 3. +Exact expressions.— For the dependence of the co- +efficients, we introduce the list of a diagram Ψ by +λΨ = {λ0 ≡ λL; λ1, . . . , λw; λw+1 ≡ λR} where λL(R) is +the length of the leftmost(rightmost) coast of Ψ and +λi + 1(1 ≤ i ≤ w ≡ lΨ − 2) is the length of the i-th +inner coast from the left. We give a example of a con- +nected diagram Ψ of a list λΨ = {λL; λ1, λ2, λ3, λ4; λR}: +I I +I I I I I I I I +I I +I I I I +I I I I I +λ1+1 +λ4+1 +λR +λL +λ2+1 +λ3+1 +, +where Ψ is the (24, 6) diagram and the gap number is +gΨ = 2 and land number is lΨ = 6. λΨ = {2; 1, 3, 2, 2; 3} +and the length of the coasts are depicted by the arrows, +and the gap is indicated by the teal shaded area. +We show the exact expression of Qk. Q0 +k is the (k, 0) +diagram: Q0 +k = I +I + ↕. For Qj +k(j ≥ 1), we obtain the +following result. +Theorem 1. For j ≥ 1, +Qj +k = +⌊ k−1−j +2 +⌋ +� +n=0 +⌊ k−1−j +2 +⌋−n +� +d=0 +⌊ j−1 +2 ⌋ +� +m=0 +d +� +g=0 +× +� +Ψ∈Sk,j,m +n,d,g +(−1)n+m+gCj,m +n,d (λΨ) Ψ, +(10) +where Sk,j,m +n,d,g is the set of (k − j − 2n − d, d) connected +diagrams satisfying lΨ = j + 1 − 2m and gΨ = g. +Cj,m +n,d (λ = {λ0 . . . λw+1}) ∈ Z>0 is invariant with the per- +mutation of λi(1 ≤ i ≤ w ≡ j −1−2m), and with the +exchange of λL and λR, and with the replacement of λa +with min(λa, n) for 0 ≤ a ≤ w + 1. +We note that the freedom to add Qk′ +0 and Cj,m +n,d=0 ({−1; λ, . . .}) ≡ Cj,m +n−1,1 ({0; λ−1, . . .}) + +Cj−1,m +n,0 +({λ+1; . . .}). Cj,m +n,d (λ) = 0 for λi < 0(1 ≤ i ≤ w) +or j−1−2m < 0 or n < 0 or m < 0. Cj=1,m=0 +n,d +(λ) = 1. +We +obtained +the +general +explicit +expressions +of +Cj,m +n,d (λ) for some cases. +For n = 0 and m = 0 case, +Qj=0 +6 +Qj=1 +6 +Qj=2 +6 +Qj=3 +6 +Qj=4 +6 +Qj=5 +6 +Support +Double +(6,0) +(5,0) +(4,1) +(1,0) +(3,0) +(3,2) +(2,1) +ACZnich +VHLSsNAFD2Nr1q1rYouCmWiqsyEaniqujGZR/2Ab +WUJI41NE1CkhZq8QcEt3bhSkFE/Aw3/oCL/oHisoI +bF96mAdGi3mFmzpy586ZGdnUVNthrOsTRkbHxif +8k4Gp6ZlgKDw7l7eNhqXwnGJohlWUJZtrqs5zjupo +vGhaXKrLGi/Itd3+fqHJLVs19H2nZfJyXarq6pGqS +A5R2XQlUQlHWZy5ERkGogei8CJlhG9xgEMYUNBAH +Rw6HMIaJNjUShDBYBJXRps4i5Dq7nOcIkDaBmVxyp +CIrdFYpVXJY3Va92varlqhUzTqFikjiLEndsd67JH +dsxf28Wutluj76VFszQcrMSOlvKv+rqtPs4Ph +L9adnB0fYcr2q5N10mf4tlIG+edLpZbczsfYqu2av +5P+KdkD3UBvik3aZ65RIA+QPz53Mgvx4XE3Ex +vRFN7nhf4cyVrBG72JPaQo7OreIcF+j4noWgs +CAsDlIFn6eZx7cQIp8gJYqAQ6 +Structure of +FIG. 3. Structure of Qk for k = 6. Each plane represents the +structure of Qj +k in Fig. 2. The axis of support and double are +omitted. The commutator of diagrams in the circle at (s, d) +in Qj−1 +k +with Hint generates diagrams in crosses at (s, d) in +Qj +k, indicated by the tip of the vertical dotted arrow. The +diagrams generated in the crosses are to be canceled. +we have +Cj,m +n=0,d (λ) = +�j − 1 + d +m +� +− +�j − 1 + d +m − 1 +� +, +(12) +Cj,m=0 +n,d +(λ) = +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +i=1 +xi +� +, +(13) +where θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0. +Cj,m +n=0,d (λ) is independent of λ and Cj,m=0 +n,d +(λ) is inde- +pendent of λL,R, d. We note that Cj,m +n=0,d (λ) is the gen- +eralized Catalan number[22, 39, 50]. The expressions are +more complicated for the n, m > 0 case. For j = 3, m = 1 +case, we have +Cj=3,m=1 +n,d +(λL; λR) = +� +η=λL,λR +η +� +x=1 +�n + 3 − x +3 +� ++ 2 +�n + 4 +4 +� ++ (d − 1) +� +2 +�n + 3 +3 +� +− +�n + 2 +2 +�� +. +(14) +We can obtain the explicit expression of Q2 +k, Q3 +k for all k +from (13) and (14). +Qk commute with total magnetization and total num- +ber operator because lands commute with them, and Qk +is constructed of products of lands. +The fermionic form of the local conserved quantities Qk +can be easily obtained by Jordan Wigner transformation: +changing the lands in the diagram to the hopping term +of the same support, flavor, and mirror parity. +There are no k-support local conserved quantities other +than our Qk. This completeness is shown as follows: with +some normalization, we can prove the k-support operator +in Qk is Q0 +k, and once this (k, 0) diagram is determined, +the other components are uniquely determined to calcu- +late the cancellation at the crosses in Fig. 3 from top to +bottom. The detail of the proof is shown in [49]. +The SO(4) symmetry of Qk is ensured by its unique- +ness proved in this work and the SO(4) symmetry of the +transfer matrix [51]. +We may be able to confirm the +SO(4) symmetry of Qk directly by rewriting the fermionic +form of Qk to the SO(4) invariant form using the Clifford +algebra [51]. +Summary and outlook.— We have presented the exact +expression for the local conserved quantities of the one- +dimensional Hubbard model Qk = �jf +j=0 U jQj +k. In The- +orem 1, We proved the minimum operator basis for Qk +is the connected diagram, which is the product of the lo- +cal conserved density of XX chain (lands in our notation) +satisfying the condition (i)–(iii). This also means we have +affirmatively proved the conjecture of Grabowski and +Mathieu for the operator basis [22]. We found nontriv- +ial coefficients other than ±1 appear from Qj +k for k ≥ 6, +contrary to the conjecture of Grabowski and Mathieu for +the coefficients [22]. +The nontrivial coefficients can be calculated by the +recursion equation in Theorem 2 and have rich proper- +ties: they have the permutation invariance over the list of + +5 +the connected diagrams and some simple identities [49] +between them, which are helpful in the proof of Theo- +rem. 1. Some of them are the generalized Catalan num- +bers (12), which are also the coefficients in the local con- +served quantities of the Heisenberg chain [22, 38, 39, 50]. +Finding the general formulae of the coefficients is the re- +maining task, which may be some further generalization +of the Catalan number. The general expression of Q1 +k, +which was the terms in Qk linear in the coupling con- +stant U, was derived in [22], where all coefficients are +trivially ±1. We also obtained the general expressions of +Q2 +k and Q3 +k, where nontrivial coefficients (13) and (14) +appear. The fermionic form of Qk is also easily obtained. +Our result is valid in both finite systems and the ther- +modynamic limit L → ∞. +To our knowledge, this is the first time revealing the +structure of the local conserved quantities of the inte- +grable systems without the boost operator [52], i.e., with- +out a recursive way to construct them. Thus our work +opens new avenues to determine the structure of conser- +vation laws in the broader range of integrable systems, +including models without boost operator, such as the +models whose R-matrix do not have the difference prop- +erty like the Hubbard model [53–55], and open boundary +system [56] and long-range interaction models like the +Haldane-Shastry model [57–60]. +An integrable quantum field theory has also an infi- +nite number of local conservation laws [61–65], and is +often a scaling limit of an integrable lattice system [66– +70]. For example, there exists a particular scaling limit, +in which the Hubbard model at half-filling becomes the +SU(2) Thirring model [37, 71, 72], that is also solved by +the Bethe ansatz [73–75]. It may be interesting to derive +the local conserved quantity in the integrable quantum +field theory by taking the continuous limit of those in +lattice integrable models. +Acknowledgments.— The work was supported by Fore- +front Physics and Mathematics Program to Drive Trans- +formation (FoPM), a World-leading Innovative Gradu- +ate Study (WINGS) Program, the University of Tokyo, +and also supported by JSR Fellowship, the University +of Tokyo, and also supported by KAKENHI Grants No. +JP21J20321 from the Japan Society for the Promotion of +Science (JSPS). +∗ k.fukai@issp.u-tokyo.ac.jp +[1] H. Bethe, Zur Theorie der Metalle, Zeitschrift f¨ur Physik +71, 205 (1931). +[2] T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum +Newton’s cradle, Nature 440, 900 (2006). +[3] I. Bloch, J. Dalibard, and W. Zwerger, Many-body +physics with ultracold gases, Rev. Mod. Phys. 80, 885 +(2008). +[4] A. Polkovnikov, K. Sengupta, A. Silva, and M. 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B 443, 553 (1995). +[72] F. Woynarovich and P. Forg´acs, Scaling limit of the one- +dimensional attractive Hubbard model: The half-filled +band case, Nucl. Phys. B 498, 565 (1997). +[73] B. Berg and P. Weisz, Exact S-matrix of the chiral in- +variant SU(N) thirring model, Nucl. Phys. B 146, 205 +(1978). +[74] A. A. Belavin, Exact solution of the two-dimensional +model with asymptotic freedom, Phys. Lett. B 87, 117 +(1979). +[75] N. Andrei and J. H. Lowenstein, Diagonalization of the +Chiral-Invariant Gross-Neveu Hamiltonian, Phys. Rev. +Lett. 43, 1698 (1979). + +8 +Supplemental Material for “All Local Conserved Quantities of the +One-Dimensional Hubbard Model” +Kohei Fukai∗ +The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan +CONTENTS +S1. Commutation relations of diagram with hamiltonian +8 +S2. Identities of Cjm +nd (λ) +12 +S3. Proof of conservation law of Qk +14 +S4. Proof of identities of Cjm +nd (λ) +19 +S5. Proof of identities of Aσµ +i +35 +S6. Proof of the completeness of Qk +46 +S7. Examples of higher order Qk +59 +S8. Examples of coefficients Cjm +nd (λ) +66 +S1. COMMUTATION RELATIONS OF DIAGRAM WITH HAMILTONIAN +This section explains the commutation relation of a diagram and the hamiltonian. +The hamiltonian is H = +H0 + UHint where H0 = I + ↕ is the hopping term and Hint = is the interaction term. U is the coupling constant. +A land is symmetric under mirror reflection: the parity of a land is t(−1)s where t is the type of land, and s is the +support of the land (its length +1). A connected diagram is also symmetric under mirror reflection, and the product +of the parity of the land in it determines its parity. The commutator of connected diagrams with H0 or Hint changes +the mirror parity. +commutator with the hopping term H0 +The commutator of a land of flavor s from j-th site and H0 increases or decreases the length of the land: +� +l +s(j), H0 +� += +l + 1 +s(j) − +l + 1 +s(j − 1) + +l − 1 +s(j) − +l − 1 +s(j + 1) +(S1) += AdR,+ +0 +ψ + AdL,+ +0 +ψ + AdR,− +0 +ψ + AdL,− +0 +ψ, +(S2) +where ψ ≡ +l +s(j) is a land of + type, and the length of the land is increased or decreased by one. AdX,α +0 +ψ’s +in the second line are defined by the RHS of the first line, respectively. The type the lands in AdX,α +0 +ψ are the same as +ψ. AdR(L),+ +0 +ψ is the land made by adding +to the right(left) end of ψ. AdR(L),+ +0 +ψ is the land made by erasing the +on the right(left) end of ψ. The sign factor of AdX,α +0 +ψ is +(−) for X = R(L). In the case of ψ ≡ +l +s(j) , +which is a land of − type, the action of AdX,α +0 +ψ is defined in the same way. + +9 +The commutator of H0 with the land of length 1 and the land of length 0 is special case: +� +s(j), H0 +� += +s(j) − +s(j − 1), +(S3) +� +s(j), H0 +� += +s(j) − +s(j − 1) + 2 s(j) − 2 s(j − 1), +(S4) +� +s(j), H0 +� += +s(j) − +s(j − 1). +(S5) +Accordingly, the action of the special cases is defined by +AdX,− +0 +s(j) = AdX,− +0 +s(j) = 0, +(S6) +AdR,− +0 +s(j) = 2 s(j), +(S7) +AdL,− +0 +s(j) = −2 s(j − 1), +(S8) +where X = L, R. +The commutator of a diagram and H0 is now defined since a diagram is the product of lands and its site translational +summation. +[Ψ, H0] = +� +s=↑,↓ +Ns +� +r=1 +� +X=L,R +� +α=± +Ads,X,α +0,r +Ψ, +(S9) +where N↑(↓) is the number of the land on the upper(lower) row of Ψ and +Ads,X,α +0,r +Ψ := +L +� +i=1 +� +AdX,α +0,r Ψs,i +� +Ψ−s,i, +(S10) +AdX,α +0,r Ψs,i := +� +� +� +� +Ns +� +q=1 +q̸=r +Ψq +s,i +� +� +� +� AdX,α +0 +Ψr +s,i, +(S11) +where − ↑≡↓, − ↓≡↑. +For the proof of Theorem 1 and Theorem 2, we introduce the diagrammatic notation for the nonzero action of +Ads,X,α +0,r +Ψ: +I +I += Ads,L,+ +0,r +[ +I +I +] = +I +× (−1), +I +I +I += Ads,L,+ +0,r +[ +I +I +I +] = +I +I +× (−1), +I += Ads,L,− +0,r +[ +I +] = +I +I +× (−1), +I +I += Ads,L,− +0,r +[ +I +I +] = +I +I +I +× (−2), +I I += Ads,R,+ +0,r +[ +I I +] = +I +, +I I I += Ads,L,+ +0,r +[ +I I I +] = +I +I +, +I += Ads,R,− +0,r +[ +I +] = +I I +, +I +I += Ads,R,− +0,r +[ +I +I +] = +I I I +× 2, +where the row of the flavor −s of the diagram is omitted, and the land of LHS of each equation is the r-th land from +the left on the row of flavor s. The dotted line indicates the rest of the diagram. +The commutator with H0 increases or decreases the support or the double by one. For the support increasing case +on the upper row, we introduce the following notation: +I +I += Ad↑,L,+ +0,rL +� +� += I +× (−1), +I +I +I += Ad↑,L,+ +0,rL +� +I +� += I +× (−1), +I +I = Ad↑,R,+ +0,rR +� +� += +I , +I +I +I += Ad↑,R,+ +0,rR +� +I +� += +I , +where rL = 1 and rR = N↑. The same is true for the case of Ad↓,X,+ +0,rX . +For the support decreasing case on the upper row, we introduce the following notation: +I += Ad↑,L,− +0,rL +� +I +� += +× (−1), +I = Ad↑,R,− +0,rR +� +I +� += +. +We introduce the notation for Ad↓,X,− +0,rX +in the same way. + +10 +When the support is not changed by the action of Ad↓,X,α +0,r +, the double is increased or decreased by one. +The special case is that the action of Ads,X,+ +0,r +makes a hole. For the case that there is one I between r-th and +r + 1-th land on the same row, +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r +� +Ψ becomes +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I +] =Ads,R,+ +0,r +[ +I +] + Ads,L,+ +0,r+1 [ +I +] += ( ++ +) + (− +− +) += +− +, +(S12) +where the left(right) land of the first line is r(r + 1)-th land of flavor s of Ψ, and the row of flavor −s is omitted, and +the type of the land with a hole on the second and last line is −trtr+1, where tr, tr+1 = ± is the type of r-th and +r + 1-th land on the row of flavor s in Ψ and the type of hole-less land on the second line is trtr+1 and it is canceled +in the last equality. For the case of the length of the r-th land is zero, we have +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I I +] =Ads,R,+ +0,r +[ +I I +] + Ads,L,+ +0,r+1 [ +I I +] += ( +I ++ +I +) + (− +I +) += +I +, +(S13) +where the type of the land with hole is −trtr+1 = tr+1. In the same way, for the case that the length of the r + 1-th +land is zero, we have +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I +I +] = +I +× (−1), +(S14) +where the type of the land with hole is −trtr+1 = tr. +For these hole-making cases, we introduce the following diagrammatic notation: +I += Ads,L,+ +0,r+1 [ +I +] = +× (−1), +I I += Ads,L,+ +0,r+1 [ +I I +] = +I +× (−1), +I I += Ads,L,+ +0,r+1 [ +I I +] = 0, +I += Ads,R+ +0,r +[ +I +] = +, +I I += Ads,R+ +0,r +[ +I I +] = +I +, +I I += Ad↑,r,R+ [ +I I +] = 0, +and using this diagrammatic notation, the actions is written by +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I +] = +I ++ +I +, +(S15) +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I I +] = +I I ++ +I I +, +(S16) +� +Ads,R,+ +0,r ++ Ads,L,+ +0,r+1 +� +[ +I +I +] = +I I ++ +I I +. +(S17) +commutator with the interaction term Hint +The commutator of a land of flavor s from j-th site and Hint increases or decreases the length of the land: +� +l +↑(j), Hint +� += +l +I +I (j) − +l +I +I (j) = AdR +intψ↑ + AdL +intψ↑, +(S18) +� +l +↓(j), Hint +� += +l +I +I (j) − +l +I +I (j) = AdR +intψ↓ + AdL +intψ↓, +(S19) +� +s(j), Hint +� += 0, +(S20) + +11 +where ψ ≡ +l +s(j) whose type is +, and +l +I +I (j) ≡ +l +↑(j) × +↓(j + l) = AdR +intψ↑ and +l +I +I (j) ≡ +l +↑(j) × +↓(j) = −AdR +intψ↑ and +l +I +I (j) ≡ +l +↓(j) × +↑(j + l) = AdR +intψ↓ and +l +I +I (j) ≡ +l +↓(j) × +↑(j) = −AdR +intψ↓ +and AdX +int +s(j) = 0 for X = L, R. The type of the land in AdX +intψ of the same flavor with ψ differs from the type of +ψ. In the case of ψ ≡ +l +s(j) whose type is −, the action of AdX +intψ is defined in the same way. +The commutator of a diagram and Hint is now defined since a diagram is the product of lands and its site translational +summation. +[Ψ, Hint] = +� +s=↑,↓ +Ns +� +r=1 +� +X=L,R +Ads,X +0,r Ψ, +(S21) +where N↑(↓) is the number of the land on the upper(lower) row of Ψ and +Ads,X +int,rΨ := +L +� +i=1 +� +AdX +int,rΨs,i +� +Ψ−s,i, +(S22) +AdX +int,rΨs,i := +� +� +� +� +Ns +� +q=1 +q̸=r +Ψq +s,i +� +� +� +� AdX +intΨr +s,i. +(S23) +For the proof of Theorem 1 and Theorem 2, we introduce the diagrammatic notation for the nonzero action of +Ads,X +int,rΨ: +I +I +I += Ad↑,L +int,r +� +I +I +I +� += +I +I +I +× (−1), +I +I +I += Ad↑,L +int,r +� +I +I +I +� += +I +I +I +× (−1), +I += Ad↑,L +int,r +� +I +� += +I +, +I +I I += Ad↑,R +int,r +� +I +I I +� += +I +I I +, +I +I I += Ad↑,R +int,r +� +I +I I +� += +I +I I +, +I += Ad↑,R +int,r +� +I +� += +I +× (−1), +where the lands on the upper rows are the r-th land from the left. In the last line, holes are generated in the RHS. +For the action of Ad↓,X +int,r, the diagrammatic notation is introduced in the same way. +Note that Ads,X +int,r does not +change the support or double of a diagram, but increases or decreased the land number by one. + +12 +S2. IDENTITIES OF Cjm +nd (λ) +In this section we show the identities Cjm +nd (λ) satisfies. The proofs of the identities are given in section S4. +We introduce some notations of list λ = {λ0 ≡ λL; λ1, λ2, . . . , λw; λw+1 ≡ λR}. We denote λa↔b by the configuration +where λa and λb in λ are swapped. We denote λˆa by the configuration where λa is removed from λ. We denote +λλa→A by the configuration where λa in λ is replaced by A. We define λa:±δ ≡ λλa→λa±δ. +Cjm +nd (λ) satisfies the following identities. +Cj,m +n,d (λ) = Cj,m +n,d (λL↔R) , +(S24) += Cj,m +n,d (λi1↔j2) , +(S25) += Cj,m +n,d +� +λλa→min(λa,n) +� +, +(S26) += Cj,m +n−1,d+2 +� +λL(R):−1,i:−1 +� ++ Cj−1,m +n,d+1 +� +λλL(R)→λL(R)+λi1,ˆi +� +, +(S27) += Cj,m +n−1,d+2 (λi1:−1,i2:−1) + Cj−1,m +n,d+1 +� +λλi1→λi1+λi2,ˆi2 +� +, +(S28) += Cj,m +n,d (λa:−1,b:+1) + Cj,m +n−1,d+2 (λa:−1,b:−1) − Cj,m +n−1,d+2 (λa:−2) +(λa > 0), +(S29) +where 1 ≤ i, i1, i2 ≤ w and 0 ≤ a, b ≤ w + 1 and w ≡ j − 1 − 2m. (S24), (S25),(S26) are mentioned in the theorem 2. +For i1 = 0, i2 = i case in (S28), we have +Cj,m +n,d (λλi→0) = Cj−1,m +n,d+1 (λˆi) . +(S30) +We review the recursion equation of Theorem 2. +Cjm +nd (λ) = Cjm +nd (λL − 1; λ1, . . . , λw; λR + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR + 1) +� +. +(S31) +The λL = −1 case is defined by +Cj,m +n,d (−1; λ1, . . .) := +� +Cj,m +n,d−1 (1; λ1, . . .) +(d > 0) +Cj,m +n−1,1 (0; λ1 − 1, . . .) + Cj−1,m +n,0 +(λ1 + 1; . . .) +(d = 0) . +In the same way, we also define the λR = −1 case by +Cj,m +n,d (. . . , λw; −1) := +� +Cj,m +n,d−1 (. . . , λw; 1) +(d > 0) +Cj,m +n−1,1 (. . . , λw − 1; 0) + Cj−1,m +n,0 +(. . . ; λw + 1) +(d = 0) . +This recursion equation can be rewritten as +∆Cj,m +n,d (λ) = 2∆Cj,m +n−1,d+1 (λ) − ∆Cj,m +n−2,d+2 (λ) + Cj−1,m−1 +n,d +((λR:−1)←0) − Cj−1,m−1 +n,d +(0→(λL:−1)) , +(S32) +where ∆Cj,m +n,d (λ) = Cj,m +n,d (λR:−1) − Cj,m +n,d (λL:−1). +The recursion equation reflects the local cancellation in [Qk, H] = 0 (Fig. S1). To see this, we further rewrite (S32) +as +0 = +� +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +� +− +� +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +� +− +� +Cj,m +n−1,d+1 (λ) − Cj,m +n−1,d+1 (λL:−1,R:+1) +� ++ +� +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +� +− +� +Cj−1,m−1 +n,d +(λ←0) − Cj−1,m−1 +n,d +(0→λ) +� +, +(S33) +where +we +change +λR +→ +λR + 1 +and +used +the +relation +obtained +from +(S29): +Cj,m +n,d−1 (λL:+1,R:+1) − +Cj,m +n,d−1 (λR:+2) = Cj,m +n−1,d+1 (λ) − Cj,m +n−1,d+1 (λL:−1,R:+1) and Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) = Cj,m +n−2,d+2 (λ) − +Cj,m +n−2,d+2 (λL:−1,R:+1). The first, second, third, and fourth terms in the RHS of (S33) are the coefficients of the + +13 +(s, d), (s + 1, d − 1), (s + 1, d + 1), (s + 2, d) diagrams in Qj +k respectively (s = k − j − 2n − d), which are represented by +the circles in the Qj +k plane in Fig. S1. The commutator of these diagrams in Qj +k with H0 generates the diagrams at +(s + 1, d) in the Qj +k plane in Fig. S1. The last term in the RHS of (S33) is the coefficients of the (s + 1, d) diagrams in +Qj−1 +k +, which are represented by the circle in the Qj−1 +k +plane in Fig. S1. The commutator of these diagrams in Qj−1 +k +with Hint generate the diagrams at (s+1, d) in the Qj +k plane in Fig. S1. These diagrams are represented by the circles +in Fig. S1. +Support +Double +Cj,m +n,d (λ) − Cj,m +n,d (T λ) +(1) +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +(2) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(3) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(4) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(5) +Cj,m +n,d (λL:+1) − Cj,m +n,d (λR:+1) +(6) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(7) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(8) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(9) +1 +Cj,m +n,d (λ) − Cj,m +n,d (T λ) +(1) +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +(2) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(3) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(4) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(5) +Cj,m +n,d (λL:+1) − Cj,m +n,d (λR:+1) +(6) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(7) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(8) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(9) +1 +Cj,m +n,d (λ) − Cj,m +n,d (T λ) +(1) +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +(2) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(3) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(4) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(5) +Cj,m +n,d (λL:+1) − Cj,m +n,d (λR:+1) +(6) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(7) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(8) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(9) +1 +ACaHichVHLgRBFD3T3uPVWCA2YkKsJtUiNWEDTuDRjLGpLvVjDL9SnfPJEz8gI0lYkUiIj7Dxg9Y+AQsSWws3O6eRBDc +SlWdOnXPrVNVumsKP2DsISE1NDY1t7S2Jds7Oru65Z7eNd+peAZXDcd0vA1d87kpbK4GIjD5hutxzdJNvq6X58P9Sr3fOHYq8 +Gey/OWVrJFURhaQJSaLZS3dgtyiqVZFM/gVIHqcwiolhy5CtsYhsODFRgcNGQNiEBp9aDgoYXOLyqBHnERLRPscBkqStUBan +DI3YMo0lWuXqrE3rsKYfqQ06xaTukXIYo+yeXbMXdsdu2CN7/7VWLaoRetmjWY+13C10Hw6svP2rsmgOsPOp+tNzgCJmIq+CvL +sRE97CiPXV/eOXldnl0doYu2DP5P+cPbBbuoFdfTUus3z5DEn6AOX7c/8EaxNpZSo9mZ1MZebin0ArhjCcXrvaWSwgCWodK7A +EU5wmniSZKlfGoxTpURd04cvIY18AGT1jB0= +Qj +k +ACbHichVG7SgNBFD1ZXzG+4qMQghAMERvDrAQVq6CNdk0iaIx7K5jXLMvdjeBGPIDthYWaqEgIn6GjT9g4SeIYBPBxsKb +TUBU1DvMzJkz9w5MyNbmuq4jD36hI7Oru4ef2+gr39gcCg4PJ1zLKt8Ixiaqa9IUsO1SDZ1zV1fiGZXNJlzWek0vLzf1chdu +OahrbtXieV0qGuqeqkguUZupQmndjAj1gvBCIsxL8I/gdgGkcQqvEiawWtsYxcmFJShg8OAS1iDBIfaFkQwWMTlUSPOJqR6+x +x1BEhbpixOGRKxJRqLtNpqswatmzUdT63QKRp1m5RhRNkDu2ENds9u2RN7/7VWzavR9FKlW5puVUYOhpfe/tXpdPsYv9T9adn +F3tY8Lyq5N3ymOYtlJa+cnjSWFtMR2tT7JI9k/8L9sju6AZG5VW5SvH0KQL0AeL35/4JsrMxcS4WT8UjiaXWT8CPECYxTe89jwR +WkESGztVxjDOc+16EMSEkTLRSBV9bM4ovIUx9ALBnjZs= +Qj�1 +k +Cj,m +n,d (λ) − Cj,m +n,d (T λ) +(1) +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +(2) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(3) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(4) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(5) +Cj,m +n,d (λL:+1) − Cj,m +n,d (λR:+1) +(6) +Cj,m +n−1,d+1 (λ) − Cj,m +n−1,d+1 (λL:−1,R:+1) +(7) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(8) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(9) +Cj−1,m−1 +n,d +(0→λ) − Cj−1,m−1 +n,d +(λ←0) +(10) +1 +Cj,m +n,d (λ) − Cj,m +n,d (T λ) +(1) +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) +(2) +Cj,m +n,d+1 (λ) − Cj,m +n,d+1 (λL:−1,R:+1) +(3) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(4) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(5) +Cj,m +n,d (λL:+1) − Cj,m +n,d (λR:+1) +(6) +Cj,m +n−1,d+1 (λ) − Cj,m +n−1,d+1 (λL:−1,R:+1) +(7) +Cj,m +n,d−1 (λL:+1,R:+1) − Cj,m +n,d−1 (λR:+2) +(8) +Cj,m +n−1,d (λL:+1,R:+1) − Cj,m +n−1,d (λR:+2) +(9) +Cj−1,m−1 +n,d +(λ←0) − Cj−1,m−1 +n,d +(0→λ) +(10) +1 +𝑛 +𝑛 +(𝑠 + 1, 𝑑 + 1) +(𝑠 + 1, 𝑑) +(𝑠, 𝑑) +(𝑠 + 1, 𝑑 − 1) +(𝑠 + 2, 𝑑) +𝑛 − 1 +FIG. S1. The general structure of the cancellation of diagrams in Qk. The upper plane represents Qj−1 +k +and lower plane +represents Qj +k. +We show only the components of Qk which are relevant to the cancellation of diagrams in the crosses at +(s+1, d) in the plane of Qj +k. The coefficients encircled with the orange solid (blue dotted) line is the coefficients of the diagrams +in the circles indicated by the arrow tip and contribute to the recursion relation in the form of (S33) with the factor of +1(−1). + +14 +S3. PROOF OF CONSERVATION LAW OF Qk +In this section, we prove Theorem 1 and Theorem 2 in the main manuscript by showing that Qk constructed from +the result of Theorem 1 and Theorem 2 is actually conserved, i.e., we prove [Qk, H] = 0. +[Qk, H] = +jf +1 +� +j=0 +U j �� +Qj +k, H0 +� ++ +� +Qj−1 +k +, Hint +�� += +jf +1 +� +j=0 +U j +⌊ k−1−j +2 +⌋ +� +n=0 +⌈ j−1 +2 ⌉ +� +m=0 +⌊ k−1−j +2 +⌋−n +� +d=0 +d +� +g=0 +� +�Ψ∈Fk,j,m +n,d,g +�Dk,j,m +n,d,g (�Ψ)�Ψ +(S34) +where 0 ≤ j ≤ jf ≡ 2⌊k/2⌋ − 1 and Q−1 +k += Qjf +1 +k += 0 and Fk,j,m +n,d,g is the set of (k − j − 2n − d + 1, d) diagrams �Ψ +whose land number is j + 1 − 2m and gap number is g. Note that Fk,j,m +n,d,g include not only connected diagrams but +also disconnected diagrams. What we should prove is �Dk,j,m +n,d,g (�Ψ) = 0 for all �Ψ ∈ Fk,j,m +n,d,g . We consider the cancellation +of the terms at (j, k − j − 2n − d + 1, d). +cancellation of connected diagram +We prove �Dk,j,m +n,d,g (�Ψ) = 0 for the case that �Ψ ∈ Fk,j,m +n,d,g is a connected diagram and l�Ψ > 1,i.e., j + 1 − 2m > 0. We +denote the list of �Ψ by λ. +�Dk,j,m +n,d,g (�Ψ) = AσR +0 +0 (�Ψ) + +w +� +i=1 +AσL +i ,σR +i +i +(�Ψ) + A +σL +w+1 +w+1 (�Ψ) +(S35) +where σL(R) +a += ± is determined depending on the configuration around the a-th coast of �Ψ. +AσL +i ,σR +i +i +(�Ψ) is the +contribution to �Dk,j,m +n,d,g (�Ψ) from the diagram in Qj +k whose list is λi:±1. AσR +0 +0 (�Ψ) is the contribution to �Dk,j,m +n,d,g (�Ψ) from +the diagram in Qj +k whose list is λL:±1 and the diagram in Qj−1 +k +whose list is 0→(λL:−1). AσL +0 +w+1(�Ψ) is the contribution +to �Dk,j,m +n,d,g (�Ψ) from the diagram in Qj +k whose list is λR:±1 and the diagram in Qj−1 +k +whose list is (λR:−1)←0. There is +no other contribution to the cancellation of �Ψ other than the above contribution. We do not write explicitly the �Ψ +dependence of AσR +0 +0 (�Ψ), AσL +i ,σR +i +i +(�Ψ), A +σL +w+1 +w+1 (�Ψ) in the following. +σL +i = −(+) if the i-th coast and i − 1-th coast are adjacent(not adjacent). σR +i = −(+) if the i-th coast and i + 1-th +coast are adjacent(not adjacent). Therefore, σR +i = σL +i+1 holds. +We give the examples of the case of σR +i = σL +i+1 = −, i.e., the case that i−th and i + 1-th coasts are adjacent: +λi + 1 +λi+1 + 1 +, +λi + 1 +λi+1 + 1 +. +We give the example of the case of σR +i = σL +i+1 = +, i.e., the case that i−th and i + 1-th coasts are not adjacent: +λi + 1 +λi+1 + 1 +, +λi + 1 +I I +I I +λi+1 + 1 +. + +15 +The values of AσL +i σR +i +i +are +A++ +i += A−− +i += 0, +(S36) +A+− +i += −A−+ +i += (−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +, +(S37) +A+ +0 = (−1)n+m+g � +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) ++Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +, +(S38) +A+ +w+1 = (−1)n+m+g+1 � +Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d (λR:−1) ++Cj,m +n−1,d (λR:+1) − Cj,m +n,d−1 (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +, +(S39) +A− +0 = (−1)n+m+g � +2Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +, +(S40) +A− +w+1 = (−1)n+m+g+1 � +2Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d (λR:−1) + Cj,m +n−1,d (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +. +(S41) +The proof of these equations is given in section S5. +Using the identity (S29), we have +A+− +i1 ++ A−+ +i2 += (−1)n+m+g � +Cj,m +n−1,d+1 (λi1:−1) + Cj,m +n,d−1 (λi1:+1) − +� +Cj,m +n−1,d+1 (λi2:−1) + Cj,m +n,d−1 (λi2:+1) +�� += (−1)n+m+g � +Cj,m +n−1,d+1 (λi1:−1) − Cj,m +n−1,d+1 (λi2:−1) − +� +Cj,m +n,d−1 (λi2:+1) − Cj,m +n,d−1 (λi1:+1) +�� += 0, +(S42) +where we used (S29) in the last equality. In the same way, we have +A−+ +i1 ++ A+− +i2 += 0. +(S43) +A− +0 + A−+ +i += (−1)n+m+g � +2Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +− +� +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +�� += (−1)n+m+g � +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +− +� +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) − Cj,m +n−1,d+1 (λL:−1) +�� += (−1)n+m+g � +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) ++Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) − Cj,m +n,d−1 (λL:+1) +� += A+ +0 , +(S44) +In the same way, we have +A+− +i ++ A− +w+1 = A+ +w+1, +(S45) +and also we have +A− +0 + A− +w+1 += (−1)n+m+g � +2Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +− +� +2Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d (λR:−1) + Cj,m +n−1,d (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +�� += (−1)n+m+g � +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d−1 (λL:+1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +− +� +Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d−1 (λR:+1) − Cj,m +n,d (λR:−1) + Cj,m +n−1,d (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +�� += A+ +0 + A+ +w+1, +(S46) + +16 +Therefore, in any case of a connected diagram �Ψ, we have +�Dk,j,m +n,d,g (�Ψ) = AσR +0 +0 ++ +w +� +i=1 +AσL +i ,σR +i +i ++ A +σL +w+1 +w+1 += A+ +0 + A+ +w+1 += (−1)n+m+g � +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +− +� +Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d (λR:−1) + Cj,m +n−1,d (λR:+1) − Cj,m +n,d−1 (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +�� += (−1)n+m+g � +−∆Cj,m +n,d (λ) + 2∆Cj,m +n−1,d+1 (λ) − ∆Cj,m +n−2,d+2 (λ) ++Cj−1,m−1 +n,d +(0→(λL:−1)) − Cj−1,m−1 +n,d +((λR:−1)←0) +� += 0, +(S47) +where we used (S32) in the last equality. +We next prove �Dk,j,m +n,d,g (�Ψ) = 0 for the case that �Ψ ∈ Fk,j,m +n,0,0 is a connected diagram and the land number of �Ψ = 1,i.e., +j = 2m. In this case, the double and the gap number of �Ψ is zero. We consider the case of +�Ψ = I +I +l +. +(S48) +In this case, the contributions to the cancellation of �Ψ in [Qk, H] comes only from +� +Qj−1 +k +, Hint +� +because Qj +k does not +have the diagram of land number 1 by the fix of freedom to add lower Qk′ 0 cases +and if we suppose P(j − 1, m, n, 0) and P(j − 1, m − 1, n, 0) and P(j, m, n − 1, 1) holds, then P(j, m, n, d = 0) also +holds for d = 0 cases. Going back through the induction steps, it comes down to the base case P(j, m = 0, n, d) and +P(j, m, n = 0, d). +Proof of Base case +Proof of P(j, m = 0, n, d) +The explicit solution of the m = 0 case is +Cj,m=0 +n,d +(λ) = +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +i=1 +xi +� +, +where θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0. +We prove this explicit solution actually satisfies the recursion equation. The recursion equation for the m = 0 case +is +Cj,m=0 +nd +(λ) =Cj,m=0 +nd +(λL − 1; λ1, . . . , λj−1; λR + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,−1 +˜n,n+d−˜n (λL; λ1, . . . , λj−1, λR; 0) − Cj−1,−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λj−1; λR + 1) +� +=Cj,m=0 +nd +(λL − 1; λ1, . . . , λj−1; λR + 1) . +(S61) +The equation we have to prove is +Cj,m=0 +nd +(λ) = Cj,m=0 +nd +(λL − 1; λ1, . . . , λj−1; λR + 1) . +(S62) +For λL > 0 case, (S62) holds trivially because Cj,m=0 +nd +(λ) does not depend on λL and λR. +For λL = 0 case, +Cj,m=0 +nd +(−1; λ1, . . . , λj−1; λR + 1) = +� +� +� +Cj,0 +n,d−1 (1; λ1, . . . , λj−1; λR + 1) +(d > 0) +Cj,0 +n−1,1 (0; λ1 − 1, . . . , λj−1; λR + 1) + Cj−1,0 +n,0 +(λ1 + 1; λ2, . . . , λj−1; λR + 1) +(d = 0) +. +For λL = 0 and d > 0 case, (S62) holds trivially because Cj,m=0 +nd +(λ) does not depend on λL and d. +For λL = 0 and d = 0 case, +Cj,0 +n−1,1 (0; λ1 − 1, . . . , λj−1; λR + 1) + Cj−1,0 +n,0 +(λ1 + 1; λ2, . . . , λj−1; λR + 1) += +λ1−1 +� +x1=0 +λ2 +� +x2=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� ++ +λ2 +� +x2=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=2 +xl +� += +λ1 +� +x1=1 +λ2 +� +x2=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ2 +� +x2=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=2 +xl +� += +λ1 +� +x1=0 +λ2 +� +x2=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,0 +n,0 (0; λ1, . . . , λj−1; λR) , +(S63) + +20 +and we can see (S62) holds. We have confirmed (S62) also holds for λL = 0 and d = 0 case, and we have proved that +the explicit expression for m = 0 case satisfies the recursion equation. +In the following, we prove that the explicit expression for m = 0 satisfies the identity (S24)-(S29). (S24) and +(S25) holds trivially for m = 0 case because Cj,m=0 +nd +(λ) does not depend on λL and λR and is symmetric under the +permutation of λi’s (1 ≤≤ j − 1). +We show (S26) for m = 0 case. Cj,m +n=0,d does not depend on λL and λR, and a = L, R case is trivially holds. We +consider a = i(1 ≤ i ≤ w = j − 1) case. We suppose λi > n. +Cj,m=0 +n,d +(λ) = +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ(n − +j−1 +� +l=1 +xl) += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +� +n +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λi +� +xi=n+1 +θ +� +n − +j−1 +� +l=1 +xl +�� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +min(n,λi) +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λi−n−1 +� +xi=0 +θ +� +� +� +�−1 − xi − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +min(n,λi) +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� += Cj,m=0 +n,d +� +λλi→min(λi,n) +� +, +(S64) +where +ˆi· · · means that the summation over xi is absent. We have proved (S26) for m = 0 base case. +We show (S27) for m = 0 case. +Cj,m=0 +n−1,d+2 +� +λL(R):−1,i:−1 +� ++ Cj−1,m=0 +n,d+1 +� +λλL(R)→λL(R)+λi1,ˆi +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−1 +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +θ +� +� +� +�n − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi−1 +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� ++ θ +� +� +� +�n − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi +� +xi=1 +θ +� +� +� +�n − xi − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� + θ +� +� +� +�n − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,m=0 +n,d +(λ) . +(S65) +We have proved (S27) for m = 0 base case. + +21 +We prove (S28) for m = 0 base case. +Cj,m=0 +n−1,d+2 (λi1:−1,i2:−1) + Cj−1,m=0 +n,d+1 +� +λλi1→λi1+λi2,ˆi2 +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2−1 +� +xi2=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1+λi2 +� +xp=0 +θ +� +� +� +�n − xp − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi1−1 +� +xi1=0 +λi2−1 +� +xi2=0 +θ +� +� +� +�n − 1 − xi1 − xi2 − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� + +λi1+λi2 +� +xp=0 +θ +� +� +� +�n − xp − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi1−1 +� +xi1=0 +λi2 +� +xi2=1 +θ +� +� +� +�n − xi1 − xi2 − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� + +λi1−1 +� +xi1=0 +θ +� +� +� +�n − xi1 − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� ++ +λi1+λi2 +� +xp=λi1 +θ +� +� +� +�n − xp − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi1−1 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +� +� +�n − xi1 − xi2 − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� + +λi1+λi2 +� +xp=λi1 +θ +� +� +� +�n − xp − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi1−1 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +� +� +�n − xi1 − +� +1≤l≤j−1 +l̸=i1 +xl +� +� +� +� + +λi2 +� +xi2=0 +θ +� +� +� +�n − λi1 − xi2 − +� +1≤l≤j−1 +l̸=i1,i2 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi2 +� +xi2=0 +� +� +� +� +� +� +� +λi1−1 +� +xi1=0 +θ +� +� +� +�n − xi1 − +� +1≤l≤j−1 +l̸=i1 +xl +� +� +� +� + θ +� +� +� +�n − λi1 − +� +1≤l≤j−1 +l̸=i1 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +ˆi1 +· · · +λj−1 +� +xj−1=0 +� +� +� +� +� +� +� +λi1 +� +xi1=0 +θ +� +� +� +�n − xi1 − +� +1≤l≤j−1 +l̸=i1 +xl +� +� +� +� +� +� +� +� +� +� +� += +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,m=0 +n,d +(λ) . +(S66) +Thus we have proved (S28) for m = 0 base case. +We prove (S29) for m = 0 case. a = R(L), b = L(R) case holds trivially because Cj,m=0 +n,d +(λ) does not depend on d +and λL and λR. + +22 +a = L(R), b = i(1 ≤ i ≤ j − 1) case is +Cj,m=0 +n,d +� +λL(R):−1,i:+1 +� ++ Cj,m=0 +n−1,d+2 +� +λL(R):−1,i:−1 +� +− Cj,m=0 +n−1,d+2 +� +λL(R):−2 +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi+1 +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−1 +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi+1 +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +θ +� +� +� +�n − 1 − λi − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,m=0 +n,d +(λ) . +(S67) +a = i, b = L(R)(1 ≤ i ≤ j − 1) case is +Cj,m=0 +n,d +� +λi:−1,L(R):+1 +� ++ Cj,m=0 +n−1,d+2 +� +λi:−1,L(R):−1 +� +− Cj,m=0 +n−1,d+2 (λi:−2) += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−1 +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−1 +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−2 +� +xi=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi−1 +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +θ +� +� +� +�n − 1 − (λi − 1) − +� +1≤l≤j−1 +l̸=i +xl +� +� +� +� += +λ1 +� +x1=0 +ˆi· · · +λj−1 +� +xj−1=0 +λi +� +xi=0 +θ +� +n − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,m=0 +n,d +(λ) . +(S68) + +23 +We consider a = i1, b = i2(1 ≤ i1, i2 ≤ j − 1) case. +Cj,m=0 +n,d +(λi1:−1,i2:+1) + Cj,m=0 +n−1,d+2 (λi1:−1,i2:−1) − Cj,m=0 +n−1,d+2 (λi1:−2) += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2+1 +� +xi2=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2−1 +� +xi2=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−2 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2+1 +� +xi2=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2−1 +� +xi2=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +n − 1 − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi2 +� +xi2=0 +θ +� +n − λi1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2+1 +� +xi2=0 +θ +� +n − +j−1 +� +l=1 +xl +� +− +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +θ +� +� +� +�n − 1 − λi2 − +� +1≤l≤j−1 +l̸=i2 +xl +� +� +� +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi2 +� +xi2=0 +θ +� +n − λi1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1−1 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +n − +j−1 +� +l=1 +xl +� ++ +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi2 +� +xi2=0 +θ +� +n − λi1 − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +ˆi1,ˆi2 +· · · +λj−1 +� +xj−1=0 +λi1 +� +xi1=0 +λi2 +� +xi2=0 +θ +� +n − +j−1 +� +l=1 +xl +� += +λ1 +� +x1=0 +· · · +λj−1 +� +xj−1=0 +θ +� +n − +j−1 +� +l=1 +xl +� +=Cj,m=0 +n,d +(λ) . +(S69) +Thus, we have proved (S29) of m = 0 base case. +We have proved P(j, m = 0, n, d). +Proof of P(j, m, n = 0, d) +The explicit solution for n = 0 case is +Cj,m +n=0,d (λ) = Cj−1−m+d,m = +�j − 1 + d +m +� +− +�j − 1 + d +m − 1 +� +, +where Cn,k := +�n+k +k +� +− +�n+k +k−1 +� +is the generalized Catalan number, which satisfies the following equation. +Cn,k = Cn,k−1 + Cn−1,k for 1 < k < n, +Cn,n = Cn,n−1 for n ≥ 2. +(S70) +We confirm this solution actually satisfies the recursion equation. The recursion equation for n = 0 case is +Cj,m +n=0,d (λ) =Cj,m +n=0,d (λL − 1; λ1, . . . , λj−1; λR + 1) ++ Cj−1,m−1 +0,d +(λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +0,d +(0; λL − 1, λ1, . . . , λw; λR + 1) . +(S71) + +24 +For λL > 0 case, the RHS of (S71) becomes +Cj−1−m+d,m + Cj−1−m+d,m−1 − Cj−1−m+d,m−1 = Cj−1−m+d,m = Cj,m +n=0,d (λ) , +(S72) +and we can see (S71) holds. +For λL = 0 case, the RHS of (S71) becomes +Cj,m +n=0,d (−1; λ1, . . . , λj−1; λR + 1) + Cj−1,m−1 +0,d +(0; λ1, . . . , λw, λR; 0) += +� +� +� +Cj,m +0,d−1 (1; λ1, . . . , λj−1; λR + 1) + Cj−1−m−d,m−1 +(d > 0) +Cj−1,m +0,0 +(λ1 + 1; . . . , λj−1; λR + 1) + Cj−1−m,m−1 +(d = 0) += +� +� +� +Cj−2−m−d,m + Cj−1−m−d,m−1 +(d > 0) +Cj−2−m,m + Cj−1−m,m−1 +(d = 0) += +� +� +� +Cj−1−m−d,m +(d > 0) +Cj−1−m,m +(d = 0) +=Cj,m +n=0,d (λ) , +(S73) +where we used (S70) in the third equality We can see (S71) holds. We have proved the solution for n = 0 case. +We will prove Cj,m +n=0,d (λ) satisfies (S24)-(S29). +Cj,m +n=0,d (λ) does not depend on λ, +thus we can see +(S24),(S25),(S26),(S29) trivially hold. +We prove (S27) for n = 0 case. +Cj,m +n=−1,d+2 +� +λL(R):−1,i:−1 +� ++ Cj−1,m +n=0,d+1 +� +λλL(R)→λL(R)+λi1,ˆi +� += 0 + Cj−1−m+d,m = Cj,m +n=0,d (λ) . +(S74) +We can also prove (S28) for n = 0 case in the same way. We have proved P(j, m, n = 0, d). +Proof for Induction step +We show that if we suppose P(j − 1, m − 1, n, d) and P(j, m, n, d − 1) holds, then P(j, m, n, d) also holds for d > 0 +cases and if we suppose P(j − 1, m, n, 0) and P(j − 1, m − 1, n, 0) and P(j, m, n − 1, 1) holds, then P(j, m, n, d = 0) +also holds for d = 0 cases. + +25 +induction for (S24) +Using the recursion equation repeatedly, we have +Cjm +nd (λ) += Cjm +nd (λλL:−1,λR:+1) + +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λ←0) − Cj−1,m−1 +˜n,n+d−˜n (0→λ) +� += Cjm +nd (λL − 1; λ1, . . . , λw; λR + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR + 1) +� += Cjm +nd (λL − 2; λ1, . . . , λw; λR + 2) ++ +1 +� +i=0 +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL − i; λ1, . . . , λw, λR + i; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − i − 1, λ1, . . . , λw; λR + i + 1) +� +... += Cjm +nd (0; λ1, . . . , λw; λR + λL) ++ +λL−1 +� +i=0 +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL − i; λ1, . . . , λw, λR + i; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − i − 1, λ1, . . . , λw; λR + i + 1) +� += Cjm +nd (−1; λ1, . . . , λw; λR + λL + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λL +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λR + λL − i, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL + λR + 1 − i; i − 1, λ1, . . . , λw; 0) +� +, +(S75) +where we used (S24) of P(j − 1, m − 1, n, d) in the last equality. If λL > λR(λL < λR), +λL(R) +� +i=λR(L)+1 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λR + λL − i, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL + λR + 1 − i; i − 1, λ1, . . . , λw; 0) +� += +λL(R) +� +i=λR(L)+1 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λR + λL − i, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λR + λL − i, λ1, . . . , λw; 0) +� +=0, +(S76) +and we have +Cjm +nd (λ) = Cjm +nd (−1; λ1, . . . , λw; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . , λw; 0) +� +, +(S77) +where λL+R ≡ λL + λR, and ”λLorλR” means we can choose λL or λR as the upper limit of summation. This is the +statement of (S24) of P(j, n, m, d). + +26 +induction for (S25) +We assume j ≥ 2m + 3(w ≥ 2). We first consider the 1 < i1 < i2 case. +Cj,m +n,d (λ) − Cj,m +n,d (λi1↔j2) +=Cjm +nd (−1; . . . λi1, . . . , λi2, . . . ; λL+R + 1) − Cjm +nd (−1; . . . λi2, . . . , λi1, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, . . . λi1, . . . , λi2, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, . . . λi2, . . . , λi1, . . . ; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, . . . , λi1, . . . , λi2, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, . . . , λi1, . . . , λi2, . . . ; 0) +�� +=Cjm +nd (−1; . . . λi1, . . . , λi2, . . . ; λL+R + 1) − Cjm +nd (−1; . . . λi2, . . . , λi1, . . . ; λL+R + 1) += +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; . . . λi1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n,d−1 (1; . . . λi2, . . . , λi1, . . . ; λL+R + 1) +(d > 0) +Cjm +n−1,1 (0; λ1 − 1, . . . λi1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λ1 − 1, . . . λi2, . . . , λi1, . . . ; λL+R + 1) ++Cj−1,m +n,0 +(λ1 + 1; . . . λi1, . . . , λi2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(λ1 + 1; . . . λi2, . . . , λi1, . . . ; λL+R + 1) +(d = 0) +=0. +(S78) +We next consider i1 = 1 < i2 case. +Cj,m +n,d (λ) − Cj,m +n,d (λi1=1↔i2) +=Cjm +nd (−1; λ1, . . . , λi2, . . . ; λL+R + 1) − Cjm +nd (−1; λi2, . . . , λ1, . . . ; λL+R + 1) += +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λ1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n,d−1 (1; λi2, . . . , λ1, . . . ; λL+R + 1) = 0 +(d > 0) +Cjm +n−1,1 (0; λ1 − 1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λi2 − 1, . . . , λ1, . . . ; λL+R + 1) ++Cj−1,m +n,0 +(λ1 + 1; . . . , λi2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(λi2 + 1; . . . , λ1, . . . ; λL+R + 1) +(d = 0) +. +(S79) +For d = 0 and λ1 > 0 case, +Cjm +n−1,1 (0; λ1 − 1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λi2 − 1, . . . , λ1, . . . ; λL+R + 1) ++ Cj−1,m +n,0 +(λ1 + 1; . . . , λi2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(λi2 + 1; . . . , λ1, . . . ; λL+R + 1) +=Cjm +n−1,1 (0; λ1 − 1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λi2 − 1, . . . , λ1, . . . ; λL+R + 1) ++ Cj−1,m +n−1,2 (λ1; . . . , λi2 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (λi2; . . . , λ1 − 1, . . . ; λL+R + 1) +=Cjm +n−1,1 (0; λi2, . . . , λ1 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (λi2; . . . , λ1 − 1, . . . ; λL+R + 1) +− +� +Cjm +n−1,1 (0; λ1, . . . , λi2 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (λ1; . . . , λi2 − 1, . . . ; λL+R + 1) +� +=Cjm +n−2,3 (−1; λi2 − 1, . . . , λ1 − 1, . . . ; λL+R + 1) − Cjm +n−2,3 (−1; λ1 − 1, . . . , λi2 − 1, . . . ; λL+R + 1) +=Cjm +n−2,2 (1; λi2 − 1, . . . , λ1 − 1, . . . ; λL+R + 1) − Cjm +n−2,2 (1; λ1 − 1, . . . , λi2 − 1, . . . ; λL+R + 1) +=0, +(S80) +where we use (S29) of P(j − 1, m, n, d) on the first equality and (S25) of P(j, m, n − 1, 1) on the second equality and +(S28) of P(j, m, n − 1, 1) on the third equality. For d = 0 and λ1 = 0 case, +Cjm +n−1,1 (0; −1, . . . , λi2, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λi2 − 1, . . . , 0, . . . ; λL+R + 1) ++ Cj−1,m +n,0 +(1; . . . , λi2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(λi2 + 1; . . . , 0, . . . ; λL+R + 1) +=Cj−1,m +n,0 +(1; λi2, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (0; λi2 − 1, . . . ; λL+R + 1) − Cj−2,m +n,1 +(λi2 + 1; . . . ; λL+R + 1) +=0, +(S81) +where we use (S30) of P(j, m, n − 1, 1) and P(j − 1, m, n, 0) on the first equation and (S28) of P(j − 1, m, n, 0) on the +second equation. +We have proved (S25) of P(j, m, n, d). In the following, we can use (S25) of P(j, n, m, d) and can freely swap the +λi1 and λi2 (1 ≤ i1 ≤ i2 ≤ w). Without mentioning, we rearrange the order of λ for easy viewing. + +27 +induction for (S26) +We next consider (S26). It is enough to consider the λa > n case. +For the a = L case, we have +Cj,m +n,d (λ) − Cj,m +n,d (λλL→n) +=Cjm +nd (−1; λ1, . . . , λw; λR + λL + 1) − Cjm +nd (−1; λ1, . . . , λw; λR + n + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λR + λL − i, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λR + n − i, λ1, . . . , λw; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (λL + λR + 1 − i; i − 1, λ1, . . . , λw; 0) − Cj−1,m−1 +˜n,n+d−˜n (n + λR + 1 − i; i − 1, λ1, . . . , λw; 0) +�� +=Cjm +nd (−1; λ1, . . . , λw; λR + λL + 1) − Cjm +nd (−1; λ1, . . . , λw; λR + n + 1) += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λ1, . . . , λw; λR + λL + 1) − Cjm +n,d−1 (1; λ1, . . . , λw; λR + n + 1) +(d > 0) +Cjm +n−1,1 (0; λ1 − 1, . . . , λw; λR + λL + 1) − Cjm +n−1,1 (0; λ1 − 1, . . . , λw; λR + n + 1) ++Cj−1,m +n0 +(λ1 + 1; . . . , λw; λR + λL + 1) − Cj−1m +n0 +(λ1 + 1; . . . , λw; λR + n + 1) +(d = 0 ∧ j > 2m + 1) +Cjm +n−1,1 (0; λR + λL) − Cjm +n−1,1 (0; λR + n) +(d = 0 ∧ j = 2m + 1) +=0, +(S82) +where we used (S26) of P(j, m, n, d−1) for d > 0 case and used (S26) of P(j, m, n−1, 1) and P(j −1, m, n, 0) for d = 0 +case and used min(λR+λL+1, n) = min(λR+n+1, n) = n and min(λR+λL+1, n−1) = min(λR+n+1, n−1) = n−1 +and min(lambdaR + λL) = min(lambdaR + n) = lambdaR + n for λL > n. +We have also proved the a = R case due to (S24) of P(j, m, n, d). +For a = i(1 ≤ i ≤ w) case, we can set i = 1 without losing generality due to (S25), and we have, +Cj,m +n,d (λ) − Cj,m +n,d (λλ1→n) +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; n, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λL +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, n, . . . ; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, n, . . . ; 0) +�� +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; n, . . . ; λL+R + 1) += +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λ1, . . . ; λL+R + 1) − Cjm +n,d−1 (1; n, . . . ; λL+R + 1) +(d > 0) +Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cjm +n−1,1 (0; n − 1, . . . ; λL+R + 1) ++Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(n + 1; . . . ; λL+R + 1) +(d = 0) +=0. +(S83) +We have proved (S26) of P(j, m, n, d). + +28 +induction for (S27) +Because we have proved (S24) of P(j, n, d), we only consider the L case. Due to (S25), we can set i = 1 without +losing generality. +Cj,m +n,d (λ) − Cj,m +n−1,d+2 +� +λL(R):−1,1:−1 +� +− Cj−1,m +n,d+1 +� +λλL(R)→λL(R)+λ1,ˆ1 +� +=Cj,m +n,d (λL; λ1, . . . ; λR) − Cj,m +n−1,d+2 (λL − 1; λ1 − 1, . . . ; λR) − Cj−1,m +n,d+1 (λL + λ1; . . . ; λR) +=Cj,m +n,d (−1; λ1, . . . ; λL+R + 1) − Cj,m +n−1,d+2 (−1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,d+1 (−1; . . . ; λL+R + λ1 + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +� +− +n−1 +� +˜n=0 +(n − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−1−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−1−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� +− +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−2,m−1 +˜n,n+d−˜n (i; λL+R + λ1 − i, . . . ; 0) − Cj−2,m−1 +˜n,n+d−˜n (λL+R + λ1 + 1 − i; i − 1, . . . ; 0) +� +=Cj,m +n,d (−1; λ1, . . . ; λL+R + 1) − Cj,m +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,d +(1; . . . ; λL+R + λ1 + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +− +� +Cj−1,m−1 +˜n−1,n+d−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� +− +� +Cj−2,m−1 +˜n,n+d−˜n (i; λL+R + λ1 − i, . . . ; 0) − Cj−2,m−1 +˜n,n+d−˜n (λL+R + λ1 + 1 − i; i − 1, . . . ; 0) +�� ++ n +λR +� +i=0 +� +Cj−1,m−1 +0,n+d−1 (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +0,n+d−1 (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� +=Cj,m +n,d (−1; λ1, . . . ; λL+R + 1) − Cj,m +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,d +(1; . . . ; λL+R + λ1 + 1) += +� +� +� +� +� +� +� +� +� +Cj,m +n,d−1 (1; λ1, . . . ; λL+R + 1) − Cj,m +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,d +(1; . . . ; λL+R + λ1 + 1) = 0 +(d > 0) +Cj,m +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) + Cj−1,m +n,0 +(λ1 + 1; . . . ; λL+R + 1) +−Cj,m +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,0 +(1; . . . ; λL+R + λ1 + 1) +(d = 0) +, +(S84) +where we used (S28), (S27) of P(j − 1, n, d) and explicit solution of n = 0 case. +For d = 0 case, +Cj,m +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) + Cj−1,m +n,0 +(λ1 + 1; . . . ; λL+R + 1) +− Cj,m +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n,0 +(1; . . . ; λL+R + λ1 + 1) +=Cj,m +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) + Cj−1,m +n−1,2 (λ1; . . . ; λL+R) +− Cj,m +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n−1,2 (0; . . . ; λL+R + λ1) +=Cj,m +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) + Cj−1,m +n−1,2 (λ1; . . . ; λL+R) +− Cj,m +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n−1,2 (0; . . . ; λL+R + λ1) += +� +Cj,m +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (0; . . . ; λL+R + λ1) +� +− +� +Cj,m +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n−1,2 (λ1; . . . ; λL+R) +� +=Cj,m +n−1,3 (0; λ1 − 2, . . . ; λL+R) − Cj,m +n−1,3 (0; λ1 − 2, . . . ; λL+R) +=0, +(S85) +where we used (S29) of P(j − 1, m, n, 0) and (S27) of P(j, m, n − 1, 1). + +29 +We have proved (S27) of P(j, m, n, d). +induction for (S28) +Without losing generality, we can set i1 = 1, i2 = 2 due to (S25) of P(j, m, n, d). +Cj,m +n,d (λ) − Cj,m +n−1,d+2 (λ1:−1,2:−1) − Cj−1,m +n,d+1 +� +λλ1→λ1+λ2,ˆ2 +� +=Cjm +nd (−1; λ1, λ2, . . . ; λL+R + 1) − Cjm +n−1,d+2 (−1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n,d+1 (−1; λ1 + λ2, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, λ2, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, λ2, . . . ; 0) +� +− +n−1 +� +˜n=0 +(n − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d+1−˜n (i; λL+R − i, λ1 − 1, λ2 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (λL+R + 1 − i; i − 1, λ1 − 1, λ2 − 1, . . . ; 0) +� +− +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−2,m−1 +˜n,n+d+1−˜n (i; λL+R − i, λ1 + λ2, . . . ; 0) − Cj−2,m−1 +˜n,n+d+1−˜n (λL+R + 1 − i; i − 1, λ1 + λ2, . . . ; 0) +� +=Cjm +nd (−1; λ1, λ2, . . . ; λL+R + 1) − Cjm +n−1,d+2 (−1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n,d+1 (−1; λ1 + λ2, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λLorλR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, λ2, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − i, λ1 − 1, λ2 − 1, . . . ; 0) +− Cj−2,m−1 +˜n,n+d+1−˜n (i; λL+R − i, λ1 + λ2, . . . ; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, λ2, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R + 1 − i; i − 1, λi1 − 1, λi2 − 1, . . . ; 0) +−Cj−2,m−1 +˜n,n+d+1−˜n (λL+R + 1 − i; i − 1, λ1 + λ2, . . . ; 0) +�� ++n +λLorλR +� +i=0 +� +Cj−1,m−1 +0,n+d+1 (i; λL+R − i, λi1 − 1, λi2 − 1, . . . ; 0) − Cj−1,m−1 +0,n+d+1 (λL+R + 1 − i; i − 1, λi1 − 1, λi2 − 1, . . . ; 0) +� +=Cjm +nd (−1; λi1, λi2, . . . ; λL+R + 1) − Cjm +n−1,d+1 (1; λi1 − 1, λi2 − 1, . . . ; λL+R + 1) − Cj−1,m +n,d +(1; λi1 + λi2, . . . ; λL+R + 1) += +� +� +� +� +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λi1, λi2, . . . ; λL+R + 1) − Cjm +n−1,d+1 (1; λi1 − 1, λi2 − 1, . . . ; λL+R + 1) +−Cj−1,m +n,d +(1; λi1 + λi2, . . . ; λL+R + 1) += 0 +(d > 0) +Cj−1,m +n0 +(λi1 + 1, λi2, . . . ; λL+R + 1) + Cjm +n−1,1 (0; λi1 − 1, λi2, . . . ; λL+R + 1) +−Cjm +n−1,1 (1; λi1 − 1, λi2 − 1, . . . ; λL+R + 1) − Cj−1,m +n,0 +(1; λi1 + λi2, . . . ; λL+R + 1) +(d = 0) +, +(S86) +where we used (S28) of P(j − 1, m − 1, m, d) and the explicit form of n = 0 case in the third equality. For d = 0 case, +we have +Cj−1,m +n0 +(λ1 + 1, λ2, . . . ; λL+R + 1) + Cjm +n−1,1 (0; λ1 − 1, λ2, . . . ; λL+R + 1) +− Cjm +n−1,1 (1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n,0 +(1; λ1 + λ2, . . . ; λL+R + 1) +=Cj−1,m +n0 +(λ1 + 1, λ2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(1; λ1 + λ2, . . . ; λL+R + 1) ++ Cjm +n−1,1 (0; λ1 − 1, λ2, . . . ; λL+R + 1) − Cjm +n−1,1 (1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) +=Cj−1,m +n−1,2 (λ1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (0; λ1 + λ2 − 1, . . . ; λL+R + 1) ++ Cjm +n−1,1 (0; λ1 − 1, λ2, . . . ; λL+R + 1) − Cjm +n−1,1 (1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) +=Cjm +n−2,3 (0; λ1 − 2, λ2 − 1, . . . ; λL+R + 1) − Cjm +n−2,3 (0; λ1 − 2, λ2 − 1, . . . ; λL+R + 1) +=0, +(S87) + +30 +where we used (S29) of P(j − 1, m, n, 0) in the second equality +Cj−1,m +n0 +(λ1 + 1, λ2, . . . ; λL+R + 1) − Cj−1,m +n,0 +(1; λ1 + λ2, . . . ; λL+R + 1) +=Cj−1,m +n−1,2 (λ1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (0; λ1 + λ2 − 1, . . . ; λL+R + 1) , +and used (S28) and (S27) of P(j, m, n − 1, 1) in the third equality, +Cjm +n−1,1 (0; λ1 − 1, λ2, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (0; λ1 + λ2 − 1, . . . ; λL+R + 1) = Cjm +n−2,3 (0; λ1 − 2, λ2 − 1, . . . ; λL+R + 1) , +Cjm +n−1,1 (1; λ1 − 1, λ2 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (λ1, λ2 − 1, . . . ; λL+R + 1) = Cjm +n−1,1 (0; λ1 − 2, λ2 − 1, . . . ; λL+R + 1) . +We have proved (S28) of P(j, m, n, d). +induction for (S29) +We first consider a = L, b = i(1 ≤ i ≤ w) case. Without losing generality, we can set i = 1 due to (S25) of +P(j, m, n, d). +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,1:+1) − Cj,m +n−1,d+2 (λL:−1,1:−1) + Cj,m +n−1,d+2 (λL:−2) +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 + 1, . . . ; λL+R) +− Cjm +n−1,d+2 (−1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+2 (−1; λ1, . . . ; λL+R − 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +� +− +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − 1 − i, λ1 + 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R − i; i − 1, λ1 + 1, . . . ; 0) +� +− +n−1 +� +˜n=0 +(n − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d+1−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� ++ +n−1 +� +˜n=0 +(n − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d+1−˜n (i; λL+R − 2 − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (λL+R − 1 − i; i − 1, λ1, . . . ; 0) +� +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 + 1, . . . ; λL+R) +− Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+1 (1; λ1, . . . ; λL+R − 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − 1 − i, λ1 + 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R − i; i − 1, λ1 + 1, . . . ; 0) +� +− +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� ++ +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 2 − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − 1 − i; i − 1, λ1, . . . ; 0) +�� ++ (n + 1) +λR +� +i=0 +� +Cj−1,m−1 +0,n+d+1 (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +0,n+d+1 (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +− +� +Cj−1,m−1 +0,n+d+1 (i; λL+R − 2 − i, λ1, . . . ; 0) − Cj−1,m−1 +0,n+d+1 (λL+R − 1 − i; i − 1, λ1, . . . ; 0) +�� +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 + 1, . . . ; λL+R) +− Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+1 (1; λ1, . . . ; λL+R − 1) , +(S88) + +31 +where we used the identity (S29) of P(j − 1, m − 1, n, d) +0 =Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − 1 − i, λ1 + 1, . . . ; 0) +− Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 2 − i, λ1, . . . ; 0) , +0 =Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R − i; i − 1, λ1 + 1, . . . ; 0) +− Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) + Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − 1 − i; i − 1, λ1, . . . ; 0) , +and we used the expression of the n = 0 case, +�j + n + d − 1 +m − 1 +� +− +�j + n + d − 1 +m − 2 +� += Cj−1,m−1 +0,n+d+1 (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) = Cj−1,m−1 +0,n+d+1 (λL+R − i; i − 1, λ1 − 1, . . . ; 0) += Cj−1,m−1 +0,n+d+1 (i; λL+R − 2 − i, λ1, . . . ; 0) = Cj−1,m−1 +0,n+d+1 (λL+R − 1 − i; i − 1, λ1, . . . ; 0) , +in the lase equality. Thus we can see +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,1:+1) − Cj,m +n−1,d+2 (λL:−1,1:−1) + Cj,m +n−1,d+2 (λL:−2) +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 + 1, . . . ; λL+R) +− Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) − Cjm +n−1,d+1 (1; λ1, . . . ; λL+R − 1) += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λ1, . . . ; λL+R + 1) − Cjm +n,d−1 (1; λ1 + 1, . . . ; λL+R) +−Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+1 (1; λ1, . . . ; λL+R − 1) = 0 +(d > 0) +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1 + 2; . . . ; λL+R) ++Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λ1, . . . ; λL+R) +−Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1, . . . ; λL+R − 1) +(d = 0) +. +(S89) +For d = 0 case, we have +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1 + 2; . . . ; λL+R) + Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) +− Cjm +n−1,1 (0; λ1, . . . ; λL+R) − Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1, . . . ; λL+R − 1) +=Cj−1,m +n−1,2 (λ1; . . . ; λL+R) − Cj−1,m +n−1,2 (λ1 + 1; . . . ; λL+R − 1) + Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) +− Cjm +n−1,1 (0; λ1, . . . ; λL+R) − Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1, . . . ; λL+R − 1) += − Cjm +n−2,3 (−1; λ1 − 1, . . . ; λL+R) + Cjm +n−2,3 (0; λ1 − 1, . . . ; λL+R − 1) ++ Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) +=0, +(S90) +where we used (S29) of P(j − 1, m, n, 0) in the first equality, +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1 + 2; . . . ; λL+R) = Cj−1,m +n−1,2 (λ1; . . . ; λL+R) − Cj−1,m +n−1,2 (λ1 + 1; . . . ; λL+R − 1) , +and used the identity of (S27) of P(j, m, n − 1, 1) in the second equality, +Cjm +n−1,1 (0; λ1, . . . ; λL+R) − Cj−1,m +n−1,2 (λ1; . . . ; λL+R) = Cjm +n−2,3 (−1; λ1 − 1, . . . ; λL+R) , +Cjm +n−1,1 (1; λ1, . . . ; λL+R − 1) − Cj−1,m +n−1,2 (λ1 + 1; . . . ; λL+R − 1) = Cjm +n−2,3 (0; λ1 − 1, . . . ; λL+R − 1) , +and used the identity of (S29) of P(j, m, n − 1, 1) at the last equality. +We next consider b = L, a = i(1 ≤ i ≤ w) case. Without losing generality, we can set i = 1 due to (S25) of +P(j, m, n, d). + +32 +Cj,m +n,d (λ) − Cj,m +n,d (λL:+1,1:−1) − Cj,m +n−1,d+2 (λL:−1,1:−1) + Cj,m +n−1,d+2 (λ1:−2) +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 − 1, . . . ; λL+R + 2) +− Cjm +n−1,d+2 (−1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+2 (−1; λ1 − 2, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +� +− +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R + 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 2 − i; i − 1, λ1 − 1, . . . ; 0) +� +− +n−1 +� +˜n=0 +(n − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d+1−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� ++ +n−1 +� +˜n=0 +(n − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d+1−˜n (i; λL+R − i, λ1 − 2, . . . ; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (λL+R + 1 − i; i − 1, λ1 − 2, . . . ; 0) +� +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 − 1, . . . ; λL+R + 2) +− Cjm +n−1,d+2 (−1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+2 (−1; λ1 − 2, . . . ; λL+R + 1) ++ +n +� +˜n=0 +(n + 1 − ˜n) +λR +� +i=0 +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) +− +� +Cj−1,m−1 +˜n,n+d−˜n (i; λL+R + 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 2 − i; i − 1, λ1 − 1, . . . ; 0) +� +− +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +� ++ +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − i, λ1 − 2, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R + 1 − i; i − 1, λ1 − 2, . . . ; 0) +�� ++ (n + 1) +λR +� +i=0 +� +Cj−1,m−1 +0,n+d+1 (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +0,n+d+1 (λL+R − i; i − 1, λ1 − 1, . . . ; 0) +− +� +Cj−1,m−1 +0,n+d+1 (i; λL+R − i, λ1 − 2, . . . ; 0) − Cj−1,m−1 +0,n+d+1 (λL+R + 1 − i; i − 1, λ1 − 2, . . . ; 0) +�� +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 − 1, . . . ; λL+R + 2) +− Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+1 (1; λ1 − 2, . . . ; λL+R + 1) , +(S91) +where we used the identity (S29) of P(j − 1, m − 1, n, d) in the last equality, +0 =Cj−1,m−1 +˜n,n+d−˜n (i; λL+R − i, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (i; λL+R + 1 − i, λ1 − 1, . . . ; 0) +− Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (i; λL+R − i, λ1 − 2, . . . ; 0) , +0 =Cj−1,m−1 +˜n,n+d−˜n (λL+R + 1 − i; i − 1, λ1, . . . ; 0) − Cj−1,m−1 +˜n,n+d−˜n (λL+R + 2 − i; i − 1, λ1 − 1, . . . ; 0) +− Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R − i; i − 1, λ1 − 1, . . . ; 0) + Cj−1,m−1 +˜n−1,n+d+2−˜n (λL+R + 1 − i; i − 1, λ1 − 2, . . . ; 0) , +and we used +�j + n + d +m − 1 +� +− +�j + n + d +m − 2 +� += Cj−1,m−1 +0,n+d+1 (i; λL+R − 1 − i, λ1 − 1, . . . ; 0) = Cj−1,m−1 +0,n+d+1 (λL+R − i; i − 1, λ1 − 1, . . . ; 0) += Cj−1,m−1 +0,n+d+1 (i; λL+R − i, λ1 − 2, . . . ; 0) = Cj−1,m−1 +0,n+d+1 (λL+R + 1 − i; i − 1, λ1 − 2, . . . ; 0) , + +33 +in the last equality. Thus we have +Cj,m +n,d (λ) − Cj,m +n,d (λL:+1,1:−1) − Cj,m +n−1,d+2 (λL:−1,1:−1) + Cj,m +n−1,d+2 (λ1:−2) +=Cjm +nd (−1; λ1, . . . ; λL+R + 1) − Cjm +nd (−1; λ1 − 1, . . . ; λL+R + 2) +− Cjm +n−1,d+2 (−1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+2 (−1; λ1 − 2, . . . ; λL+R + 1) += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Cjm +n,d−1 (1; λ1, . . . ; λL+R + 1) − Cjm +n,d−1 (1; λ1 − 1, . . . ; λL+R + 2) +−Cjm +n−1,d+1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,d+1 (1; λ1 − 2, . . . ; λL+R + 1) = 0 +(d > 0) +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1; . . . ; λL+R + 2) ++Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cjm +n−1,1 (0; λ1 − 2, . . . ; λL+R + 2) +−Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1 − 2, . . . ; λL+R + 1) +(d = 0) +, +(S92) +where we have used (S29) of P(j, m, n − 1, d + 1) in the last equality of d > 0 case. For d = 0 case, we have +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1; . . . ; λL+R + 2) + Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) +− Cjm +n−1,1 (0; λ1 − 2, . . . ; λL+R + 2) − Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1 − 2, . . . ; λL+R + 1) +=Cj−1,m +n−1,2 (λ1; . . . ; λL+R) − Cj−1,m +n−1,2 (λ1 − 1; . . . ; λL+R + 1) + Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) +− Cjm +n−1,1 (0; λ1 − 2, . . . ; λL+R + 2) − Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) + Cjm +n−1,1 (1; λ1 − 2, . . . ; λL+R + 1) +=Cjm +n−2,3 (−1; λ1 − 2, . . . ; λL+R + 1) − Cjm +n−2,3 (0; λ1 − 2, . . . ; λL+R) +− Cjm +n−1,1 (0; λ1 − 2, . . . ; λL+R + 2) + Cjm +n−1,1 (1; λ1 − 2, . . . ; λL+R + 1) +=0, +(S93) +where we used the identity of (S29) of P(j − 1, m, n, 0) in the first equality +Cj−1,m +n0 +(λ1 + 1; . . . ; λL+R + 1) − Cj−1,m +n0 +(λ1; . . . ; λL+R + 2) = Cj−1,m +n−1,2 (λ1; . . . ; λL+R) − Cj−1,m +n−1,2 (λ1 − 1; . . . ; λL+R + 1) , +and used the identity of (S27) of P(j, m, n − 1, 1) in the second equality +Cjm +n−1,1 (1; λ1 − 1, . . . ; λL+R) − Cj−1,m +n−1,2 (λ1; . . . ; λL+R) = Cjm +n−2,3 (0; λ1 − 2, . . . ; λL+R) , +Cjm +n−1,1 (0; λ1 − 1, . . . ; λL+R + 1) − Cj−1,m +n−1,2 (λ1 − 1; . . . ; λL+R + 1) = Cjm +n−2,3 (−1; λ1 − 2, . . . ; λL+R + 1) , +and used the identity of (S29) of P(j, m, n − 1, 1) in the last equality. +For a = L, b = R case, using above result +Cj,m +n,d (λ) − Cj,m +n,d (λL:−1,R:+1) − Cj,m +n−1,d+2 (λL:−1,R:−1) + Cj,m +n−1,d+2 (λL:−2) += +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR + 1) +� +− +n−1 +� +˜n=0 +(n − ˜n) +� +Cj−1,m−1 +˜n,n+d+1−˜n (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d+1−˜n (0; λL − 2, λ1, . . . , λw; λR) +� += +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR + 1) +− +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (0; λL − 2, λ1, . . . , λw; λR) +�� ++ (n + 1) +� +Cj−1,m−1 +0,n+d+1 (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +0,n+d+1 (0; λL − 2, λ1, . . . , λw; λR) +� += +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (0; λL − 2, λ1, . . . , λw; λR) +− +� +Cj−1,m−1 +˜n−1,n+d+2−˜n (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (0; λL − 2, λ1, . . . , λw; λR) +�� +=0, +(S94) + +34 +where we used the following identity from (S29) of P(j − 1, m − 1, n, d) in the third equality +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR + 1) += Cj−1,m−1 +˜n−1,n+d+2−˜n (λL − 1; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n−1,n+d+2−˜n (0; λL − 2, λ1, . . . , λw; λR) . +a = R, b = L case is attributed to a = L, b = R case using (S25). +For a = i1, b = i2(1 ≤ i1, i2 ≤ w) case, we can set i1 = 1, i2 = 2 without losing generality due to (S25) of P(j, m, n, d) +and we have +Cj,m +n,d (λ) − Cj,m +n,d (λi1:−1,i2:+1) − Cj,m +n−1,d+2 (λi1:−1,i2:−1) + Cj,m +n−1,d+2 (λi1:−2) +=Cj,m +n,d (λ) − Cj,m +n,d (λi1:−1,i2:+1) − Cj,m +n−1,d+2 (λi1:−1,i2:−1) + Cj,m +n−1,d+2 (λi1:−2) +− Cj,m +n,d (λL:+1,i1:−1) + Cj,m +n,d (λL:+1,i1:−1) + Cj,m +n−1,d+2 (λL:−1,i1:−1) − Cj,m +n−1,d+2 (λL:−1,i1:−1) +=Cj,m +n,d (λ) − Cj,m +n,d (λL:+1,i1:−1) − Cj,m +n−1,d+2 (λL:−1,i1:−1) + Cj,m +n−1,d+2 (λi1:−2) ++ Cj,m +n,d ((λL:+1,i1:−1)) − Cj,m +n,d ((λL:+1,i1:−1)L:−1,i2:+1) − Cj,m +n,d ((λL:+1,i1:−1)L:−1,i2:−1) + Cj,m +n,d ((λL:+1,i1:−1)L:−2) +=0, +(S95) +where we use the following equation, which is the result of a = L, b = i case, +0 = Cj,m +n,d (λ) − Cj,m +n,d (λL:+1,i1:−1) − Cj,m +n−1,d+2 (λL:−1,i1:−1) + Cj,m +n−1,d+2 (λi1:−2) , +0 = Cj,m +n,d ((λL:+1,i1:−1)) − Cj,m +n,d ((λL:+1,i1:−1)L:−1,i2:+1) − Cj,m +n,d ((λL:+1,i1:−1)L:−1,i2:−1) + Cj,m +n,d ((λL:+1,i1:−1)L:−2) . +We have proved (S29) of P(j, m, n, d). +proof of (S32) +We prove (S32). From the recursion equation, we have +∆Cj,m +n,d (λ) = Cj,m +n,d (λR:−1) − Cj,m +n,d (λL:−1) += +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR) +� +, +(S96) +using this equation, we have +∆Cj,m +n,d (λ) − 2∆Cj,m +n−1,d+1 (λ) + ∆Cj,m +n−2,d+2 (λ) += +n +� +˜n=0 +(n + 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR) +� +− 2 +n−1 +� +˜n=0 +(n − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR) +� ++ +n−2 +� +˜n=0 +(n − 1 − ˜n) +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR) +� += +n−2 +� +˜n=0 +(n + 1 − ˜n − 2(n − ˜n) + n − 1 − ˜n) +× +� +Cj−1,m−1 +˜n,n+d−˜n (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +˜n,n+d−˜n (0; λL − 1, λ1, . . . , λw; λR) +� ++ Cj−1,m−1 +n,d +(λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +n,d +(0; λL − 1, λ1, . . . , λw; λR) ++ 2 +� +Cj−1,m−1 +n−1,d+1 (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +n−1,d+1 (0; λL − 1, λ1, . . . , λw; λR) +� +− 2 +� +Cj−1,m−1 +n−1,d+1 (λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +n−1,d+1 (0; λL − 1, λ1, . . . , λw; λR) +� +=Cj−1,m−1 +n,d +(λL; λ1, . . . , λw, λR − 1; 0) − Cj−1,m−1 +n,d +(0; λL − 1, λ1, . . . , λw; λR) . +(S97) +Then we have proved (S32). + +35 +S5. PROOF OF IDENTITIES OF Aσµ +i +In this section, we prove the equation (S36)–(S41). Note that the diagrams generated by the commutator of H +and connected diagrams satisfy rule(ii) for the connected diagram. In the following, we assume the type of land in a +diagram satisfies rule(ii), and we do not explicitly write the type of land if not mentioned. We call the two coasts are +(not) looking in the same direction if the lands of the two coasts are (not) on the same row. We omit I ’s in the coast +and indicate the length of the coast by an arrow, for example. +I I I += +3 +. +proof of (S36) +In this subsection, we prove A++ +i += A−− +i += 0. We firstly prove A++ +i += 0. Let �Ψ be a diagram where the i-th coast +and i ± 1-th coasts are not adjacent. +We consider the case that the left and right side of i-th coast is an overlap, such as +�Ψ = +λi + 1 +, +(S98) +where +represents the sequence of +and +represents the rest of the diagram, and the I’s of the coast +are not shown. In this case, the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list +is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +λi +− Cj,m +n−1,d+1 (λi:−1) +λi ++Cj,m +n,d−1 (λi:+1) +λi + 2 ++ Cj,m +n,d−1 (λi:+1) +λi + 2 +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) − Cj,m +n,d−1 (λi:+1) +� +�Ψ +=0, +(S99) +where +λi +, +λi +∈ Sk,j,m +n−1,d,g, +λi + 2 +, +λi + 2 +∈ +Sk,j,m +n,d−1,g. +We next consider the case that the left and right side of the i-th coast is both gaps, such as +�Ψ = +I I +I I +I I +I I +λi + 1 +. +(S100) +In this case, the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +I I I +I I I +I I +I I +λi +− Cj,m +n−1,d+1 (λi:−1) +I I +I I +I I I +I I I +λi +−Cj,m +n,d−1 (λi:+1) +I +I +I I +I I +λi + 2 +− Cj,m +n,d−1 (λi:+1) +I I +I I +I +I +λi + 2 +� +� +� +=(−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) − Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:−1) − Cj,m +n,d−1 (λi:−1) +� +�Ψ +=0, +(S101) +where +I I I +I I I +I I +I I +λi +, +I I +I I +I I I +I I I +λi +∈ Sk,j,m +n,d,g+1 and +I +I +I I +I I +λi + 2 +, +I I +I I +I +I +λi + 2 +∈ +Sk,j,m +n−1,d,g−1. + +36 +We next consider the case that left side of the i-th coast is a gap and the right side is a overlap, such as +�Ψ = +I I +I I +λi + 1 +. +(S102) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�Cj,m +n−1,d+1 (λi:−1) +I I I +I I I +λi ++ Cj,m +n−1,d+1 (λi:−1) +I I +I I +λi +−Cj,m +n,d−1 (λi:+1) +I +I +λi + 2 ++ Cj,m +n,d−1 (λi:+1) +I I +I I +λi + 2 +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:−1) − Cj,m +n,d−1 (λi:−1) +� +�Ψ +=0, +(S103) +where +I I I +I I I +λi +∈ +Sk,j,m +n,d,g+1, +I I +I I +λi +∈ +Sk,j,m +n−1,d+1,g, +I +I +λi + 2 +∈ +Sk,j,m +n−1,d,g−1, +I I +I I +λi + 2 +∈ Sk,j,m +n,d−1,g. +In the same way, for the case that the right side of the i-th coast is a gap, and the left side of that is a land of +non-zero length, and i − 1-th coast and i-th coast are not horizontally adjacent, such as +�Ψ = +I I +I I +λi + 1 +, +(S104) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is 0. +Thus we have proved A++ +i += 0 for all cases. +We next prove A−− +i += 0. Let �Ψ be a diagram where the i-th coast and i ± 1-th coasts are adjacent. +We consider the case that the i-th and i ± 1-th coast are adjacent and they are looking in the same direction, such +as +�Ψ = +I +I +λi + 1 +. +(S105) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +I +I +λi +− Cj,m +n−1,d+1 (λi:−1) +I +I +λi +� +� +� +=(−1)n+m+g2 +� +Cj,m +n−1,d+1 (λi:−1) − Cj,m +n−1,d+1 (λi:−1) +� +=0, +(S106) +where +I +I +λi +, +Cj,m +n−1,d+1 (λi:−1) +I +I +λi +∈ Sk,j,m +n−1,d,g. +We consider the case that the i-th and i ± 1-th coast are adjacent and the i-th and i + 1(i − 1)-th coast are (are +not) looking in the same direction, such as +�Ψ = +λi−1+1 +I +λi + 1 +. +(S107) + +37 +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +I +λi +− Cj,m +n−1,d+1 (λi:−1) +I +λi ++Cj,m +n−1,d+1 (λi:−1) +I +I +I +λi +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + 2Cj,m +n−1,d+1 (λi:−1) − Cj,m +n−1,d+1 (λi:−1) +� +�Ψ +=0, +(S108) +where +I +λi +, +I +λi +∈ Sk,j,m +n−1,d+1,g, +I +I +I +λi +∈ Sk,j,m +n,d,g+1 . +In the same way, for the case that the i-th and i ± 1-th coast are adjacent and the i-th and i − 1(i + 1)-th coast are +(not) looking in the same direction, such as +�Ψ = +λi+1+1 +I +λi + 1 +, +(S109) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is 0. +We consider the case that the i-th and i ± 1-th coast are adjacent and they are not looking in the same direction, +such as +�Ψ = +λi−1+1 +λi+1+1 +λi + 1 +. +(S110) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +λi +− Cj,m +n−1,d+1 (λi:−1) +λi ++Cj,m +n−1,d+1 (λi:−1) +I +I +λi ++ Cj,m +n−1,d+1 (λi:−1) +I +I +λi +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) − Cj,m +n−1,d+1 (λi:−1) +� +�Ψ +=0, +(S111) +where +λi +, +λi +∈ Sk,j,m +n,d+1,g, +I +I +λi +, +I +I +λi +∈ +Sk,j,m +n,d+1,g+1. +Thus we have proved A−− +i += 0 for all cases. +proof of (S37) +We prove (S37). Let �Ψ be a diagram where i-th coast and i − 1(i + 1)-th coast are (are not) adjacent. +We first consider the case that the i-th and i + 1-th coasts are adjacent and looking in the same direction, and the +left side of the i-th coast is an overlap , such as +�Ψ = +I +λi + 1 +. +(S112) + +38 +In this case, the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +I +λi +− Cj,m +n−1,d+1 (λi:−1) +I +λi ++Cj,m +n,d−1 (λi:+1) +I +λi + 2 +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + 2Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ, +(S113) +where +I +λi +, +I +λi +∈ Sk,j,m +n−1,d+1,g and +I +λi + 2 +∈ Sk,j,m +n,d−1,g. +In the same way, for the case that the i-th and i − 1-th coasts are adjacent and looking in the same direction and +the right side of the i-th coast is a overlap, such as +�Ψ = +I +λi + 1 +, +(S114) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g+1 � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ. +(S115) +We next consider the case that the i-th and i + 1-th coasts are adjacent and looking in the same direction and the +left side of the i-th coast is a gap, such as +�Ψ = +I I +I I +I +λi + 1 +. +(S116) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�Cj,m +n−1,d+1 (λi:−1) +I I I +I I I +I +λi +− Cj,m +n−1,d+1 (λi:−1) +I I +I I +I +λi +−Cj,m +n,d−1 (λi:+1) +I +I +I +λi + 2 +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + 2Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ, +(S117) +where +I I I +I I I +I +λi +∈ Sk,j,m +n,d,g+1, +I I +I I +I +λi +∈ Sk,j,m +n−1,d+1,g, +I +I +I +λi + 2 +∈ Sk,j,m +n,d−1,g. +In the same way, for the case that the i-th and i − 1-th coasts are adjacent and looking in the same direction and +the right side of the i-th coast is a gap, such as +�Ψ = +I +I +I +I +I +λi + 1 +, +(S118) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g+1 � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ. +(S119) + +39 +We next consider the case that the i-th and i + 1-th coasts are adjacent and looking in the different direction and +the left side of the i-th coast is a overlap, such as +�Ψ = +λi + 1 +λi+1+1 +. +(S120) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d+1 (λi:−1) +λi +− Cj,m +n−1,d+1 (λi:−1) +λi ++Cj,m +n−1,d+1 (λi:−1) +I +I +λi ++ Cj,m +n,d−1 (λi:+1) +λi + 2 +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ, +(S121) +where +λi +∈ +Sk,j,m +n−1,d+1,g, +λi +∈ +Sk,j,m +n−1,d+1,g, +I +I +λi +∈ +Sk,j,m +n,d,g+1, +λi + 2 +∈ Sk,j,m +n,d−1,g. +In the same way, for the case that the i-th and i − 1-th coasts are adjacent and looking in the different direction +and the right side of the i-th coast is a overlap, such as +�Ψ = +λi + 1 +λi−1+1 +, +(S122) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g+1 � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ. +(S123) +We next consider the case that the i-th and i + 1-th coasts are adjacent and looking in the different direction and +the left side of the i-th coast is a gap, such as +�Ψ = +I I +I I +λi + 1 +λi+1+1 +. +(S124) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g +� +� +�Cj,m +n−1,d+1 (λi:−1) +I I I +I I I +λi ++ Cj,m +n−1,d+1 (λi:−1) +I I +I I +I +I +λi +−Cj,m +n,d−1 (λi:+1) +I +I +λi + 2 +− Cj,m +n−1,d+1 (λi:+1) +I I +I I +λi +� +� +� +=(−1)n+m+g � +−Cj,m +n−1,d+1 (λi:−1) + Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) − Cj,m +n−1,d+1 (λi:−1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ, +(S125) +where +I I I +I I I +λi +, +I I +I I +I +I +λi +∈ +Sk,j,m +n−1,d+1,g+1, +I +I +λi + 2 +∈ +Sk,j,m +n,d−1,g−1, +I I +I I +λi +∈ Sk,j,m +n,d−1,g. + +40 +In the same way, for the case that the i-th and i − 1-th coasts are adjacent and looking in the different direction +and the right side of the i-th coast is a gap, such as +�Ψ = +I +I +I +I +λi + 1 +λi−1+1 +. +(S126) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λi:±1 is +(−1)n+m+g+1 � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +�Ψ. +(S127) +Thus we have proved A+− +i += −A−+ +i += (−1)n+m+g � +Cj,m +n−1,d+1 (λi:−1) + Cj,m +n,d−1 (λi:+1) +� +for all cases. +proof of (S38) and (S39) +We prove (S38). Let �Ψ be a diagram where the leftmost coast(0-th coast) and 1-th coast are not adjacent. +We consider the contribution to the cancellation of �Ψ from the diagram in Qj +k whose list is λL:±1 and from the +diagram in Qj−1 +k +whose list is 0→(λL:−1) = {0; λL − 1, . . .}. +(S39) is proved in the same way. +We consider the case that the right side of the leftmost coast is overlap and λL > 0, such as +�Ψ = +λL +. +(S128) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:±1 and from the diagram +in Qj−1 +k +whose list is 0→(λL:−1) is +(−1)n+m+g +� +� +�Cj,m +n,d (λL:−1) I +I +λL − 1 +− Cj,m +n−1,d+1 (λL:−1) +λL − 1 +−Cj,m +n−1,d (λL:+1) +λL + 1 ++ Cj,m +n,d−1 (λL:+1) +λL + 1 +−Cj−1,m−1 +n,d +(0→(λL:−1)) +λL +� +� +� +=(−1)n+m+g � +−Cj,m +n,d (λL:−1) + Cj,m +n−1,d+1 (λL:−1) ++Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S129) +where I +I +λL − 1 +∈ Sk,j,m +n,d,g , +λL − 1 +∈ Sk,j,m +n−1,d+1,g, +λL + 1 +∈ Sk,j,m +n−1,d,g, +λL + 1 +∈ +Sk,j,m +n,d−1,g, +λL +∈ Sj−1,m−1 +n,d,g +. The gap enclosed by the dotted line is virtual gap that is on the leftmost or +rightmost of the diagram. A virtual gap is not counted for the value of gΨ and dΨ. +In the same way, for the case that the left side of the rightmost coast is an overlap, and λR > 0, such as +�Ψ = +λR +. +(S130) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qk whose list is λR:±1 is +(−1)n+m+g+1 � +−Cj,m +n,d (λR:−1) + Cj,m +n−1,d+1 (λR:−1) ++Cj,m +n−1,d (λR:+1) − Cj,m +n,d−1 (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S131) + +41 +We consider the case that λL > 0 and the right side of the leftmost coast is a gap, such as, +�Ψ = +I I +I I +λL +. +(S132) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d (λL:+1) +I I +I I +λL + 1 ++ Cj,m +n,d (λL:−1) I +I +I I +I I +λL − 1 ++Cj,m +n−1,d+1 (λL:−1) +I I I +I I I +λL − 1 +− Cj,m +n,d−1 (λL:+1) +I +I +λL + 1 +−Cj−1,m−1 +n,d +(0→(λL:−1)) +I I +I I +λL +� +� +� +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) − Cj,m +n,d (λL:−1) ++Cj,m +n−1,d−1 (λL:−1) − Cj,m +n,d−1 (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S133) +where +I I +I I +λL + 1 +∈ Sk,j,m +n−1,d,g, +I +I +I I +I I +λL − 1 +∈ Sk,j,m +n,d,g , +I I I +I I I +λL − 1 +∈ Sk,j,m +n,d,g+1, +I +I +λL + 1 +∈ +Sk,j,m +n,g−1,d, +I I +I I +λL +∈ Sk,j−1,m−1 +n,d,g +. +In the same way, for the case that λR > 0 and the left side of the rightmost coast is a gap, such as, +�Ψ = +I +I +I +I +λR +. +(S134) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 is +(−1)n+m+1 � +Cj,m +n−1,d (λR:+1) + Cj,m +n−1,d+1 (λR:−1) + Cj,m +n,d−1 (λR:+1) − Cj,m +n,d−1 (λR:+1) ++Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S135) +We consider the case that λL = 0 and the leftmost of the diagram is overlap, such as +�Ψ = +. +(S136) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 is +(−1)n+m+g � +−Cj,m +n−1,d (λL:+1) I +− Cj,m +n−1,d (λL:+1) I ++Cj,m +n,d−1 (λL:+1) I ++ Cj,m +n,d−1 (λL:+1) I +� +=(−1)n+m+g2 +� +Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) +� +�Ψ +=(−1)n+m+g � +−Cj,m +n,d (λL:−1) + Cj,m +n−1,d+1 (λL:−1) ++Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S137) +where I +, +I +∈ Sk,j,m +n−1,d,g, +I +, +I +∈ Sk,j,m +n,d−1,g and we used Cj,m +n−1,d (λL:+1) = Cj,m +n−1,d+1 (λL:−1) and +Cj,m +n,d−1 (λL:+1) = Cj,m +n,d (λL:−1) for λL = 0 and Cj−1,m−1 +n,d +(0→(λL:−1)) = 0 for λL = 0. +In the same way, for the case that λR = 0 and the rightmost of the diagram is a overlap, such as +�Ψ = +. +(S138) + +42 +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λR:+1 is +(−1)n+m+12 +� +Cj,m +n−1,d (λR:+1) − Cj,m +n,d−1 (λR:+1) +� +�Ψ +=(−1)n+m+1 � +−Cj,m +n,d (λR:−1) + Cj,m +n−1,d+1 (λR:−1) ++Cj,m +n−1,d (λR:+1) − Cj,m +n,d−1 (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S139) +We consider the case that λL = 0 and the leftmost of the diagram is gap, such as +�Ψ = I I +I I +. +(S140) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 is +(−1)n+m+g +� +−Cj,m +n−1,d (λL:+1) I +I I +I I +− Cj,m +(n−1)−(g−1),d−1 (λL:+1) I +I +I +� +=(−1)n+m+g2 +� +Cj,m +n−1,d (λL:+1) − Cj,m +n,d−1 (λL:+1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d−1 (λL:+1) − Cj,m +n,d (λL:−1) ++Cj−1,m−1 +n,d +(0→λ) +� +�Ψ, +(S141) +where I +I I +I I +∈ Sk,j,m +n−1,d,g, +I +I +I +∈ Sk,j,m +n,g−1,d and we used Cj,m +n−1,d (λL:+1) = Cj,m +n−1,d+1 (λL:−1) and Cj,m +n,d−1 (λL:+1) = +Cj,m +n,d (λL:−1) and Cj−1,m−1 +n,d +(0→λ) = 0 for λL = 0. +In the same way, for the case that λR = 0 and the rightmost of the diagram is a gap and a z-bar, such as +�Ψ = +I +I +I +I +, +(S142) +the contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λR:+1 is +(−1)n+m+g+1 � +Cj,m +n−1,d (λR:+1) + Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d−1 (λR:+1) − Cj,m +n,d (λR:−1) ++Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S143) +Thus we have proved (S38) and (S39) for all cases. +proof of (S40) and (S41) +We prove (S40). Let �Ψ be a diagram where the leftmost coast(0-th coast) and 1-th coast are adjacent. +We consider the contribution to the cancellation of �Ψ from the diagram in Qj +k whose list is λL:±1 and from the +diagram in Qj−1 +k +whose list is 0→(λL:−1) = {0; λL − 1, . . .}. For λL = 0 case, We also consider the contribution from +the diagram in Qj +k whose list is λ1:−1 = {0; λ1 − 1, . . .}. We note that we incorporate this contribution to A− +L, not to +A−σR +i +1 +for convenience. +(S41) is proved in the same way. +We firstly consider the case that the leftmost coast(0-th coast) and 1-th coast are adjacent and looking in the same +direction and λL > 0, such as +�Ψ = +I +λL +. +(S144) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:±1 and from the diagram + +43 +in Qj−1 +k +whose list is 0→(λL:−1) is +(−1)n+m+g +� +� +�Cj,m +n,d (λL:−1) I +I +I +λL − 1 +− Cj,m +n−1,d+1 (λL:−1) +I +λL − 1 +−Cj,m +n−1,d (λL:+1) +I +λL + 1 +− Cj−1,m−1 +n,d +(0→(λL:−1)) +I +λL +� +� +� +=(−1)n+m+g � +−Cj,m +n,d (λL:−1) + 2Cj,m +n−1,d+1 (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S145) +where +I +λL − 1 +∈ Sk,j,m +n,d,g , +I +λL − 1 +∈ Sk,j,m +n−1,d+1,g, +I +λL + 1 +∈ Sk,j,m +n,d−1,g, +I +λL +∈ Sj−1,m−1 +n,d,g +. +In the same way, for the case that the leftmost coast(w + 1-th coast) and w-th coast are adjacent and looking in +the same direction and λR > 0, such as +�Ψ = +I +λR +. +(S146) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λR:±1 and from the diagram +in Qj−1 +k +whose list is 0→(λL:−1) is +(−1)n+m+g+1 � +−Cj,m +n,d (λR:−1) + 2Cj,m +n−1,d+1 (λR:−1) + Cj,m +n−1,d (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S147) +We next consider the case that the leftmost coast(0-th coast) and 1-th coast are adjacent and looking in the different +direction and λL > 0, such as +�Ψ = +λL +λ1 + 1 +. +(S148) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:±1 and from the diagram +in Qj−1 +k +whose list is 0→(λL:−1) is +(−1)n+m+g +� +� +�Cj,m +n,d (λL:−1) I +I +λL − 1 +− Cj,m +n−1,d+1 (λL:−1) +λL − 1 +−Cj,m +n−1,d (λL:+1) +λL + 1 ++ Cj,m +n−1,d+1 (λL:+1) +I +I +λL − 1 +−Cj−1,m−1 +n,d +(0→(λL:−1)) +λL +� +� +� +=(−1)n+m+g � +−Cj,m +n,d (λL:−1) + 2Cj,m +n−1,d+1 (λL:−1) + Cj,m +n−1,d (λL:+1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S149) +where I +I +λL − 1 +∈ Sk,j,m +n,d,g , +λL − 1 +∈ Sk,j,m +n−1,d+1,g, +λL + 1 +∈ Sk,j,m +n,d−1,g, +I +I +λL − 1 +∈ +Sk,j,m +n−1,d+1,g+1, +λL +∈ Sj−1,m−1 +n,d,g +. +In the same way, for the case that the rightmost coast(w + 1-th coast) and w-th coast are adjacent and looking in +the different directions and λR > 0, such as +�Ψ = +λR +λw + 1 +. +(S150) + +44 +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λR:±1 and from the diagram +in Qj−1 +k +whose list is 0→(λR:−1) is +(−1)n+m+g+1 � +−Cj,m +n,d (λR:−1) + 2Cj,m +n−1,d+1 (λR:−1) + Cj,m +n−1,d (λR:+1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S151) +We consider the case that λL = 0 and the left most of the diagram is 1-th coast and w > 0, such as +�Ψ = +λ1+1 +. +(S152) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 and from the diagram +in Qj−1 +k +whose list is 0→(λL:−1) = {0; λL − 1, . . .} and from the diagram in Qj +k whose list is λ1:−1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,d (λL:+1) I +λ1+1 +− Cj,m +n−1,d (λL:+1) I +λ1+1 ++Cj,m +n−1,d+1 (λ1:−1) I +I +λ1 ++ Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +� +λ1+1 +� +� +� +=(−1)n+m+g � +3Cj,m +n−1,d (λL:+1) − Cj,m +n−1,d+1 (λ1:−1) − Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +�� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + Cj,m +n−1,d+1 (λL:−1) ++ +� +Cj,m +n−1,d+1 (λL:−1) − Cj,m +n−1,d+1 (λ1:−1) +� +− Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +�� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + Cj,m +n−1,d+1 (λL:−1) ++ +� +Cj,m +n,d−1 (λi:+1) − Cj,m +n,d−1 (λL:+1) +� +− Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +�� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d−1 (λL:+1) ++ +� +Cj,m +n,d−1 (λi:+1) − Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +��� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) ++Cj,m +n−1,d+1 (λL:−1) +� +�Ψ +=(−1)n+m+g � +Cj,m +n−1,d (λL:+1) + 2Cj,m +n−1,d+1 (λL:−1) − Cj,m +n,d (λL:−1) + Cj−1,m−1 +n,d +(0→(λL:−1)) +� +�Ψ, +(S153) +where +I +λ1+1 +, +I +λ1+1 +∈ +Sk,j,m +n−1,d,g, +I +I +λ1 +∈ +Sk,j,m +n,d,g+1, +λ1+1 +∈ +Sk,j−1,m +n,d,g +and +we +used +Cj−1,m−1 +n,d +(0→(λL:−1)) = 0 for λL = 0 and used Cj,m +n−1,d (λL:+1) = Cj,m +n−1,d+1 (λL:−1) for λL = 0 and Cj,m +n−1,d+1 (λL:−1)− +Cj,m +n−1,d+1 (λ1:−1) += +Cj,m +n,d−1 (λi:+1) − Cj,m +n,d−1 (λL:+1) for (S29) and Cj,m +n,d−1 (λi:+1) − Cj−1,m +n,d +� +λλL→1+λ1,ˆ1 +� += +Cj,m +n−1,d+1 (λL:−1) which can be derived using (S27), +Cj,m +n,d−1 (λ1:+1) = Cj,m +n−1,d+1 ((λ1:+1)L:−1,1:−1) + Cj−1,m +n,d +� +(λ1:+1)λL→0+(λ1+1),ˆ1 +� += Cj,m +n−1,d+1 (λL:−1) + Cj−1,m +n,d +� +λλL→λ1+1,ˆ1 +� +. +(S154) +In the same way, for the case that λR = 0 and the rightmost of the diagram is w-th coast and w > 0, such as +�Ψ = +λw+1 +. +(S155) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 is +(−1)n+m+g+1 � +Cj,m +n−1,d (λR:+1) + 2Cj,m +n−1,d+1 (λR:−1) − Cj,m +n,d (λR:−1) + Cj−1,m−1 +n,d +((λR:−1)←0) +� +�Ψ. +(S156) + +45 +We consider the case that λL = 0 and d = 0 and w = 0 (j = 2m + 1), such as +�Ψ = +λR +. +(S157) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λL:+1 and from the diagram +whose list is λR:−1 is +(−1)n+m+g +� +� +�−Cj,m +n−1,0 (λL:+1) I +λR +− Cj,m +n−1,0 (λL:+1) I +λR ++ Cj,m +n−1,1 (λR:−1) +I +IλR − 1 +� +� +� +=(−1)n+m+g � +3Cj,m +n−1,0 (λL:+1) − Cj,m +n−1,1 (λR:−1) +� +�Ψ +=(−1)n+m+g � +2Cj,m +n−1,1 (λL:−1) + Cj,m +n−1,0 (λL:+1) − Cj,m +n,0 (λL:−1) +� +�Ψ +=(−1)n+m+g � +2Cj,m +n−1,1 (λL:−1) + Cj,m +n−1,0 (λL:+1) − Cj,m +n,0 (λL:−1) + Cj−1,m−1 +n,0 +(0→(λL:−1)) +� +�Ψ, +(S158) +where I +λR +, +I +λR +∈ Sk,j,m +n−1,0,g, +I +IλR − 1 +∈ Sk,j,m +n−1,1,g+1 and we used Cj−1,m−1 +n,d +(0→(λL:−1)) = 0 for λL = 0 +and we used Cj,m +n−1,d (λL:+1) = Cj,m +n−1,d+1 (λL:−1) and Cj,m +n−1,1 (λR:−1) = Cj,m +n,0 (λL:−1) for λL = 0. +In the same way, for We consider the case that λR = 0 and d = 0 and w = 0 (j = 2m + 1), such as +�Ψ = +λL +. +(S159) +The contributions to the cancellation of �Ψ in [Qk, H] from the diagram in Qj +k whose list is λR:+1 and λL:−1 is +(−1)n+m+g+1 � +2Cj,m +n−1,1 (λR:−1) + Cj,m +n−1,0 (λR:+1) − Cj,m +n,0 (λR:−1) + Cj−1,m−1 +n,0 +((λR:−1)←0) +� +�Ψ. +(S160) +Thus we have proved (S40) A− +L = (−1)n+m+g � +2Cj,m +n−1,1 (λL:−1) + Cj,m +n−1,0 (λL:+1) − Cj,m +n,0 (λL:−1) + Cj−1,m−1 +n,0 +(0→(λL:−1)) +� +and (S41) A− +R = (−1)n+m+g+1 � +2Cj,m +n−1,1 (λR:−1) + Cj,m +n−1,0 (λR:+1) − Cj,m +n,0 (λR:−1) + Cj−1,m−1 +n,0 +((λR:−1)←0) +� +for all +cases. + +46 +S6. PROOF OF THE COMPLETENESS OF Qk +We prove the family of the local conserved quantities {Qk}k≥2 is complete; in other words, we prove there is no +other k-support conserved quantity in the one-dimensional Hubbard model other than our Qk. +The following proof is similar in spirit to the proof of non-integrability of the spin-1/2 XYZ chain with magnetic +field [N. Shiraishi, EPL 128, 17002 (2019)]. +We assume Fk is a k-support local conserved quantity of the one-dimensional Hubbard model. We write Fk in +terms of the polynomial of U as +Fk = +je +� +j=0 +U jF j +k. +(S161) +where je is proved to be k − 1(k − 2) for even(odd) k. +Because Fk is a conserved quantity, we can see 0 = [Fk, H] = �je+1 +j=0 U j �� +F j +k, H0 +� ++ +� +F j−1 +k +, Hint +�� +and comparing +each power of U j, we have +� +F j +k, H0 +� ++ +� +F j−1 +k +, Hint +� += 0 where F −1 +k += F je+1 +k += 0. For the j = 0 case, we have +� +F 0 +k , H0 +� += 0. +At first, we prove there is no 1-support conserved quantity F1 other than �L +i=1 Zis(s =↑, ↓), which is the conservation +of electron numbers with the flavor s =↑, ↓. Then we prove that the k-support operators (k > 1) in Fk is F 0 +k and +F 0 +k = Q0 +k = I +I +k − 1 ++ ↕ with some normalization and we next prove F j +k = Qj +k with some choice of the linear combination +that fixes the freedom to add Fk′ k −j or s+d−k −j ≡ 1( +mod 2), we can prove F j +k have a diagram whose support is more than k − j. We suppose such a diagram A and A is +(s′, d′) diagram, and A has the maximum support in F j +k and satisfies s′ > k − j. [A, H0] generates (s′ + 1, d′) diagram +and it must be canceled with the commutator of (s′ + 1, d′) diagram in F j−1 +k +, and repeating this, we can see Q0 +k have + +57 +at least s′ +j > k-support operator and this is contradiction with the assumption that Fk is k-support local conserved +quantities. +Then we can write F j +k as +F j +k = +⌊ k−1−j +2 +⌋ +� +n=0 +⌊ k−1−j +2 +⌋−n +� +d=0 +F j +k(k − j − 2n − d, d), +(S236) +where F j +k(s, d) is the (s, d) diagrams in F j +k. +We consider the cancellation of (s + 1, d) diagram (s = k − j − 2n − d) in +� +F j +k, H0 +� ++ +� +F j−1 +k +, Hint +� +, that is +� +F j +k(s, d), H0 +���� +(s+1,d) + +� +F j +k(s + 1, d − 1), H0 +���� +(s+1,d) ++ +� +F j +k(s + 1, d + 1), H0 +���� +(s+1,d) + +� +F j +k(s + 2, d), H0 +���� +(s+1,d) ++ +� +F j−1 +k +(s + 1, d), Hint +� += 0, +(S237) +where O|(s,d) is the (s, d) diagrams in O. We define F j +k(s, d) = 0 if s + d > k − j or d < 0, which is out of the range +of the structure. +We prove F j +k = Qj +k by induction. We show if we suppose F j +k(s+1, d−1), F j +k(s+1, d+1), F j +k(s+2, d), F j−1 +k +(s+1, d) +are determined uniquely, F j +k(s, d) is also determined uniquely. +We have already proved the base case: F 0 +k (k, 0) is uniquely determined to be Q0 +k up to the normalization. +We already know one solution to the F j +k, which is our Qj +k. When F j +k(s+1, d−1), F j +k(s+1, d+1), F j +k(s+2, d), F j−1 +k +(s+ +1, d) are determined uniquely, they must be Qj +k(s + 1, d − 1), Qj +k(s + 1, d + 1), Qj +k(s + 2, d), Qj−1 +k +(s + 1, d) respectively. +Thus, the equation that determine F j +k(s, d) is +0 = +� +F j +k(s, d), H0 +���� +(s+1,d) + +�� +Qj +k(s + 1, d − 1), H0 +���� +(s+1,d) + +� +Qj +k(s + 1, d + 1), H0 +���� +(s+1,d) ++ +� +Qj +k(s + 2, d), H0 +���� +(s+1,d) + +� +Qj−1 +k +(s + 1, d), Hint +�� += +� +F j +k(s, d), H0 +���� +(s+1,d) − B, +(S238) +where the RHS of (S238) is the connected diagrams because disconnected diagrams generated by the commutators +in the RHS cancel within themselves, and we denote the terms in the parenthesis in the RHS of the first line by −B. +In the following, we prove F j +k(s, d) is uniquely determined by the equation (S238). We know at least one solution to +(S238), F j +k(s, d) = Qj +k(s, d). Thus, we should prove there is no other solution to (S238). +The (s + 1, d) diagram generated by the commutator of (s, d) diagram and H0 is increased the λL or λR by one. +We give an example of (s, d) = (6, 2) case: +� +I +I I , H0 +���� +6+1-support = +I +I +I +I I ++ +I +I I +I +I += I +I I I − I I +I I , +(S239) +where the list of the diagram {λL; λR} is changed by the commutator with H0 as {1; 2} → {1; 3} and {1; 2} → {2; 2}. +We define operation T on the diagram Ψ. T Ψ is the diagram whose list is {λL − 1; . . . ; λR + 1} where the list of Ψ +is {λL; . . . ; λR}. If λL = 0, we define T Ψ = 0. We give an example of the operation T : +T +� +I I +I I +� += I +I I I . +(S240) +We define ALΨ(ARΨ) is the diagram that has the same configuration as Ψ but the λL (λR) is increased by one. +We give a example of ALΨ and ARΨ: +AL +� +I +I I +� += I I +I I , +(S241) +AR +� +I +I I +� += I +I I I . +(S242) + +58 +ALΨ(ARΨ) is well-defined for λL > 0(λR > 0). For λL(λR) = 0, there are two possibility of the ALΨ(ARΨ). For +example, if Ψ = +I , there are two diagram that are increased λL by one from Ψ, which are I +I and I +I . +Thus, for λL = 0(λR = 0) case, we define A↑ +LΨ(A↑ +RΨ) by the diagram that are increased λL(λR) by adding +to the +left(right) end of the upper row of Ψ and A↓ +LΨ(A↓ +RΨ) by adding +to lower row. We give an example of A↑,↓ +L Ψ: +A↑ +L +� +I +� += I +I , +(S243) +A↓ +L +� +I +� += I +I . +(S244) +Using these notations, the [Ψ, H0]|(s+1,d) is written as [Ψ, H0]|(s+1,d) = AR(Ψ) − AL(Ψ) for λL, λR > 0 where Ψ is +(s, d) diagram. For λL = 0, we change AL(Ψ) to A↑ +L(Ψ) + A↓ +L(Ψ) and the same is for λR = 0. +We write F j +k(s, d) as +F j +k(s, d) = +� +Ψ:(s,d) +c(Ψ)Ψ, +(S245) +where the summation over Ψ runs over all the (s, d) diagrams. We prove c(Ψ) is uniquely determined from (S238). +We denote b(Ψ) by the coefficients of Ψ in B. We denote the list of Ψ by {λΨ +L; . . . ; λΨ +R} in the following. +If λΨ +L = 0 and λΨ +R > 0, the cancellation of AR(Ψ) in (S238) is c(Ψ)−b(AR(Ψ)) = 0 and c(Ψ) is uniquely determined +as c(Ψ) = b(AR(Ψ)). In the same way, we can prove c(Ψ) is determined uniquely for the case of λΨ +R = 0 and λΨ +L > 0 +and for the case of λΨ +L = 0 and λΨ +R = 0. +If λΨ +R, λΨ +L > 0, the contribution to the cancellation of AR(Ψ) in (S238) is c(Ψ) − c(T Ψ) − b(AR(Ψ)) = 0, where we +used AR(Ψ) = AL(T Ψ), and we have +c(Ψ) − c(T Ψ) = b(AR(Ψ)). +(S246) +In the same way, we have +c(T Ψ) − c(T 2Ψ) = b(AR(T Ψ)), +c(T 2Ψ) − c(T 3Ψ) = b(AR(T 2Ψ)), +... +c(T λΨ +L−1Ψ) − c(T λΨ +LΨ) = b(AR(T λΨ +L−1Ψ)), +(S247) +where c(T λΨ +LΨ)) is already determined because λT λΨ +L Ψ +L += λΨ +L − λΨ +L = 0. adding up all the sides, we have +c(Ψ) = +λΨ +L +� +i=0 +b(AR(T iΨ)). +(S248) +Then we have proved that c(Ψ) is uniquely determined and that F j +k(s, d) is uniquely determined. +By induction, we have proved that the one-dimensional Hubbard has the unique k-support local conserved quantity, +which is our Qk. + +59 +S7. EXAMPLES OF HIGHER ORDER Qk +We present below three examples of the higher order local conserved quantity Q6, Q7, Q8 that have not been found +before. In our diagram notation, Qk is represented more simply than the usual spin operator notation. Thus, we can +read out the rule of the pattern of Qk. We write Qk as +Qk = +jf +� +j=0 +U j +⌊ k−1−j +2 +⌋ +� +n=0 +⌊ k−1−j +2 +⌋−n +� +d=0 +Qj +k(k − j − 2n − d, d), +(S249) +where Qj +k(s, d) is the (s, d) diagrams in Qj +k and jf = k − 1(k − 2) for even(odd) k. Qj +k(s, d) is more explicitly written +as +Qj +k(k − j − 2n − d, d) = +⌊ j−1 +2 ⌋ +� +m=0 +d +� +g=0 +� +Ψ∈Sk,j,m +n,d,g +(−1)n+m+gCj,m +n,d (λΨ) Ψ. +(S250) +All the expressions of Qj +k(s, d) below have + ↕ in the RHS implicitly, and they are omitted. +Explicit expressions for Q6 +We show all the expressions for the components of Q6. We explain the structure of Qj +6 for each j in Fig. S2(a) and +all the structure of Q6 in Fig. S2(b). The circle at (s, d) in Fig. S2 represents Qj +k(s, d). +Q1 +6(5, 0) = I I +I I + I +I I I + I I I +I + I I I I + I I I I + I I I I + I I I I + I I I I +Q1 +6(4, 1) = − I I +I I − I +I I I − I I I +I + I +I + +I I + I +I + I I +Q1 +6(3, 2) = I I +I I + +Q1 +6(3, 0) = − I +I − I I − I I − I I +Q1 +6(2, 1) = I +I − +Q1 +6(1, 0) = +Q2 +6(4, 0) = I I +I + +I +I +I + +I I +I ++ +I +I +I + I +I I + +I +I +I + +I +I I ++ I I I + I I I + I I I + +I +I +I ++ +I I +I ++ +I +I I +Q2 +6(3, 1) = − I I +I +− I I +I + I ++ +I +Q2 +6(2, 0) = − I +Q3 +6(3, 0) = I +I + I I − I +I − I I − I I − I I +Q3 +6(2, 1) =2 +� +I +I − +� +Q3 +6(1, 0) =5 +Q4 +6(2, 0) = − 2 I +Q5 +6(1, 0) =2 + +60 +Qj=0 +6 +Qj=1 +6 +Qj=2 +6 +Qj=3 +6 +Qj=4 +6 +Qj=5 +6 +Support +Double +(b) +Qj=1 +6 +Support +Double +n = 0 +n = 1 +n = 2 +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +0 +1 +2 +# +# +# +! +! +! + + + +Qj=2 +6 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +5 +0 +1 +2 +# +# +# +! +! +! + +Qj=3 +6 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +0 +1 +# +! + +Qj=4 +6 +Support +Double +n = 0 +" +1 +2 +3 +0 +1 +# +! +Qj=5 +6 +Support +Double +n = 0 +" +1 +2 +0 +2 +(a) +FIG. S2. The structure of Qj +6 for each j (a) and all the structure of Q6 (b). In (b), each plane represents the structure of Qj +6 +in (a), and the axis of support and double are omitted. The solid arrow in planes represents the commutator of diagrams with +H0, and the vertical dotted arrow represents the commutator of diagrams with Hint. + +61 +Explicit expressions for Q7 +We show all the expressions for the components of Q7. We explain the structure of Qj +7 for each j in Fig. S3(a) and +all the structure of Q7 in Fig. S3(b). The circle at (s, d) in Fig. S3 represents Qj +7(s, d). +Q2k+1 does not have (s, s − 1) diagrams because of the parity of the space reflection. (s, s − 1) connected diagrams +is even under space reflection and Q2k+1 is odd under space reflection: (s, s−1) connected diagrams are the diagrams +where the type of upper and lower sequence are both −, and all the region is an overlap, such as +and these +diagrams are even under space reflection because the two lands in (s, s − 1) connected diagrams are the same, and +these parities under space reflection are the same. +Q1 +7(6, 0) = I I +I I I + I I I +I I + I +I I I I + I I I I +I + I I I I I + I I I I I + I I I I I ++ I I I I I + I I I I I + I I I I I +Q1 +7(5, 1) = − I I +I I I − I I I +I I − I +I I I I − I I I I +I + I +I I + I I +I + +I I I + I +I I ++ I I +I + I I I +Q1 +7(4, 2) = I I +I I I + I I I +I I + +I + I +Q1 +7(4, 0) = − I +I I − I I +I − I I I − I I I − I I I − I I I +Q1 +7(3, 1) = I +I I + I I +I − +I − I +Q1 +7(2, 0) = I + I +Q2 +7(5, 0) = +I +I +I I + I I +I I + +I I +I +I + +I +I I +I + +I I +I I ++ I I I +I + +I +I +I I + +I I +I +I + +I I I +I ++ +I I +I +I + +I +I I +I + I +I I I + +I +I +I I + +I +I I +I + +I +I I I ++ I I I I + I I I I + I I I I + I I I I ++ +I +I +I I + +I I +I +I + +I I I +I ++ +I +I I +I + +I I +I I ++ +I +I I I +Q2 +7(4, 1) = − I I +I I − I I +I +I − +I I +I +I − +I I +I I +− I I +I +I − I I I +I +− I I I +I − +I I +I +I + I +I + +I +I + +I +I ++ +I +I ++ I +I + I I ++ +I +I ++ +I I + +I I + +I +I +Q2 +7(3, 0) = − I +I − +I +I +− I I − +I +I +− 2 I I +Q3 +7(4, 0) = I I +I + +I I +I ++ +I +I +I + +I +I I ++ I +I I + I +I I + +I +I +I ++ +I +I +I ++ I I I + I I I + I I I + +I I +I +− I +I I − I I +I − I I I − I I I − I I I − I I I +Q3 +7(3, 1) =2 +� +I +I I + I I +I − +I − I +� +Q3 +7(2, 0) =6 +� +I + I +� +Q4 +7(3, 0) =2 +� +− I +I − +I +I +− I I − I I − +I +I +� +Q5 +7(2, 0) =2 +� +I + I +� + +62 +Support +Double +(a) +(b) +Qj=0 +7 +Qj=1 +7 +Qj=2 +7 +Qj=3 +7 +Qj=4 +7 +Qj=5 +7 +Qj=1 +7 +n = 0 +n = 1 +n = 2 +Support +Double +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +7 +0 +1 +2 +3 +# +# +# +# +# +# +! +! +! +! +! +! + + + +Qj=2 +7 +n = 0 +n = 1 +Support +Double +" +" +" +1 +2 +3 +4 +5 +6 +0 +1 +# +# +# +! +! +! + +Qj=3 +7 +n = 0 +n = 1 +Support +Double +" +" +" +1 +2 +3 +4 +5 +0 +1 +2 +# +# +# +! +! +! + +Qj=4 +7 +n = 0 +Support +Double +" +1 +2 +3 +4 +0 +# +! +Qj=5 +7 +n = 0 +Support +Double +" +1 +2 +3 +0 +1 +# +! +5 +FIG. S3. The structure of Qj +7 for each j (a) and all the structure of Q7 (b). In (b), each plane represents the structure of Qj +k +in (a), and the axis of support and double are omitted. The solid arrow in planes represents the commutator of diagrams with +H0, and the vertical dotted arrow represents the commutator of diagrams with Hint. + +63 +Explicit expressions for Q8 +We show all the expressions for the components of Q6. We explain the structure of Qj +8 for each j in Fig. S4(a) and +all the structure of Q8 in Fig. S4(b). The circle at (s, d) in Fig. S4 represents Qj +8(s, d). From Q8, the general structure +of the cancellation of diagrams (Fig. S4(c)) appears. +Q1 +8(7, 0) = I I I +I I I + I I +I I I I + I I I I +I I + I +I I I I I + I I I I I +I + I I I I I I ++ I I I I I I + I I I I I I + I I I I I I + I I I I I I + I I I I I I + I I I I I I +Q1 +8(6, 1) = − I I I +I I I − I I +I I I I − I I I I +I I + I I +I I − I +I I I I I − I I I I I +I + I +I I I ++ I I I +I + +I I I I + I +I I I + I I +I I + I I I +I + I I I I +Q1 +8(5, 2) = I I I +I I I + I I +I I I I + I I I I +I I + I +I + +I I + I +I + I I +Q1 +8(4, 3) = − I I I +I I I + +Q1 +8(5, 0) = − I I +I I − I +I I I − I I I +I − I I I I − I I I I − I I I I − I I I I − I I I I +Q1 +8(4, 1) = I I +I I + I +I I I + I I I +I − I +I − +I I − I +I − I I +Q1 +8(3, 2) = − I I +I I − +Q1 +8(3, 0) = I +I + I I + I I + I I +Q1 +8(2, 1) = − I +I + +Q1 +8(1, 0) = − +Q2 +8(6, 0) = I I I +I I + +I I I +I +I + +I +I I +I I + +I I +I I +I + +I I I +I I ++ +I I +I +I I + +I +I I +I I ++ +I +I +I I I + I I +I I I + +I I +I +I I + +I I +I I +I + +I +I I I +I + +I I +I I I ++ I I I I +I + +I +I +I I I ++ +I I +I +I I + +I I I +I +I + +I I I I +I ++ +I I I +I +I + +I I +I I +I + +I +I I I +I + I +I I I I + +I +I +I I I ++ +I +I I +I I + +I +I I I +I + +I +I I I I ++ I I I I I + I I I I I + I I I I I + I I I I I + I I I I I ++ +I +I +I I I + +I I +I +I I + +I I I +I +I + +I I I I +I ++ +I +I I +I I + +I I +I I +I + +I I I +I I ++ +I +I I I +I ++ +I I +I I I ++ +I +I I I I +Q2 +8(5, 1) = − I I I +I +I − +I I +I I +I − +I I I +I I +− I I I +I I − +I I I +I +I − +I I +I +I I − I I +I I I − I I +I +I I − +I I +I +I I +− +I I +I I +I − +I I +I I +I − +I I +I I I +− I I +I +I I − I I I +I +I − I I I I +I +− I I I I +I − +I I I +I +I − +I I +I I +I + I I +I ++ +I +I +I + +I I +I ++ +I +I +I + +I I +I + +I +I +I + +I I +I ++ +I +I +I + I +I I + +I +I I + +I +I +I ++ +I +I +I + +I +I I ++ +I +I I ++ I +I I + I I +I + I I I ++ +I +I +I + +I I +I ++ +I I I + +I I I ++ +I I I + +I +I I ++ +I +I +I + +I I +I ++ +I +I I +Q2 +8(4, 2) = I I I +I I ++ I I I +I +I + I I I +I I − I I +I +− +I I +I + I ++ +I ++ +I +Q2 +8(4, 0) = − +I +I +I − +I +I +I − I +I I − +I +I +I − +I +I I +− I I I − +I +I +I − +I +I I ++ 2 +� +− I I +I − +I I +I +− I I I − I I I +− +I I +I +� + +64 +Q2 +8(3, 1) = I I +I ++ I I +I − I +− +I +Q2 +8(2, 0) = I +Q3 +8(5, 0) = I +I I +I + +I +I +I +I + +I I +I +I ++ I I +I I + I I +I I + +I +I +I +I + +I I +I +I ++ I I +I I + +I I +I +I ++ +I I +I I ++ I I I +I + I I I +I + +I I +I +I + +I I I +I ++ +I I I +I ++ +I I +I +I + +I +I +I I + +I +I I +I + +I +I I I ++ I +I I I + I +I I I + I +I I I + +I +I +I +I + +I +I I +I ++ +I +I +I +I + +I +I I +I ++ +I +I +I I ++ +I +I +I I ++ I I I I ++ I I I I + I I I I + I I I I + I I I I + I I I I + +I I +I +I + +I I I +I ++ +I I I +I ++ +I I +I I +− I I +I I +− I +I I I − I I I +I − I I I I − I I I I − I I I I − I I I I − I I I I +Q3 +8(4, 1) = − I +I I +I − I I +I I − I I +I +I +− I I I +I +− I I I +I + I +I ++ I +I + +I +I + I I ++ I +I + +I I + 2 +� +I I +I I ++ I +I I I + I I I +I − I +I − +I I − I +I − I I +� +Q3 +8(3, 2) =3 +� +− I I +I I − +� +Q3 +8(3, 0) = − I +I − I I + 6 +� +I I + I I +� ++ 7 +� +I +I + I I +� +Q3 +8(2, 1) =10 +� +− I +I + +� +Q3 +8(1, 0) = − 16 +Q4 +8(4, 0) = +I +I +I + +I +I I ++ I +I I + I I I + 2 +� +− I I +I − +I +I +I − +I I +I +− +I +I +I − I +I I − +I +I +I − +I +I I +− I I I +− I I I − I I I − +I +I +I − +I I +I +− +I +I I +� +Q4 +8(3, 1) =3 +� +I I +I ++ I I +I − I +− +I +� +Q4 +8(2, 0) =10 I +Q5 +8(3, 0) =2 +� +I +I + I I + I I + I I +� ++ 3 +� +− I +I − I I +� +Q5 +8(2, 1) =5 +� +− I +I + +� +Q5 +8(1, 0) = − 21 +Q6 +8(2, 0) =5 I +Q7 +8(1, 0) = − 5 + +65 +Qj=1 +8 +Support +Double +n = 0 +n = 1 +n = 2 +n = 3 +" +" +" +" +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +7 +8 +0 +1 +2 +3 +# +# +# +# +# +# +! +! +! +! +! +! + + + + + + +Qj=2 +8 +Support +Double +n = 0 +n = 1 +n = 2 +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +7 +0 +1 +2 +3 +# +# +# +# +# +# +! +! +! +! +! +! + + + +Qj=3 +8 +Support +Double +n = 0 +n = 1 +n = 2 +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +0 +1 +2 +# +# +# +! +! +! + + + +Qj=4 +8 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +5 +0 +1 +2 +# +# +# +! +! +! + +Qj=5 +8 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +0 +1 +# +! + +Qj=6 +8 +Support +Double +n = 0 +" +1 +2 +3 +0 +1 +# +! +Qj=7 +8 +Support +Double +n = 0 +" +1 +2 +0 +4 +Qj=1 +8 +Support +Double +n = 0 +n = 1 +n = 2 +n = 3 +" +" +" +" +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +7 +8 +0 +1 +2 +3 +# +# +# +# +# +# +! +! +! +! +! +! + + + + + + +Qj=2 +8 +Support +Double +n = 0 +n = 1 +n = 2 +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +7 +0 +1 +2 +3 +# +# +# +# +# +# +! +! +! +! +! +! + + + +Qj=3 +8 +Support +Double +n = 0 +n = 1 +n = 2 +" +" +" +" +" +" +1 +2 +3 +4 +5 +6 +0 +1 +2 +# +# +# +! +! +! + + + +Qj=4 +8 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +5 +0 +1 +2 +# +# +# +! +! +! + +Qj=5 +8 +Support +Double +n = 0 +n = 1 +" +" +" +1 +2 +3 +4 +0 +1 +# +! + +Qj=6 +8 +Support +Double +n = 0 +" +1 +2 +3 +0 +1 +# +! +Qj=7 +8 +Support +Double +n = 0 +" +1 +2 +0 +4 +Qj=0 +8 +Qj=1 +8 +Qj=2 +8 +Qj=3 +8 +Qj=4 +8 +Qj=5 +8 +Qj=6 +8 +Qj=7 +8 +Support +Double +𝑛 +𝑛 +𝑛 − 1 +Support +Double +(c) +(a) +(b) +ACaHichVHLgRBFD3T3uPVWCA2YkKsJtUiNWE +DTuDRjLGpLvVjDL9SnfPJEz8gI0lYkUiIj7Dxg9Y+AQsSWws3O6eRBDcSlWdOnXPrVNVumsKP2DsISE1NDY1t7S2Jds7Oru65Z7eNd+peAZXDcd0vA1d87kpbK4GIjD5hutxzdJNvq6X58P9Sr3fOHYq8Gey/OWVrJFURhaQJSaLZS3dgtyiq +VZFM/gVIHqcwiolhy5CtsYhsODFRgcNGQNiEBp9aDgoYXOLyqBHnERLRPscBkqStUBanDI3YMo0lWuXqrE3rsKYfqQ06xaTukXIYo+yeXbMXdsdu2CN7/7VWLaoRetmjWY+13C10Hw6svP2rsmgOsPOp+tNzgCJmIq+CvLsRE97CiPXV/eOX +ldnl0doYu2DP5P+cPbBbuoFdfTUus3z5DEn6AOX7c/8EaxNpZSo9mZ1MZebin0ArhjCcXrvaWSwgCWodK7AEU5wmniSZKlfGoxTpURd04cvIY18AGT1jB0= +Qj +k +ACbHichVG7SgNBFD1ZXzG+4qMQghAMERvDrAQV +q6CNdk0iaIx7K5jXLMvdjeBGPIDthYWaqEgIn6GjT9g4SeIYBPBxsKbTUBU1DvMzJkz9w5MyNbmuq4jD36hI7Oru4ef2+gr39gcCg4PJ1zLKt8Ixiaqa9IUsO1SDZ1zV1fiGZXNJlzWek0vLzf1chduOahrbtXieV0qGuqeqkguUZupQm +mndjAj1gvBCIsxL8I/gdgGkcQqvEiawWtsYxcmFJShg8OAS1iDBIfaFkQwWMTlUSPOJqR6+x1BEhbpixOGRKxJRqLtNpqswatmzUdT63QKRp1m5RhRNkDu2ENds9u2RN7/7VWzavR9FKlW5puVUYOhpfe/tXpdPsYv9T9adnF3tY8Lyq5N3y +mOYtlJa+cnjSWFtMR2tT7JI9k/8L9sju6AZG5VW5SvH0KQL0AeL35/4JsrMxcS4WT8UjiaXWT8CPECYxTe89jwRWkESGztVxjDOc+16EMSEkTLRSBV9bM4ovIUx9ALBnjZs= +Qj�1 +k +FIG. S4. The structure of Qj +8 for each j (a) and all the structure of Q8 (b), and the basic structure of the cancellation of +diagrams (c). In (b), each plane represents the structure of Qj +8 in (a), and the axis of support and double are omitted. The +solid arrow in planes represents the commutator of diagrams with H0, and the vertical dotted arrow represents the commutator +of diagrams with Hint. In (c), we only show the circle and arrows related to the cancellation at the crosses. + +66 +S8. EXAMPLES OF COEFFICIENTS Cjm +nd (λ) +We present several examples of the coefficients Cjm +nd (λ) in this section. The coefficients presented here include +enough information to construct up to Q16. +We present the table for the value of Cjm +nd (λ) below for m, n > 0. The explicit expressions of Cjm +nd (λ) for m = 0 or +n = 0 are presented in the main manuscript. Using the invariance of Cjm +nd (λ) (S24) and (S25) and (S26), we restrict +λ = {λL; λ1, . . . , λw; λR}(w = j − 1 − 2m) to satisfy λL ≤ λR and λ1 ≤ λ2 ≤ · · · ≤ λw and 0 ≤ λa ≤ n without losing +generality. The columns indicate (λL, λR). The rows indicate the value of d and (λ1, λ2, . . . , λw) which is represented +by φ for the case of w = 0, i.e., the case of λ = {λL; λR}. +Cj=3,m=1 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 φ +5 +6 +7 +d=1 φ +10 +11 +12 +d=2 φ +15 +16 +17 +d=3 φ +20 +21 +22 +d=4 φ +25 +26 +27 +d=5 φ +30 +31 +32 +Cj=3,m=1 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 φ +16 +20 +21 +24 +25 +26 +d=1 φ +30 +34 +35 +38 +39 +40 +d=2 φ +44 +48 +49 +52 +53 +54 +d=3 φ +58 +62 +63 +66 +67 +68 +d=4 φ +72 +76 +77 +80 +81 +82 +Cj=3,m=1 +n=3,d +(0, 0) (0, 1) (0, 2) (0, 3) (1, 1) (1, 2) (1, 3) (2, 2) (2, 3) (3, 3) +d=0 φ +40 +50 +54 +55 +60 +64 +65 +68 +69 +70 +d=1 φ +70 +80 +84 +85 +90 +94 +95 +98 +99 +100 +d=2 φ +100 +110 +114 +115 +120 +124 +125 +128 +129 +130 +d=3 φ +130 +140 +144 +145 +150 +154 +155 +158 +159 +160 +Cj=3,m=1 +n=4,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (4, 4) +d=0 φ +85 +105 +115 +119 +120 +125 +135 +139 +140 +145 +149 +150 +153 +154 +155 +d=1 φ +140 +160 +170 +174 +175 +180 +190 +194 +195 +200 +204 +205 +208 +209 +210 +d=2 φ +195 +215 +225 +229 +230 +235 +245 +249 +250 +255 +259 +260 +263 +264 +265 +Cj=3,m=1 +n=5,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (0, 5) (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (2, 2) (2, 3) (2, 4) (2, 5) (3, 3) (3, 4) (3, 5) (4, 4) (4, 5) (5, 5) +d=0 φ +161 +196 +216 +226 +230 +231 +231 +251 +261 +265 +266 +271 +281 +285 +286 +291 +295 +296 +299 +300 +301 +d=1 φ +252 +287 +307 +317 +321 +322 +322 +342 +352 +356 +357 +362 +372 +376 +377 +382 +386 +387 +390 +391 +392 +Cj=3,m=1 +n=6,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (0, 5) (0, 6) (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 3) (3, 4) (3, 5) (3, 6) (4, 4) (4, 5) (4, 6) (5, 5) (5, 6) (6, 6) +d=0 φ +280 +336 +371 +391 +401 +405 +406 +392 +427 +447 +457 +461 +462 +462 +482 +492 +496 +497 +502 +512 +516 +517 +522 +526 +527 +530 +531 +532 + +67 +Cj=4,m=1 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 (0) +10 +11 +12 +(1) +14 +15 +16 +d=1 (0) +15 +16 +17 +(1) +20 +21 +22 +d=2 (0) +20 +21 +22 +(1) +26 +27 +28 +d=3 (0) +25 +26 +27 +(1) +32 +33 +34 +d=4 (0) +30 +31 +32 +(1) +38 +39 +40 +Cj=4,m=1 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 +(0) +30 +34 +35 +38 +39 +40 +(1) +50 +55 +56 +60 +61 +62 +(2) +56 +61 +62 +66 +67 +68 +d=1 +(0) +44 +48 +49 +52 +53 +54 +(1) +69 +74 +75 +79 +80 +81 +(2) +76 +81 +82 +86 +87 +88 +d=2 +(0) +58 +62 +63 +66 +67 +68 +(1) +88 +93 +94 +98 +99 +100 +(2) +96 +101 +102 +106 +107 +108 +d=3 +(0) +72 +76 +77 +80 +81 +82 +(1) +107 +112 +113 +117 +118 +119 +(2) +116 +121 +122 +126 +127 +128 +Cj=4,m=1 +n=3,d +(0, 0) (0, 1) (0, 2) (0, 3) (1, 1) (1, 2) (1, 3) (2, 2) (2, 3) (3, 3) +d=0 +(0) +70 +80 +84 +85 +90 +94 +95 +98 +99 +100 +(1) +128 +142 +147 +148 +156 +161 +162 +166 +167 +168 +(2) +158 +173 +178 +179 +188 +193 +194 +198 +199 +200 +(3) +166 +181 +186 +187 +196 +201 +202 +206 +207 +208 +d=1 +(0) +100 +110 +114 +115 +120 +124 +125 +128 +129 +130 +(1) +172 +186 +191 +192 +200 +205 +206 +210 +211 +212 +(2) +207 +222 +227 +228 +237 +242 +243 +247 +248 +249 +(3) +216 +231 +236 +237 +246 +251 +252 +256 +257 +258 +d=2 +(0) +130 +140 +144 +145 +150 +154 +155 +158 +159 +160 +(1) +216 +230 +235 +236 +244 +249 +250 +254 +255 +256 +(2) +256 +271 +276 +277 +286 +291 +292 +296 +297 +298 +(3) +266 +281 +286 +287 +296 +301 +302 +306 +307 +308 +Cj=4,m=1 +n=4,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (4, 4) +d=0 +(0) +140 +160 +170 +174 +175 +180 +190 +194 +195 +200 +204 +205 +208 +209 +210 +(1) +270 +300 +314 +319 +320 +330 +344 +349 +350 +358 +363 +364 +368 +369 +370 +(2) +356 +390 +405 +410 +411 +424 +439 +444 +445 +454 +459 +460 +464 +465 +466 +(3) +396 +431 +446 +451 +452 +466 +481 +486 +487 +496 +501 +502 +506 +507 +508 +(4) +406 +441 +456 +461 +462 +476 +491 +496 +497 +506 +511 +512 +516 +517 +518 +d=1 +(0) +195 +215 +225 +229 +230 +235 +245 +249 +250 +255 +259 +260 +263 +264 +265 +(1) +355 +385 +399 +404 +405 +415 +429 +434 +435 +443 +448 +449 +453 +454 +455 +(2) +455 +489 +504 +509 +510 +523 +538 +543 +544 +553 +558 +559 +563 +564 +565 +(3) +500 +535 +550 +555 +556 +570 +585 +590 +591 +600 +605 +606 +610 +611 +612 +(4) +511 +546 +561 +566 +567 +581 +596 +601 +602 +611 +616 +617 +621 +622 +623 + +68 +Cj=4,m=1 +n=5,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (0, 5) (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (2, 2) (2, 3) (2, 4) (2, 5) (3, 3) (3, 4) (3, 5) (4, 4) (4, 5) (5, 5) +d=0 +(0) +252 +287 +307 +317 +321 +322 +322 +342 +352 +356 +357 +362 +372 +376 +377 +382 +386 +387 +390 +391 +392 +(1) +502 +557 +587 +601 +606 +607 +612 +642 +656 +661 +662 +672 +686 +691 +692 +700 +705 +706 +710 +711 +712 +(2) +692 +757 +791 +806 +811 +812 +822 +856 +871 +876 +877 +890 +905 +910 +911 +920 +925 +926 +930 +931 +932 +(3) +806 +875 +910 +925 +930 +931 +944 +979 +994 +999 +1000 1014 1029 1034 1035 1044 1049 1050 1054 1055 1056 +(4) +856 +926 +961 +976 +981 +982 +996 +1031 1046 1051 1052 1066 1081 1086 1087 1096 1101 1102 1106 1107 1108 +(5) +868 +938 +973 +988 +993 +994 +1008 1043 1058 1063 1064 1078 1093 1098 1099 1108 1113 1114 1118 1119 1120 +Cj=5,m=1 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 +(0, 0) +15 +16 +17 +(0, 1) +20 +21 +22 +(1, 1) +25 +26 +27 +d=1 +(0, 0) +20 +21 +22 +(0, 1) +26 +27 +28 +(1, 1) +32 +33 +34 +d=2 +(0, 0) +25 +26 +27 +(0, 1) +32 +33 +34 +(1, 1) +39 +40 +41 +d=3 +(0, 0) +30 +31 +32 +(0, 1) +38 +39 +40 +(1, 1) +46 +47 +48 +d=4 +(0, 0) +35 +36 +37 +(0, 1) +44 +45 +46 +(1, 1) +53 +54 +55 +Cj=5,m=1 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 +(0, 0) +44 +48 +49 +52 +53 +54 +(0, 1) +69 +74 +75 +79 +80 +81 +(0, 2) +76 +81 +82 +86 +87 +88 +(1, 1) +101 +107 +108 +113 +114 +115 +(1, 2) +108 +114 +115 +120 +121 +122 +(2, 2) +115 +121 +122 +127 +128 +129 +d=1 +(0, 0) +58 +62 +63 +66 +67 +68 +(0, 1) +88 +93 +94 +98 +99 +100 +(0, 2) +96 +101 +102 +106 +107 +108 +(1, 1) +126 +132 +133 +138 +139 +140 +(1, 2) +134 +140 +141 +146 +147 +148 +(2, 2) +142 +148 +149 +154 +155 +156 +d=2 +(0, 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2648 +(2) 2222 2495 2563 2572 2776 2845 2854 2914 2923 2932 +(3) 2266 2539 2607 2616 2820 2889 2898 2958 2967 2976 +d=1 +(0) 2002 2242 2309 2319 2489 2557 2567 2625 2635 2645 +(1) 3064 3371 3448 3458 3686 3764 3774 3842 3852 3862 +(2) 3406 3723 3800 3810 4048 4126 4136 4204 4214 4224 +(3) 3460 3777 3854 3864 4102 4180 4190 4258 4268 4278 +Cj=6,m=2 +n=4,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (4, 4) +d=0 +(0) 3891 4607 4882 +4957 +4968 +5351 +5633 +5709 +5720 +5916 +5992 +6003 +6068 +6079 +6090 +(1) 6849 7840 8190 +8276 +8287 +8866 +9224 +9311 +9322 +9583 +9670 +9681 +9757 +9768 +9779 +(2) 8253 9319 9680 +9766 +9777 10421 10790 10877 10888 11160 11247 11258 11334 11345 11356 +(3) 8680 9757 10118 10204 10215 10870 11239 11326 11337 11609 11696 11707 11783 11794 11805 +(4) 8745 9822 10183 10269 10280 10935 11304 11391 11402 11674 11761 11772 11848 11859 11870 +Cj=7,m=2 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 +(0, 0) +92 +98 +104 +(0, 1) +112 +118 +124 +(1, 1) +132 +138 +144 +d=1 +(0, 0) +141 +148 +155 +(0, 1) +168 +175 +182 +(1, 1) +195 +202 +209 +d=2 +(0, 0) +199 +207 +215 +(0, 1) +234 +242 +250 +(1, 1) +269 +277 +285 +d=3 +(0, 0) +266 +275 +284 +(0, 1) +310 +319 +328 +(1, 1) +354 +363 +372 +Cj=7,m=2 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 +(0, 0) +507 +558 +566 +610 +618 +626 +(0, 1) +706 +765 +773 +825 +833 +841 +(0, 2) +741 +800 +808 +860 +868 +876 +(1, 1) +940 +1007 1015 1075 1083 1091 +(1, 2) +975 +1042 1050 1110 1118 1126 +(2, 2) 1010 1077 1085 1145 1153 1161 +d=1 +(0, 0) +763 +822 +831 +882 +891 +900 +(0, 1) 1029 1097 1106 1166 1175 1184 +(0, 2) 1073 1141 1150 1210 1219 1228 +(1, 1) 1339 1416 1425 1494 1503 1512 +(1, 2) 1383 1460 1469 1538 1547 1556 +(2, 2) 1427 1504 1513 1582 1591 1600 +d=2 +(0, 0) 1062 1129 1139 1197 1207 1217 +(0, 1) 1404 1481 1491 1559 1569 1579 +(0, 2) 1458 1535 1545 1613 1623 1633 +(1, 1) 1800 1887 1897 1975 1985 1995 +(1, 2) 1854 1941 1951 2029 2039 2049 +(2, 2) 1908 1995 2005 2083 2093 2103 + +73 +Cj=7,m=3 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 φ +84 +98 +112 +d=1 φ +232 +252 +272 +d=2 φ +453 +480 +507 +d=3 φ +760 +795 +830 +Cj=7,m=3 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 φ +752 +953 +980 +1161 1188 1215 +d=1 φ 2010 2290 2325 2578 2613 2648 +d=2 φ 3820 4191 4235 4571 4615 4659 +Cj=7,m=3 +n=3,d +(0, 0) (0, 1) (0, 2) (0, 3) (1, 1) (1, 2) (1, 3) (2, 2) (2, 3) (3, 3) +d=0 φ +4720 +6242 +6613 +6657 +7855 +8235 +8279 +8615 +8659 +8703 +d=1 φ 12088 14167 14641 14695 16340 16824 16878 17308 17362 17416 +Cj=7,m=3 +n=4,d +(0, 0) (0, 1) (0, 2) (0, 3) (0, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (4, 4) +d=0 φ 23203 31302 33996 34585 34650 40016 42825 43425 43490 45645 46245 46310 46845 46910 46975 +Cj=8,m=2 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 +(0, 0, 0) +141 +148 +155 +(0, 0, 1) +168 +175 +182 +(0, 1, 1) +195 +202 +209 +(1, 1, 1) +222 +229 +236 +d=1 +(0, 0, 0) +199 +207 +215 +(0, 0, 1) +234 +242 +250 +(0, 1, 1) +269 +277 +285 +(1, 1, 1) +304 +312 +320 +d=2 +(0, 0, 0) +266 +275 +284 +(0, 0, 1) +310 +319 +328 +(0, 1, 1) +354 +363 +372 +(1, 1, 1) +398 +407 +416 +Cj=8,m=3 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 (0) +232 +252 +272 +(1) +280 +300 +320 +d=1 (0) +453 +480 +507 +(1) +528 +555 +582 +d=2 (0) +760 +795 +830 +(1) +870 +905 +940 +Cj=8,m=3 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 +(0) 2010 2290 2325 2578 2613 2648 +(1) 2770 3085 3120 3408 3443 3478 +(2) 2880 3195 3230 3518 3553 3588 +d=1 +(0) 3820 4191 4235 4571 4615 4659 +(1) 4986 5401 5445 5825 5869 5913 +(2) 5140 5555 5599 5979 6023 6067 + +74 +Cj=8,m=3 +n=3,d +(0, 0) (0, 1) (0, 2) (0, 3) (1, 1) (1, 2) (1, 3) (2, 2) (2, 3) (3, 3) +d=0 +(0) 12088 14167 14641 14695 16340 16824 16878 17308 17362 17416 +(1) 18358 20911 21439 21493 23568 24106 24160 24644 24698 24752 +(2) 20042 22649 23177 23231 25360 25898 25952 26436 26490 26544 +(3) 20250 22857 23385 23439 25568 26106 26160 26644 26698 26752 +Cj=9,m=3 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 +(0, 0) +453 +480 +507 +(0, 1) +528 +555 +582 +(1, 1) +603 +630 +657 +d=1 +(0, 0) +760 +795 +830 +(0, 1) +870 +905 +940 +(1, 1) +980 +1015 1050 +d=2 +(0, 0) 1166 1210 1254 +(0, 1) 1320 1364 1408 +(1, 1) 1474 1518 1562 +Cj=9,m=4 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 φ +330 +378 +426 +d=1 φ +975 +1050 1125 +d=2 φ 2035 2145 2255 +Cj=9,m=4 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 φ +4035 +5020 +5130 +6040 +6150 +6260 +d=1 φ 11605 13101 13255 14641 14795 14949 +Cj=9,m=4 +n=3,d +(0, 0) (0, 1) (0, 2) (0, 3) (1, 1) (1, 2) (1, 3) (2, 2) (2, 3) (3, 3) +d=0 φ 33913 44341 46487 46695 55419 57619 57827 59819 60027 60235 +Cj=10,m=4 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 (0) +975 +1050 1125 +(1) 1140 1215 1290 +d=1 (0) 2035 2145 2255 +(1) 2310 2420 2530 +Cj=10,m=4 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 +(0) 11605 13101 13255 14641 14795 14949 +(1) 15246 16896 17050 18590 18744 18898 +(2) 15675 17325 17479 19019 19173 19327 +Cj=11,m=5 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 φ 1287 1452 1617 +d=1 φ 3982 4257 4532 +Cj=11,m=5 +n=2,d +(0, 0) (0, 1) (0, 2) (1, 1) (1, 2) (2, 2) +d=0 φ 20152 24618 25047 29238 29667 30096 + +75 +Cj=12,m=5 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 (0) 3982 4257 4532 +(1) 4554 4829 5104 +Cj=13,m=6 +n=1,d +(0, 0) (0, 1) (1, 1) +d=0 φ 5005 5577 6149 + diff --git a/VdE2T4oBgHgl3EQfDAbs/content/tmp_files/load_file.txt b/VdE2T4oBgHgl3EQfDAbs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e58f03ba88d6bfbeee2d7761a52d6044ad3d8e1 --- /dev/null +++ b/VdE2T4oBgHgl3EQfDAbs/content/tmp_files/load_file.txt @@ -0,0 +1,7115 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf,len=7114 +page_content='All Local Conserved Quantities of the One-Dimensional Hubbard Model Kohei Fukai∗ The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Dated: January 11, 2023) We present the exact expression for all the local conserved quantities of the one-dimensional Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We identify the operators constructing the local conserved quantities and find they have non-trivial coefficients in the higher order case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We derive the recursion equation for the coefficients, and some of them are explicitly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We also prove the local conserved quantities obtained in this work are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='— Quantum integrability and local con- servation laws are two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Quantum integrable systems are solvable many-body systems by the Bethe ansatz [1] and have an extensive number of local conserved quantities {Qk}k≥2, which is the foun- dation of their solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Recently, quantum integrable systems are becoming an arena for the studies of nonequi- librium quantum dynamics, inspired by their experimen- tal realization with ultracold atoms [2–5], where local conserved quantities play a crucial role: their existence leads to the absence of thermalization [6–8] and the con- jectured long-time steady-state is the generalized Gibbs ensemble [9–11], involving all local (and also quasi-local) conserved quantities as well as hamiltonian [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The large-scale nonequilibrium behavior is described by Gen- eralized hydrodynamics [15, 16], which is based on the local continuity equations of Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' In quantum inverse scattering methods [17, 18], the existence of the local conserved quantities and their mutual commutativity are understood by the commutativity of the transfer matri- ces [T(λ), T(µ)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' This is because Qk’s are obtained from the expansion of ln T(λ) in terms of the spectral parameter λ, and usually, the leading term Q2 = H is hamiltonian itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Another way to calculate Qk is the use of the boost operator B [19–21] if it exist: they can be calculated recursively by [B, Qk] = Qk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Although the procedures to generate the local con- served quantities Qk are known, it is practically difficult to obtain their expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' This difficulty lies not only in the expensive computational cost for higher order charges but also in finding a general pattern in the huge amounts of data that emerge out of this calculation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' This problem has been investigated particularly for the spin- 1/2 XYZ chain [23–29] and the one-dimensional Hubbard model [30–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The former case is now deeply under- stood: the explicit forms for the isotropic XXX case are obtained independently in [38] and [39] and for the gen- eral XYZ case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Grabowski and Mathieu found the oper- ator basis for Qk and derive the recursion relations to construct Qk using boost operator [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' and recently the explicit expressions were obtained by Nozawa and the au- thor [40] using the doubling-product notation [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' and for the XXZ case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' independently obtained by Nienhuis and Huijgen using the Temperley-Lieb algebra [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' On the other hand, for the one-dimensional Hubbard model, the structure of the local conserved quantities Qk remains a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The problem is that there was no recursive way to construct them, unlike the XYZ case, because of the absence of the boost operator [22, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' This comes from the fact that the Hubbard model is not Lorentz invariant due to the separation of spin and charge excitations with different velocities [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The first three non-trivial charges have been found before: Q3 [31, 35], Q4 [47, 48], and Q5 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' From these ex- pressions, Grabowski and Mathieu conjectured that Qk is constructed of products of local conserved densities of spin-1/2 XX chain, and its coefficients are trivially the power of the coupling constant ±U j [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' However, what kind of products of XX conserved densities should appear in Qk was unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' In this Letter, we reveal the structure of the local conserved quantities Qk in the one-dimensional Hubbard model and present their exact expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We proved Qk is the linear combination of connected diagrams, a nota- tion for the particular kind of products of the local con- served density of XX chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' With this notation, we found that the expressions of the higher order charges Qk≥6 and the nontrivial coefficients other than ±U j appear there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We derived the recursion equation for the nontrivial coef- ficients of the connected diagrams in Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We also proved the local conserved quantities we obtained are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Hamiltonian and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='— The Hamiltonian of the one-dimensional Hubbard model is H = L � j=1 � s=↑,↓ � a=x,y σa jsσa j+1s + U L � j=1 σz j↑σz j↓, (1) where the periodic boundary condition is imposed and σa js is the usual Pauli matrix of flavor s on j-th site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' This hamiltonian can be written in terms of the usual fermion form by Jordan-Wigner transformation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We denote the k-support local conserved quantity in terms of the polynomial of U as Qk = jf � j=0 U jQj k, (2) where jf = k − 1(k − 2) for even(odd) k and Qj k is the linear combination of less than and equal to k−j-support arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='03621v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='stat-mech] 9 Jan 2023 2 operators and k-support operator acts over k adjacent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We determine Qk to satisfy [Qk, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We firstly define doubling-product [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We denote the Pauli matrices σx, σy, σz by X, Y, Z and 2×2 identity matrix as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The doubling-product of flavor s that starts from i-th site is defined by A1 · · · Ak(i)s := (A1)is (A1A2)i+1s · · · (AlAl+1)j+is · · · × (Ak−1Ak)i+k−1s (Ak)i+ks , (3) where Ai ∈ {I, X, Y, Z} and AlAl+1 is the product of Al and Al+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' (·)is means the local operator of flavor s acting on the i-th site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' In the following, we often omit the site and flavor indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We next introduce z-bar, denoted by , which can be inserted into a doubling-product, such as A1 · · · and · · · AlAl+1 · · · and · · · Ak, where these ac- tions are defined by replacing (A1) in (3) with (ZA1) and (AlAl+1) in (3) with (AlZAl+1) and (Ak) in (3) with (AkZ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We define a land of s flavor from i-th site by n (i)s := B1B2· · ·Bn(i)s + B′ 1B′ 2· · ·B′n(i)s, (4) where Bi = X(Y ) for odd(even) i and B′ i = Y (X) for odd(even) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A land can have z-bar at its right end or left end , whose actions are defined by inserting z-bar to the corresponding positions in the doubling-products of the RHS of (4), for example, = XY + Y X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We note that = and = thus we identify these equal lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We define the type of lands ( ) by +(−) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' z-bar can also be inserted between two adjacent ’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We call this type of z-bar as hole, correspond- ing to trivial action (XZY ) ∝ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The length of land is the number of in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The zero-length land is defined by (j)s := (Z)js whose type is defined by − for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A land is the local conserved density of XX chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We introduce diagram to represent products of lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A diagram is a two-row sequence of lands, for example: I I I I (i) = (i)↑× (i + 1)↓× (i + 4)↑, (5) where “I ” is called sea, which indicates the location that is not a land, and the lands on the upper(lower) row have ↑ (↓) flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' There must be at least one sea between two lands on the same row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Thus, the lands in a diagram commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We denote a diagram by Ψ, and the more concrete definition of a diagram is Ψ(i) := � ψ∈Ψ ψ(iψ), (6) where ψ is the land in Ψ and has the flavor ↑ (↓) if ψ is on the upper(lower) row on Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' ψ(iψ) is ψ from iψ-th site and iψ ≡ i + the number of columns on the left of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A diagram without a site index denotes the site translation summation, Ψ := �L i=1 Ψ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' (a) overlap I I (b) adjacent I I I I (c) gap sandwich I I I I I I I I (d) disconnecting I I I I I I I I I I I , I I I I I I I I I I , I I I I I I , I I I FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Examples of the three types of connections of lands on the upper and lower row (a)–(c) and disconnecting gaps(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The lands on the upper and lower row in (a)–(c) are con- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The lower example in (a) does not have the region of overlap, and we assume there is an overlap of zero length in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The gaps in (c(d)) are connecting(disconnecting) and are indicated by the teal(orange) shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The columns of a diagram are the part of the follow- ing regions: overlap , gap I I I I , or coast I I , I I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The leftmost and rightmost columns are not in gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The length is the number of columns, and the support is the length +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The double is the num- ber of the columns of gap and overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' (s, d) diagram is a diagram whose support and double are s and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The gap number, denoted by gΨ, is the number of the columns of gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The land number, denoted by lΨ, is the number of the lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Two lands in a diagram, one on the upper row and the other on the lower row, are connected if (a)they have overlap, or (b)they are adjacent, or (c)they are sand- wiching a gap, as explained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1(a)–(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A gap is disconnecting(connecting) if it is (is not) sandwiched by the lands on the same row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We give examples of dis- connecting gaps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The gaps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1(c) are connecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The connection of a land is the number of the lands on the other row connected with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A connected diagram is a diagram satisfying the fol- lowing condition(i)–(iii): (i)all gaps are connecting, and (ii)the type of the land is (−)C where C is its connec- tions, and (iii)the lands are hole-less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' A diagram that is not connected is a disconnected diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The dia- grams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1(a)–(c) and (5) are connected diagrams, and the diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1(d) are disconnected diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We found Qk is the linear combination of connected dia- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Structure of Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='—We show the explicit form of lower order charges previously found, in terms of connected diagrams: Q2 = H = I + I + U and Q3 = I I + U � I + I � + ↕ , where ↕ denotes the contribution of the diagrams with the upper and lower rows reversed, excluding the diagrams with the same upper and lower rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Q4 and Q5 is written as 3 Q4 = I I I + U � I I + I I + I I + I I + − I I − � + U 2 I − U 3 + ↕, (7) Q5 = I I I I + U(12 terms) + U 2 � I I + I I + I I + I I + I I � − U 3 � I + I � + ↕, (8) where we omit the 12 terms of Q1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We newly obtained the explicit forms of higher order Qk for k ≥ 6 and found nontrivial coefficients ̸= ±1 appear, for example, Qj=3 6 = I I − I I − I I − I I − I I + I I − 2 + 2 I I + 5 + ↕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' (9) We give the explicit forms of Q6, Q7, Q8 as examples of higher order Qk in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The diagrams in Qj k are classified as (s, d) connected di- agram as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 2,3, where circles represent (s, d) connected diagram in Qj k, and crosses indicated by the arrow tip represent diagrams generated by the commu- tation relations of (s, d) connected diagram in Qj k and H = H0 + UHint, where H0 = I + ↕ and Hint = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The Qj k ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ → → → → → → → → → → ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ← ← ← ← ← ← n = 0 n = 1 n = 2 n = 3 kj+1 kj kj−1 kj−2 kj−3 kj−4 kj−6 kj−8 Support s 0 1 2 3 Double d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Structure of Qj k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' kj ≡ k − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Circles represent (s, d) connected diagrams in Qj k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Circles are in the area of s > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The commutator of diagrams in the circle at (s, d) with H0 generates the diagrams in the crosses at (s ± 1, d) and (s, d ± 1), indicated by the solid arrow tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The dia- grams in crosses are to be canceled with the contribution from � Qj−1 k , Hint � (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' solid arrow in the plane indicates the commutator with H0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The vertical dotted arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 3 in- dicates the commutator with Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The diagrams at the crosses are to be canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We give an example of commutator of land with H0 and Hint: � I , H0 � = I I +2 − I I −2 I I + − I I and � I , Hint � = I − I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We note that the commutator of connected diagrams and H also generates a disconnected diagram with a hole or a disconnecting gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' The details of the commutation relation of diagrams with H are given in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We can construct Qj k recursively by calculating the cancellation at the crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Exact expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content='— For the dependence of the co- efficients, we introduce the list of a diagram Ψ by λΨ = {λ0 ≡ λL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' , λw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λw+1 ≡ λR} where λL(R) is the length of the leftmost(rightmost) coast of Ψ and λi + 1(1 ≤ i ≤ w ≡ lΨ − 2) is the length of the i-th inner coast from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We give a example of a con- nected diagram Ψ of a list λΨ = {λL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λ1, λ2, λ3, λ4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λR}: I I I I I I I I I I I I I I I I I I I I I λ1+1 λ4+1 λR λL λ2+1 λ3+1 , where Ψ is the (24, 6) diagram and the gap number is gΨ = 2 and land number is lΨ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λΨ = {2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 1, 3, 2, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' 3} and the length of the coasts are depicted by the arrows, and the gap is indicated by the teal shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We show the exact expression of Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Q0 k is the (k, 0) diagram: Q0 k = I I + ↕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' For Qj k(j ≥ 1), we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' For j ≥ 1, Qj k = ⌊ k−1−j 2 ⌋ � n=0 ⌊ k−1−j 2 ⌋−n � d=0 ⌊ j−1 2 ⌋ � m=0 d � g=0 × � Ψ∈Sk,j,m n,d,g (−1)n+m+gCj,m n,d (λΨ) Ψ, (10) where Sk,j,m n,d,g is the set of (k − j − 2n − d, d) connected diagrams satisfying lΨ = j + 1 − 2m and gΨ = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' Cj,m n,d (λ = {λ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' λw+1}) ∈ Z>0 is invariant with the per- mutation of λi(1 ≤ i ≤ w ≡ j −1−2m), and with the exchange of λL and λR, and with the replacement of λa with min(λa, n) for 0 ≤ a ≤ w + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfDAbs/content/2301.03621v1.pdf'} +page_content=' We note that the freedom to add Qk′20% +efficiency7, as well as lasing and light-emitting +devices.3,4,6,8 +The properties of perovskite derive from its +specific ABX3 architecture, where A and B are + +2 + +cations, and X are anions arranged into chemically +stable corner-sharing octahedral BX6 frameworks. +There are three main crystallographic phases in which +perovskites +exist, +namely: +orthorhombic +(), +tetragonal (), and cubic (). Transitions between +aforementioned phases contribute to the formation of +multiple structural domains and lattice imperfections, +specifically through crystal twinning. This latter +process can be understood as an immunity response to +the loss of symmetry, when, upon minimizing the +Gibbs free energy, the system relaxes into a +thermodynamically +stable +state +by +forming +ferroelastic, +near-orthogonal +domains. +These +formations have been widely studied at the nano-, +micro- and mesoscales 9–13. In perovskites, crystal +twinning is associated with lowering of the lattice +symmetry by tilting and contortion of octahedrons, +which causes spontaneous intrinsic stress. The +introduction of such structural distortions will +significantly affect local electronic structure, hence +transition probabilities. Moreover, the density of +states is considerably larger at these sites, where the +latter can be viewed as an optical nanoantenna.14 +Temperature15,16 and pressure17 perturbations +have been utilized to alter the system’s properties +through manipulation of the crystal structure and +defect density. For many perovskite systems, the +bandgap for the tetragonal phase is slightly lower +compared to that of the orthorhombic and cubic +phases.18,19 This causes free carriers to migrate from +high to low bandgap areas, an effect that is most +pronounced near phase transition sites. It has been +hypothesized +that +such +spatially +non-uniform +transport leads to a local build-up of free carriers in +tetragonal domains.20,21 Such an accumulation of free +carriers increases the probability for electrons and +holes to radiatively recombine, affecting and, +ultimately, +enhancing +the +Raman +and +photoluminescence efficiencies near the lattice +distortions +sites. +If +temporally +and +spatially +controlled, this effect could be used to actively tune +the optoelectronic properties of the material, such as +boosting the brightness of light-emitting devices1,3 or +modulating the lasing efficiency 22,23. Control of such +enhanced emission requires precise manipulation of +phase structuring within the single crystal. This +manipulation, in turn, necessitates control of the phase +transitions and thus dynamic management of the local +temperature within the material. +There are two major prerequisites for producing +multi-phase structures in a dynamically controlled +manner – (1) a mechanism for rapid and efficient +heating at the nano- to micro-scales and (2) a +mechanism for heat release from the heated location. +In this context, the heating mechanism at small spatial +scales can benefit from the thermoplasmonic effect, +through which heat can be locally generated via +absorption of incident light by a plasmonically +resonant structure.24–26 This approach has been shown +useful for efficient heat generation (up to a few +thousands of K), followed by the rapid directional heat +transfer to the material of interest.26,27 Note that all- +inorganic halide perovskites have remarkably small +thermal conductivity (0.42 W m-1K-1),28 yet possess +high photo- and thermal stability. The latter underlines +the possibility of maintaining spatial and temporal +stability of the heat pattern and gradients across a +single crystal, resulting in a stable multi-phase +semiconducting system. +In this work, we demonstrate controlled multi-phase +structuring of a single crystal of cesium lead bromine +(CsPbBr3). We achieve this level of control by placing +the perovskite crystal on a thermoplasmonic +metasurface that consists of a 2D array of stacked +titanium nitride (TiN) plasmonic nano-pads on top of +silicon (Si) nano-pillars (Figure 1a).29 When irradiated +with visible light at a wavelength resonant with the +TiN structure, the plasmonic nano-pad serves as an +optically switchable heater, while the Si pillar +provides a channel for heat dissipation. This geometry +produces sub-wavelength thermal gradients across the +perovskite +microplate, +triggering +on +demand +formation of stable phase domains within the original +single crystal. + +2. Results and discussion +2.1. Device concept +CsPbBr3 perovskite undergoes two reversible +phase transitions above the room temperature. These +are the orthorhombic-to-tetragonal (361 K) and the +tetragonal-to-cubic () phase transitions as +determined with fast scanning calorimetry (FSC, + +3 + + + +Figure 1. (a) Schematic representation of a CsPbBr3 platelet mounted on a metasurface array. (b) A +72o tilted SEM image of the edge facet of the CsPbBr3 platelet on the TiN metasurface. (c) Optical +heating of halide perovskite crystal by the TiN/Si nanosctructure. Color areas within the temperature +gradient represent the  phase of the Pnma space group (blue), the  phase of the P4/mbm space group +(green) and the  phase of the Pm3m space group (red). (d) Finite-difference time-domain (FDTD) +and finite element method (FEM) simulations of the axial temperature distribution across the TiN/Si +and CsPbBr3 crystal. + +Supplementary Figure S1 and Supplementary Section +1). As the rate of the temperature sweep increases, the +data clearly shows the lack of mirror symmetry +between heating and cooling experiments. This +observation +points +to +defect-induced +spatial +heterogeneity within the crystal and reflects an +imbalance in the potential energy barriers associated +with the conversion of structure from lower to higher +symmetries and vice versa. This makes the FSC +method highly sensitive to the crystalline imperfection +content and density. The presence of these phase +transitions at high temperatures underlines the +possibility to form a combination of different crystal +phases in the material if steep and steady temperature +gradients are introduced. Figure 1a schematically +illustrates such a device concept that operates at +ambient +laboratory +conditions. +The +CsPbBr3 +perovskite platelet (10 m x 14 m x 1 m) is placed +on a metasurface that is comprised of a hexagonal 2D +array of Si pillars with a subwavelength base (L<). +Each nanopillar is capped by a TiN plasmonic pad on +top (Figures 1a and 1b). Upon illumination +(continuous wave, 633 nm, 16 mW, 0.6 m spot size, +NA=0.7), the TiN pad functions as a photothermal +heater, while the Si pillar transfers heat down to the +bulk substrate. Silicon was chosen as the thermostat +material because of its large thermal conductivity (148 + +Orthorhombic +Tetragonal +Cubic +(c) +(y phase) +(β phase) +(aphase) +y +361 K +403K +Cs +Pb +(a) +X +Br +cwpump +heat +361K +T,=T(h) +CsPbBr +403K +H +NLI +h +c-Si (100) +metasurfaceSi +TiNCsPbBr +(d) +650 +(b) +600 +a +β +FDTD/FEM +Y +550 +Temperature +-power-lawfit +CsPbBr +500 +450 +400 +200nm +350 +TiN/Sivoxels +300 +-1000 +-500 +0 +500 +1000 +z(nm)4 + +W m-1K-1) and strong Raman response (Si-Si 521 cm- +1). Moreover, its Raman activity is temperature +sensitive, permitting its use as a probe for Raman- +based thermometry. +The light-to-heat conversion is expected to be +maximum at the plasmonic absorption resonance of +the TiN structure, as characterized by the absorption +power +0 +abs +P +I +  +, where +abs + + is the absorption cross +section and I0 is the incident intensity.24,25 The +accessible temperature range at a thermal stationary +state of the system will depend on several factors, +namely the effective thermal conductivity of Si, the +pillar’s lateral and axial dimensions, the permittivity  +of the TiN pad and the incident flux I0. The pillar +geometry, defined by base lateral size L and height h, +governs the heat dissipation efficiency and its effect +has been discussed previously for composite TiN/Si +rods,29–31 tubes and trenches.32 Taking into account the +Fröhlich resonance condition, we can derive the +temperature change at the top of the Si pillar as a +function of structure height h and incident light +intensity +0I as follows29,31: + +𝜀��� +� +�𝜆�� � �2𝜀Si, + +𝛥𝑇��ℎ, 𝐼�� � +� +� +���� +��� 𝜀��� +�� +� +� +�Si 𝐼� � +�abs +�Si +� +��Si +�� ℎ𝐼� +��, +(1) + +where 0is the wavelength at the plasmonic +resonance, +TiN=’TiN+”TiN +is +the +complex +permittivity of a TiN heater, Q=-’TiN/”TiN is a Q- +factor for the plasmon resonance, Si is the +temperature-dependent thermal conductivity of bulk +Si and  is the geometry-dependent dimensionless +thermal capacity of TiN.25 For smaller pillar heights +(<200 nm), the first term dominates and +L +T + + is +expected to show a linear dependence on +0I (Figure +S2, see Supplementary Information) 29. For taller +pillars, the contribution of the second term increases +accordingly, resulting in a quadratic dependence of +the temperature on +0I . Moreover, +L +T + + is now +dependent on the first temperature derivative of 𝜅Si. +For bulk Si, this derivative has a negative sign above +room +temperature, +and, +thus, +L +T + + +should +monotonically increase with the incident intensity. For +structures with height exceeding h>500 nm, +significant deviation from experimental observations +have been reported and explained in terms of thermal +anisotropy.32 +Because the thermal conductivity of Si ( +1 +1 +Si +148 W m K + + + + +) significantly exceeds that both +of air ( +1 +1 +air =0.0263 W m K + + + +) and CsPbBr3 +perovskite ( +3 +1 +1 +CsPbBr +0.42 W m K + + + + +), the pillar +structure becomes the dominant channel for heat +dissipation with its geometry being the key factor in +determining the steady state temperature profile. +Hence, pillars of a specific height provide access to +specific temperature ranges, while fine control within +this range can be realized by varying the incident light +intensity I0. When irradiated, an array of such nano- +heaters can generate a two-dimensional temperature +pattern formed by sub-wavelength hot spots (L<). +Induced thermal gradients along the axial direction in +the perovskite allow particular phase domains to be +formed, subject to the distance from the heating TiN +pad (Figure 1c). Figure 1d shows a combined finite- +difference time-domain (FTDT) and finite element +(FEM) method (ANSYS/Lumerical) simulation of the +axial temperature distribution. The simulation reveals +the axial heat distribution within the layered system, +comprised of a 1 m Si pillar, a 50 nm TiN pad and a +1 m CsPbBr3 crystal. A pillar of this height is +associated with a steady state temperature range of +320-520 K or a 0.23 K/nm thermal gradient within the +Si material. The maximum temperature at the +plasmonic structure is chosen to be 630 K, a critical +temperature point beyond which the CsPbBr3 +optoelectronic properties drastically change.33 The +temperature gradient in the perovskite interior follows +a +0.54 +| |z  + dependence (R2=0.997), which is mainly +determined by crystal thermal conductivity. In these +simulations, the surrounding medium is assumed to be +air. Depending on the initial nano-pad temperature +(i.e. input light flux), the crystal interior can be +comprised of a single  phase or a structure of two or +all three phases, as shown for T0=630 K in Figure 1c. + + + + + +5 + + + +Figure 2. Confocal reflection images of a CsPbBr3 crystal using polarized light at (a) 303 K +(orthorhombic phase), (b) 393 K (tetragonal phase), (c) 408 K (cubic phase) and back (d) 393 K +(tetragonal phase), (e) 303 K (orthorhombic phase). + +2.2. Optical visualization of phase transitions and +crystal twinning +The real time dynamics of domain formation and +twinning in perovskite crystal on a microscale is +shown in Supplementary Movie SM1. In this +experiment, the crystal was placed on a hot plate to +perform temperature sweeps from 340 to 410 K and +back, spanning the orthorhombic-to-tetragonal-to- +cubic phase transitions. The heating and cooling rates +were sufficiently slow to allow a uniform temperature +to establish itself throughout the crystal. Optical +imaging and other spectroscopic experiments were +performed with the aid of a sample piezo positioning +feedback system, as described in Section 3 of +Supplementary Information. This solution overcomes +experimental obstacles such as the thermal expansion +of the sample and setup elements. It also corrects for +beam defocusing by the Bragg-like grating formed +through crystal twinning within the sample volume +(see Methods and Supplementary Information, Figure +S3 and Section 3). +Figure 2 depicts confocal reflection images of the +crystal +surface +at +the +selected +steady +state +temperatures. At 303 K (Figure 2a), the crystal +consists of parallel domains of the  phase that are +oriented at a 45o angle (<110>) relative to the lab +frame. The transition to the tetragonal phase occurs +around 393 K (Figure 2b). As the temperature is +increased to 408 K, the stripes disappear completely, +indicating the formation of a homogeneous cubic +crystal ( phase, Figure 2c). As the system cools down +and crosses the cubic-to-tetragonal transition, crystal +twinning triggers the formation of multiple tetragonal +domains (Figure 2d) and a further temperature +decrease brings the crystal back into the orthorhombic +phase (Figure 2e). The data shows clear differences +between the images of the crystal at the same +temperature points, but opposite ends of the +temperature cycle (Figures 2a and 2e). This difference +in the patterns further confirms the results of the FSC +experiments (Figure S2), indicating that the potential +energy barriers are different when the phase transition +proceeds along different directions of the temperature +sweep. +The symmetries of the original and final +crystallographic phases associated with a transition +are the key factors in the evolution of the crystal +twinning. Upon careful examination, it is clear that the +resultant orthorhombic phase reveals a 70 deviation +angle relative to the previously orthogonal orientation +of the domains in the tetragonal phase (Figure 2d and +2e). This is in excellent agreement with previous +calculations by density functional theory (DFT) that +yielded a ~130 octahedral tilt for orthorhombic + +=303K +(a) +T=393K +(b) +heating +408K +(c)1μm +-CsPbBr3 +1um +β-CsPbBr3 +T=303K +(e) +T=393K +(d)cooling +0-CsPbBra +83° +90° +LLm +1um +y-CsPbBr3 +β-CsPbBr36 + +CsPbBr3.18 The rotation of the corner-sharing Br +atoms of the [PbBr6]4- octahedron in the equatorial +plane by ~ 6.50 should result in a relative re- +orientation of the domains to 900-~ 83.50, as +demonstrated in Figure 3f. + +2.3. Temperature dependence of Raman and +photoluminescence signatures +Both the FSC and the optical imaging experiments +reveal information about the phase transitions in the +perovskite crystal, and both measurements point to the +importance of lattice imperfections and distortions. To +examine their role at the microscopic level, we +performed +Raman +and +photoluminescence +experiments, which are particularly sensitive to the +electronic structure near defects and phase interfaces. +The Raman spectrum of CsPbBr3 perovskite +features two main low energy vibrational modes, +namely the 127 cm-1 TO (first-order transverse +optical) and the 312 cm-1 2LO (second-order +longitudinal optical) Pb-Br stretching phonon modes +(Figure 3a-d).34 It is important to note that the +presence of the 312 cm-1 peak in the Raman spectrum +is evidence for the more pristine CsPbBr3 structure +relative to the presence of CsPb2Br5, with the latter +being the result of exposure to water.35 +The temperature dependence of the Raman +spectra for both the TO and 2LO modes are different +for various directions of the temperature sweep +(Figures 3e and 3f) of a uniformly heated crystal. As +the temperature is increased, the intensity of the TO +phonon line (127 cm-1) undergoes two extrema +corresponding to the orthorhombic-to-tetragonal (361 +K) and tetragonal-to-cubic (403 K) phase transitions +(Figure 3a and 3e). The -CsPbBr3 phase reveals an +expected trend, namely the decrease of the Stokes +intensity with temperature, caused by the bandgap +widening and the depletion of carriers in the valence +band.18,19 Meanwhile, the temperature dependence of +the Stokes bands ascribed to the  and  phases shows +the opposite trend (Figure 3e). This observation can be +explained by the interplay between the thermal +volumetric expansion and the tilt of the [PbBr6]4- +octahedra.18 It has been predicted that both +mechanisms are capable of significant widening of the +bandgap19, estimated to be <2.0 eV for -CsPbBr3 +versus ~2.36 eV for -CsPbBr3 and ~2.4 eV for - +CsPbBr3. These bandgap variations offer possible +explanations for the observed positive temperature +trends. For example, for the given experiments the +Raman process in -phase is closer to the resonance +for the used excitation photon energy (633 nm, 1.96 +eV). This may lead to a signal increase when more - +phase sites are introduced. Another potential +mechanism derives from the contribution of the +shallow and deep states to the free carriers population +at the conduction band is expected to increase with +temperature,36 enabling to change the Raman +polarizability.37 +When the temperature is lowered, the Raman +intensity of the TO mode decreases continuously and +does not exhibit any extrema in this temperature +range. We speculate that such behavior can be +understood +from +the +dominant +role +of +the +crystallographic +deformation +of +the +[PbBr6]4- +backbones while cooling down. Spatially resolved +Raman intensity maps for each phonon mode at +different temperatures are presented in Figure S4. The +images agree well with the confocal reflection images +(Figure 2). They demonstrate a clear variation in the +domain pattern as a function of the directionality of +the temperature sweep across particular phase +transitions - a result of the unequal potential energy +barriers of the high-to-low and low-to-high symmetry +conversions. +The difference in crystal twinning also impacts +the temperature trend of the TO and 2LO phonon +lines, resulting in their characteristically different +behavior (Figure 3e and 3f). For the TO phonon mode, +electron-phonon scattering at the  point is more +sensitive to twin domain formation due to the overall +momentum restrictions for the one-phonon process. +This is opposite for the 2LO mode, for which there is +a simpler path to fulfill momentum conservation due +to the involvement of two phonons to scatter light +inelastically. While the 2LO line clearly shows the +→ transition (red curve, Figure 3f), at the same time +it appears insensitive to the → transition. The +cooling curve exhibits similar behavior for both +transitions. Whereas the multi-phonon mode can be +utilized as a temperature probe for a defect-free +crystal, the single-phonon TO mode is more sensitive +to the orthorhombic-to-tetragonal and tetragonal-to- +cubic phase transitions. + +7 + + +Figure 3. Temperature-dependent Raman spectroscopy of a CsPbBr3 crystal at thermal equilibrium for +TO phonon mode at 127 cm-1 (a, c) and LO two-phonon mode at 312 cm-1 (b, d) upon heating and +cooling at a rate of 0.4 K/s. (e,f) Cross-sections at peak center for TO and 2LO modes (dashed lines in +(a-d). + +Photoluminescence +(PL) +microspectroscopy +provides additional information on the carrier +dynamics and the origin of the emission mechanism. +The latter has been investigated through power +dependence and fluorescence lifetime studies and is +discussed in detail in Supplementary Information, +Section 5. Here, we focus primarily on the temperature +trends of perovskite photoemission. When the +temperature is raised, the PL intensity drops +dramatically (Figure 4a), reaching minimum at T→ +at 361 K. Further increase of the lattice temperature +gives rise to a higher PL intensity for the tetragonal +() and cubic () phases (red curve, Figure 4c). The +overall PL spectral shape reveals complex behavior +through the sweep, showing splitting-like behavior at +high temperatures (Figure 4b). First, a blueshift of the +mean of the spectral distribution (~16 meV) is +observed (Figure 4d), indicating the bandgap +expansion of the cubic phase at 423 K.19,37 Second, a +red-shifted signature (~18 meV) is observed, which is +suggested to originate from the competition between +surface and interior contributions of the crystals +(Figure 4d).37,38 +A radically different trend is observed when +cooling is performed, with fluorescence showing a +strong local maximum at the →transition (Figure +4b and Figure 4c). A similar observation has been +reported +for +methylammonium +lead +triiodide +(CH3NH3PbI3 or MAPbI3), upon cooling from 160 K +to 140 K.20 The nature of the PL enhancement across +this phase transition can be understood as resulting +from the funneling effect,20 when mobile carriers +migrate to the “defect-free” low-bandgap tetragonal +phase. +The observed hysteresis agrees well with the +confocal reflection and Raman studies, and can be +understood in a similar manner - lattice reconstruction +and the dependence of crystal twinning on the sign of +the temperature change T. Since twinning requires +the base of one domain to be matched to and shared +with the side of another, its probability will strongly +depend on the presence of inherent crystal +imperfections and the geometry of the original and +resultant phases. This leads to significant differences +in the overall pattern of the multi-domain assembly as +a function of the sign of the temperature trend, and, in +turn, the number of structural defects and phase +interfaces being formed. This phenomenon also +explains the striking contrast in the PL quantum +efficiency. It suggests that the presence of point + +heatingat0.4K/s +420 +(a) +420 +(b) +Pb-Br mode at 127 cm +Pb-Brmode at 312 cml +β-& +400 +32 +6.0 +(e) +heating +(f) +Y-β +360 +-β +heating +per +340 +30 +888 +320 +320 +5.2 +ounts, +100150200250 +260 +310 +360 +Ramanshift(cm) +Ramanshift(cm") +coolingat0.4K/s420 +(c) +0zt +(d) +26 +β-α +400 +β-α +4.4 +380 +cooling +cooling +360 +24 +4.0 +340 +300330360390420 +300330360390420 +320 +320 +Temperature (K) +Temperature (K) +100150200 +250 +260 +310 +360 +Ramanshift(cm) +Ramanshift(cm8 + + + +Figure 4. (a,b) False color PL maps for different sign of temperature sweep. (c) Cross-sections along +the vertical dashed straight lines at the center of PL spectrum. (d) PL spectra for different temperatures. + +defects and crystal twinning favor the →transition +and hinder PL for the reverse direction of the +transition. + +2.4. Optical properties of multiple phase single +crystal. +After the optical characterization of pervoskite +platelets held at a uniform temperature, using +reflection, +Raman +and +photoluminescene +microspectroscopy, we next used these optical tools to +study crystals subjected to a temperature gradient. For +this purpose, we employed the metasurface heating +device discussed in Section 2.1 to maintain stable +temperature gradients in the crystal and control the +distribution of phase domains in the axial (z) direction. +Figure 5a shows a scanning electron microscopy +(SEM) image of a CsPbBr3 microplate placed on the +metasurface. Figures 5b1 to 5b3 (cyan, yellow and +magenta) depict 72o tilted images of the corners +marked with an arrow of the corresponding color +(Figure 5a). As is clear from the images, the structure +is formed by two stacked crystal plates, most clearly +observed through their exfoliation at one of the +corners (Figure 5b1 and Supplementary Figure S6). +The metasurface is comprised of a 2D hexagonal array +of TiN/Si voxels with a pillar height estimated to be +approximately 900 nm as shown in Supplementary +Figure S7. The top of the voxel is visualized in the +inset of Figure 5a. It is clear that, upon illumination, +the TiN pads become damaged for intensities +exceeding 3 MW/cm2 (green and blue contoured +images in the inset of Figure 5a). +The confocal reflection image at 633 nm of the +perovskite platelet is shown in Figure 5c. In this +image, the uncovered TiN/Si voxels have been placed +at the focal plane of the objective. For such an +arrangement, the voxels that are covered by the +perovskite appear out of focus as light has to penetrate +through the 1 m-thick material of refractive index +n=2.5.37 This effect not only prevents efficient heating +of the TiN pads, but also limits the efficient collection +of the Raman signal from the voxel. The collection +efficiency is instrumental, as the Raman response was +utilized as a remote temperature probe. In all further + +Energy(eV) +cooling +2.452.40 +2.35 +2.30 +5 +2.5 +d +(b) +T=303K +410- +T=358K +(d) +403K +T=403K +3 +390 +sity, x10 +T=423K +2.0元 +B +sity +370 +361K +blueshift +redshiftinten +350 +intel +2 +E +330- +1.0 +e +310 +510515520525530535540 +500 +510 +520 +530 +540 +550 +Wavelength(nm) +Wavelength（nm)heating +5 +a +(a) +heating +410 +C +403K +cooling +390 +nsity,x1 +e +B +361K +3Q350 +inte +B++α +?330- +P +310 +510515520525530535540 +300320340360380400420 +Wavelength(nm) +Temperature(K)9 + + + +Figure 5. CsPbBr3 platelet on the thermoplasmonic TiN/Si metasurface. (a) SEM image of the CsPbBr3 +plate over the metasurface. The insets show TiN/Si voxels, marked with the red, green and blue squares, +exposed to 633 nm cw illumination with the intensity of 0, 3.5 and 5.0 MW/cm2. (b1)-(b3) SEM images +(side views at the tilt angles of 72o (b1) and 48o (b2), (b3) from the sides marked with cyan, yellow and +magenta arrows in Figure 5a. (с) A confocal reflection image at 633 nm. (d) False color PL spectra +central frequency map. (e) PL spectra taken at spots marked in Figure 5d with red, blue and green filled +circles, respectively. (f, g) Raman maps at 521 cm-1 (c-Si) and 127 cm-1 (Pb-Br mode). (h) Raman +spectra of Si pillar as a function of input light intensity. The inset shows a cross section along a dashed +white line and numerical deconvolution of the composite band into Lorentzian and Gaussian +components. (i) Temperature map measured based on Raman thermometry. (j) Simulated cumulative +Raman signal from phase-structured crystal of different thicknesses. (k) Raman intensity vs the +pumping intensity or gradient initial temperature T0 temperature for 127 cm-1 (green) and 312 cm-1 (red) +of perovskite phonon modes and 521 cm-1 Si line. + +experiments, the light was focused on the top of the +TiN/Si voxels that are under the CsPbBr3 microplate. +Figure 5d displays the results of confocal PL +imaging. PL spectra as a function of spectral position +on the sample were collected using 1.7 W cm-2 of 473 +nm excitation. It is important to note that such low +fluxes did not introduce any meaningful temperature +gradients. In addition, the excitation wavelength used +is far away from the absorption resonance of the +plasmonic structures. The false color PL map can be +divided into three characteristic regions according to +PL spectral shape and central frequency position +(Figure 5e) - blue (522 nm, 2.375 eV), red (531 nm, +2.335 eV) and an intermediate green region. We +observe a clear correlation between the PL spectrum +and the sample thickness and/or stacking. Higher +energy PL, centered around 522 nm (2.375 eV), is +observed in areas where two thin ~400 nm plates are +stacked (blue spot in Figure 5d and Figure 5b2). +However, the spectrum is red-shifted by 40 meV +at the position where the sample appears to consist of +a 1 m-thick single plate (red spot in Figure 5d and + +(a)TiN:Simetasurface +OMWI +(C) +521cm(c-Si)() +metasurface +127cm（Pb-Br）(g) +100nm +C-3.5MW +00nm +CsPbBr,plate +CsPbBr.plate +2m +2 μm +LIT +100nm +AT(K) +RamanspectraofSi +20 +40 +60 +80 +100 +120140 +um +PLspectralpositionmap540 +5 +780K +(h) +Calculatedintensity +() +(b1) +P +B60K +4 +85 +520 +2 +200nm +400nm +200nm +836K +800nm +600 nm +JS +320K +1000nm +2 +305K +500 +500 +510 +520 +530 +0.5 +Pumpingintensity(MW/cm） +1.5 +2.5 +3.5 +Energy(eV) +Ramanshifts(cm) +2.452.402.352.302.25 +AT (K) +50 +200(e) +Ramanshiftbased +3.5 +temperaturemap +600 +(k) +uim +intensity +127 cm(Pb-Br) +312cm(Pb-Br) +2.0 +521 cm(c-Si) +PL +450 +40meV +1.0 +500510520530540550560 +2m +0.0 +um +Wavelength(nm) +300 +Pumpingintensity(MW/cm210 + +Figure 5b3). It has been suggested that the observed +phenomenon is caused by the excitation of waveguide +modes within the Fabry-Perot resonator through the +absorption-emission-absorption mechanism 34. If true, +monitoring of the PL spectral position offers a means +to probe the distribution of the perovskite thickness. +Figures 5f and 5g show confocal Raman maps for +the 521 cm-1 (c-Si peak of the pillar) and 127 cm-1 (TO +Pb-Br phonon mode of CsPbBr3) lines. It is evident +that not all voxels under the platelet can be clearly +differentiated in the image. This is caused by the +damage while positioning the perovskite on the +metasurface and/or by the poor contact at certain +positions. The enhanced Raman scattering of the TO +mode at the crystal edges originates from structural +inhomogeneities, where the density of surface states is +higher (Figure 5g). For quantitative monitoring and +visualization of the temperature at the voxel, we used +Raman thermometry. This method, thoroughly +described +elsewhere29–31 +(see +Supplementary +Information, Section 8), utilizes the temperature +dependent behavior of the c-Si Raman signal (521 cm- +1) as a remote probe. Through the use of an Echelle +grating, the spectral resolution of the imaging system +reaches +0.1 +cm-1 +and +enables +temperature +measurements with 5 K accuracy. +A detailed analysis of the open voxel temperature +(blue voxel, Figure 5a) versus input flux is shown in +Figure 5h. Note that the c-Si mode is asymmetrically +broadened (inset, Figure 5h). This effect originates +from the non-uniform heat distribution in the +structure, resulting in the presence of contributions +from both hot and cold portions of the material.31 To +further simplify the analysis, the spectrum was fitted +with Lorentzian (hot medium contribution) and +Gaussian (cold medium contribution) spectral line +shapes, using a regularized least squares method +(R2=0.998). The intensity map in Figure 5i clearly +indicates that the contribution from hot domains +deviates significantly from a linear incident intensity +dependence for intensities exceeding 4 MW cm-2 (550 +K). We attribute this effect to temperature dependent +changes in the TiN permittivity, which affects the +plasmon resonance frequency. In addition, the thermal +conductivity of Si decreases when the temperature is +raised.39 +The signs of degradation of TiN pad appear at 750 +K, where the Raman intensity peaks at about 5 MW +cm-2 and then shifts back to the higher energy side. +This is also confirmed by previous experiments using +ellipsometry on TiN films.29 Thus, in our experiments, +the incident light flux enabled access to the 293 K to +473 K temperature range – sufficient to activate all +necessary structural transitions in CsPbBr3 while +preventing photo-damage of the plasmonic structures. +Figure 5h shows the resulting temperature map +derived from the Raman shift using Equation S1 (see +Figure S9) and measured at 3.5 MW cm-2. +The local generation of hot spots produced +thermal gradients throughout the perovskite crystal. +This effect resulted in the simultaneous formation of +multiple phase domains, as illustrated in Figure 1. +However, there are significant and fundamental +differences between the temperature trends of the +Raman signals when (1) heating of the whole crystal +by a hot plate to achieve a uniform temperature +profile, as opposed to (2) establishing a temperature +gradient in the crystal with the metasurface. For the +first case, the upward temperature trend discussed in +Section 2.3 is shown by the red curve in Figures 3e. +This profile shows clear extrema at phase transitions +with an overall signal intensity decrease across the +tetragonal phase. In the second case, when the crystal +is locally heated by the plasmonic structures, the +temperature gradient induces multiple phases in the +axial direction. For this case, the Raman response is +the cumulative signal from all the phases in the +collection volume. +The trend of the cumulative Raman signal +RI +versus the plasmonic pad temperature +(0) +m +T + should +directly reflect the process of multi-phase structuring +of the perovskite. The trend can be modeled as +discussed in Supplementary Information, Section 10. +For simplicity, one can assume one-dimensional heat +dissipation in a homogeneous perovskite crystal, in +which +the +temperature +profile +obeys +a +0.54 +( ) +(0) | +| +m +m +T +z +T +z  + + power law in the axial +direction (Figure S10). The resulting Raman response +can then written be as: +0 +( ) +z +RI +I z +dz + +  + (2), +where +( ) +I z + is the average Raman signal of the +homogeneous media at a given z-plane (Figure S10) + +11 + + + +Figure 6. (a) Linearly corrected Raman response versus the incident light flux. (b, c, d) represents +simulated temperature points where single, double and triple phase structure occur as seen in Raman +signatures from (a). The blue dashed line represents the subtracted linear contribution. + +and z + is the total crystal thickness. Figure 5j shows +plots of IR vs I0 for different +z + of perovskite. As +expected, for very thin crystals (<200 nm), the +temperature trend of Raman signal closely follows the +one previously observed for a thermally equilibrated +crystal on a hot plate (green curve Figure 5j, red curve +Figure 3e). For thicker crystals, multiple phases can +contribute to the observed Raman signal. The local +maximum remains highly pronounced over the +monotonically increasing Raman response, indicating +the formation and growth of a two-phase structure ( +and ) in the axial direction. +This model agrees well with experimental +observations. Figure 5k shows the intensity evolution +of the phonon modes (TO 127 cm-1, 2LO 312 cm-1) +along with the c-Si line (521 cm-1) as a function of the +incident light flux/pad temperature. As expected, the +trends for the 127 cm-1 (Pb-Br) and 312 cm-1 (Pb-Br) +modes are inherently different from the case of the +thermally equilibrated system (red and green curves, +Figures 5k). For the TO mode, a clear presence of +local maximum around the temperature of -to- +transition is observed, indicating the formation of a +two-phase structure. Upon further increase of the pad +temperature +(0) +m +T +, another shallow bump at T ~ +140 K indicates triple phase formation (,  and ). +To visually emphasize these signatures, the linear +contribution to the Raman-temperature trend has been +subtracted in Figure 6a. The linear contribution has +been determined from a simple linear fit over 0 – 2.5 +MW cm-2 range (Figure 6a point b, Figure 6b), where +the whole perovskite crystal remains in the single +orthogonal  phase and the intensity of the TO Raman +peak should linearly increase with the incident +excitation flux and temperature (Figure 3e). Upon +increasing the incident intensity, the formation of the +tetragonal phase at the interface of the perovskite and + +△T (K) +Raman enhancement +0 +50 +100 +150 +200 +1.6 +(a) +Y,β +α,β, +1.4 +b +c +d +1.0 +Q +0 +1 +2 +3 +4 +5 +Pumping intensity (MW/cm +TiN CsPbBr +TiN CsPbBrs +TiN CsPbBr +550 +550 +550 +(b) +(c) +(d) +Temperature (K) +500 +500 +500 +450 +450 +450 +β,? +α,β, +400 +400 +400 +350 +350 +350 +300 +300 +300 +0 +500 +1000 +0 +500 +1000 +0 +500 +1000 +z (nm) +z (nm) +z (nm)12 + +TiN is expected to occur at point c (Figure 6a). At this +point of the trend, the steep temperature gradient +creates a spatially sharp defect area where the crystal +is in a transitional form between the orthorhombic and +tetragonal phase (Figure 6c). This scenario manifests +itself as a shallow Raman intensity maximum. Further +increase of the plasmonic pad temperature drives the +-phase deeper into the crystal bulk. At T~130 K +another shallow maximum of the TO Raman peak +indicates the formation of the -phase in close +proximity to the TiN structure. Upon subsequent +increase of the incident light flux, the  and  phases +extend further into the crystal and significantly +broaden (Figure 6d). This is expected to result in the +smearing of the boundaries between the different +phases. This effect is spatially asymmetric, i.e. +different for the left (-) and the right (-) sides of + phase, following the highly nonlinear temperature +gradient. At higher temperatures, the signal is an +interplay of several contributions, in particular the +phase layer thickness, the temperature, position along +the gradient, and the sharpness of the phase boundary. +We hypothesize the third maximum at T = 170 K is +the result of such a cumulative effect and may be +associated with the delocalized (disordered or +randomly located) phase boundary. Among other +instrumental contributors to the spatial phase +formation are the crystal intrinsic defects. Their +presence can trigger the spontaneous formation of +different phases, resulting in a highly irregular +structural front. At higher TiN temperatures, the - +and - boundaries broaden significantly and may +capture more defects into these areas where the phases +are highly mixed. +These experiments demonstrate that the Raman +signal from perovskite subjected to a stable +temperature gradient, as shown in Figure 5k and +emphasized in Figure 6, reveals distinct behavior that +is in contrast to a bulk crystal held at uniform +temperature. The dependence of the Raman signal on +the incident intensity shows clear signatures of +particular phase formations, their extension into the +bulk of the material, and, overall, the multi-phase +structuring process. Moreover, it shows that the +detected Raman signal exhibits notable gain across +T=150 K. Using Raman microspectroscopy as a +probe, these results indicate that it is possible to +generate on demand a distribution of single, double +and triple phase structures in perovskite by simply +controlling the incident light intensity. + +3. Conclusions +In this work, we have demonstrated the proof-of- +principle multiphase structured single crystal CsPbBr3 +halide perovskite. We have shown, the single, double +and triple phase systems can be created in optically +controlled +fashion +on +the +thermoplasmonic +metasurface using the continuous wave illumination +of modest intensities. Light-induced heat from +plasmonic TiN nanopads forms strong temperature +gradients within crystal bulk that are followed by +sequence of corresponding phase transitions. +Lattice distortions, defects and impurities operate +like an optical nanoantenna, increasing the density of +states. Thus, multi-phase perovskite structures hold +many interesting properties and open exciting +possibilities. In such system, charge carriers migrate +from lower symmetry lattice with large bandgap +(orthorhombic and cubic) to higher symmetry, but +lower bandgap crystal parts (tetragonal). There, highly +concentrated and in close proximity to the boundaries, +carriers efficiently recombine, leading to areas with +significant enhancement of the optical emission. This +multi-structured system promises to be highly +beneficial to the development of next-generation +ultracompact +broadband +light-emitting +diodes +showing high PL quantum yields above room +temperatures. + +Methods +Synthesis of CsPbBr3 structures +Perovskite microcrystals on glass substrates were +synthesized by using a protocol similar to the +previously reported.40 PbBr2 (110 mg) and CsBr (62 +mg) were mixed and dissolved in 3 ml of anhydrous +dimethyl sulfoxide (DMSO) inside a nitrogen-filled +glovebox. Droplet of the prepared solution (volume 2 +µl) was drop-casted on the substrate at ambient +conditions. After that, the substrate was sealed in a +preheated up to 60 oC Petri dish containing 200 µl of +liquid mixture. The solution was dried in the presence +of azeotropic vapor for 5 min. As a result, the +randomly +oriented separate +CsPbBr3 micro- +crystals were formed on the substrate. + + +13 + +Synthesis, nanofabrication and characterization of a +TiN/Si metasurface +TiN thin films on c-Si (100) substrates were DC +magnetron sputtered from a Ti target in the Ar/N2 +environment with a volume proportion of 30:70 at +elevated temperature of 350 oC and base pressure of +9 +3 10 + + mbar and power of 200 W. Prior to the film +growth, the c-Si substrate was sonicated in acetone for +15 min. The thickness of the TiN films, equal to +50 +5 + + nm, was measured with a contact profilometer +Alpha Step 200. +A 2D array of TiN/Si voxels were engraved with +the help of focused ion beam (FIB) milling at a lower +current of 1 pA by using Quanta 3D FEG (FEI). Since +the higher TiN/Si voxels are exposed to FIB for a +longer time, their lateral size is reduced due to edge +melting. To avoid this detrimental effect, we used +different mask templates for short and long voxels so +that their lateral size is the same regardless of height. +The temperature-dependent permittivity of TiN +thin films were measured with a spectroscopic +ellipsometer (VASE, J. A. Woollam) within the +spectral range of 250-2500 nm. The incident angle +was 70°. The TiN sample was exposed to thermal +annealing at the fixed temperature, whereas its +permittivity was probed at room temperature. The +temperature increment for each subsequent cycle was +100°C. The temperature ranged from 25 oC to 600 oC. +The samples were annealed at ambient air for 30 min +using a heating stage (Linkam Scientific Model +THMS600). The heating and cooling rates were 150 +°C/min and 100 °C/min, respectively. + +Fast Scanning Calorimetry +The Fast Scanning Calorimetry (FSC) curves +were registered on FlashDSC2+ (Mettler-Toledo, +Greifensee, Switzerland) equipped with TC100MT +intracooler with UFH1 sensor. The temperature +calibration +was +performed +using +biphenyl +( +o +69.2 C +m +T  +) +and +benzoic +acid +( +o +122.3 C +m +T  +) as standards to ±1 °C. +A single perovskite crystal was placed in the +center of the calorimetric sensor. To improve signal- +to-noise ratio the crystal size was chosen such as to +almost match the active area of the sensor. Within the +temperature range from 20 °C to 180 °C the sample +was chemically stable, and the curves were repeatable, +which allowed for averaging multiple scans to further +improve signal-to-noise ratio. + +Atomic force microscopy +The multimode scanning probe microscope +NTEGRA PRIMA (NT-MDT) was utilized for +visualizing a topography of the CsPbBr3 microplate +surface and the thermoplasmonic metasurface. The +AFM probes of the “VIT_P” series with resonant +frequencies around 350 kHz were used in AFM +measurements. The CsPbBr3 microplate mounted on +the metasurface fabricated by focused ion beam +milling was measured in tapping mode with a free +amplitude +0 +A of 10-20 nm and a set-point value of +0 2 +A +. + +Far- and near-field Raman spectroscopy and +microscopy +Raman spectra and maps were captured with a +multi-purpose +analytical +instrument +NTEGRA +SPECTRA™ (NT-MDT) in inverted configuration. +The confocal spectrometer was wavelength calibrated +with a crystalline silicon (100) wafer by registering the +first-order Raman band at 521 cm-1. A sensitivity of +the spectrometer was as high as ca. 3000 photon +counts per 0.1 s provided that we used a 100× +objective (N.A.=0.7), an exit slit (pinhole) of 100 m +and a linearly polarized light with the wavelength of +632.8 nm and the power at the sample of 16 mW. No +signal amplification regimes of a Newton EMCCD +camera (ANDOR) was used. +128x128 pixel Raman maps were raster scanned +with an exposure time per pixel of 0.1 s and were +finally collected with the EMCCD camera cooled +down to -95oC. Raman spectra within the range of +from -2000 to 2000 cm-1 were registered with a +spectral resolution of 0.1 cm-1 using the Echelle +grating. + +Fluorescence Lifetime Imaging Microscopy + To measure the PL decay time we used a system +built-in the confocal optical spectrometer (NTEGRA +SPECTRA) that includes a picosecond diode laser +(BDL-SMN) generatig pulses of 473 nm wavelength, +30 ps pulse duration, and 80, 50, or 20 MHz repetition +rate, a Simple-Tau 150 TCSPC FLIM module +(Becker&Hickl), and a HPM-100-40 GaAsP hybrid + +14 + +detector (Becker&Hickl). The detector has a detection +efficiency of about 50% and is free of afterpulsing. + +FDTD/FEM calculation +3D simulation of optical absorption of a TiN/Si +voxels consisting of stacked TiN and Si cylinders +under cw illumination was performed by using an +Ansys/Lumerical FDTD solver. The height of the TiN +pad was 50 nm, whereas the height of the Si pillar 900 +nm. To avoid anomalous electric fields near the TiN +pad edge we used disks with rounded edges (10 nm +rounding). A mesh overlayer of 1 nm was utilized +around the TiN pad and a rougher 10 nm mesh for the +rest of the structure. Perfectly matching layers were +used as boundary conditions for three directions. The +optical and thermal properties of Si and air were +imported +from +the +Ansys/Lumerical +material +database. The TiN pad was exposed to a 632.8 nm +focused laser light (NA= 0.7) with the intensity of 5 +MW/cm2. The temperature profile was calculated +through an Ansys/Lumerical FEM solver in the steady +state regime. The thermal conductivity of all +constituents +is +assumed +to +be +temperature- +independent. The boundary condition of +300 K +T  + +was set at the +min +20000 nm +z +  + of the 20×20×5 +m3 simulation region. + +Conflicts of interests +There are no conflicts to declare. + +Acknowledgement +This work was supported by grant No. 19-12-00066- +P of the RSF. The PL decay time measurements were +granted by the Kazan Federal University Strategic +Academic Leadership Program (PRIORITY-2030). +The authors acknowledge a technical support from our +industrial partners: SCANSENS (GmbH, Germany) +and NT-MDT BV (The Netherlands). + +References + +1. A. Dey, J. Ye, A. De, E. Debroye, S. K. Ha, E. Bladt, A. +S. Kshirsagar, Z. Wang, J. Yin, Y. Wang, L. N. Quan, +F. Yan, M. Gao, X. Li, J. Shamsi, T. Debnath, M. Cao, +M. A. Scheel, S. Kumar, J. A. Steele, M. Gerhard, L. +Chouhan, K. Xu, X. Wu, Y. Li, Y. Zhang, A. Dutta, C. +Han, I. Vincon, A. L. Rogach, A. Nag, A. Samanta, B. +A. Korgel, C.-J. 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Interfaces +2019, 11, 1040 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/_tE4T4oBgHgl3EQf4g2G/content/tmp_files/load_file.txt b/_tE4T4oBgHgl3EQf4g2G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8dd9deb2bfaad8c5d1d3eb1aac872c50656dca5b --- /dev/null +++ b/_tE4T4oBgHgl3EQf4g2G/content/tmp_files/load_file.txt @@ -0,0 +1,1031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf,len=1030 +page_content='1 Light-controlled multi-phase structuring of perovskite crystal enabled by thermoplasmonic metasurface Sergey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kharintsev1*, Elina I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Battalova1, Timur A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mukhametzyanov2, Anatoly P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Pushkarev3, Ivan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Scheblykin4, Sergey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Makarov3,5, Eric O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Potma6, and Dmitry A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Fishman6* 1Department of Optics and Nanophotonics, Institute of Physics, Kazan Federal University, Kremlevskaya, 16, Kazan, 420008, Russia 2Department of Physical Chemistry, Institute of Chemistry, Kazan Federal University, Kremlevskaya, 18, Kazan, 420008, Russia 3School of Physics and Engineering, ITMO University, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Petersburg 197101, Russia 4Department of Chemistry, Lund University, 221 00 Lund, Sweden 5Qingdao Innovation and Development Center, Harbin Engineering University, Qingdao 266000, Shandong, China 6Department of Chemistry, University of California, Irvine, CA 92697, USA Halide perovskites belong to an important family of semiconducting materials with unique electronic properties that enable a myriad of applications, especially in photovoltaics and optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Their optical properties, including photoluminescence quantum yield, are affected and notably enhanced at crystal imperfections where the symmetry is broken and the density of states increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' These lattice distortions can be introduced through structural phase transitions, allowing charge gradients to appear near the interfaces between phase structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In this work, we demonstrate controlled multi-phase structuring in a single perovskite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The concept uses cesium lead bromine (CsPbBr3) placed on a thermoplasmonic TiN/Si metasurface and enables single, double and triple phase structures to form on demand above the room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This approach opens up application horizons of dynamically controlled heterostructures with distinctive electronic and enhanced optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' KEYWORDS: halide perovskite, thermoplasmonics, metasurface, optical heating, phase transition, twin domains, Raman scattering, photoluminescence, fast differential scanning calorimetry, piezoresponse reflection microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Introduction Perovskite-structured direct bandgap semiconductors form an important class of materials with equally important applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The presence of antibonding states near the maximum of the valence band gives rise to a defect-tolerant semiconductor material with unique electronic and optical properties, including fast charge transport, an extended free carrier diffusion length, a high exciton binding energy, and bandgap tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1–6 These properties have already enabled promising applications in photovoltaics and solar energy conversion with >20% efficiency7, as well as lasing and light-emitting devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='3,4,6,8 The properties of perovskite derive from its specific ABX3 architecture, where A and B are 2 cations, and X are anions arranged into chemically stable corner-sharing octahedral BX6 frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' There are three main crystallographic phases in which perovskites exist, namely: orthorhombic (\uf067), tetragonal (\uf062), and cubic (\uf061).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Transitions between aforementioned phases contribute to the formation of multiple structural domains and lattice imperfections, specifically through crystal twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This latter process can be understood as an immunity response to the loss of symmetry, when, upon minimizing the Gibbs free energy, the system relaxes into a thermodynamically stable state by forming ferroelastic, near-orthogonal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' These formations have been widely studied at the nano-, micro- and mesoscales 9–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In perovskites, crystal twinning is associated with lowering of the lattice symmetry by tilting and contortion of octahedrons, which causes spontaneous intrinsic stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The introduction of such structural distortions will significantly affect local electronic structure, hence transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Moreover, the density of states is considerably larger at these sites, where the latter can be viewed as an optical nanoantenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='14 Temperature15,16 and pressure17 perturbations have been utilized to alter the system’s properties through manipulation of the crystal structure and defect density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For many perovskite systems, the bandgap for the tetragonal phase is slightly lower compared to that of the orthorhombic and cubic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='18,19 This causes free carriers to migrate from high to low bandgap areas, an effect that is most pronounced near phase transition sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It has been hypothesized that such spatially non-uniform transport leads to a local build-up of free carriers in tetragonal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='20,21 Such an accumulation of free carriers increases the probability for electrons and holes to radiatively recombine, affecting and, ultimately, enhancing the Raman and photoluminescence efficiencies near the lattice distortions sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' If temporally and spatially controlled, this effect could be used to actively tune the optoelectronic properties of the material, such as boosting the brightness of light-emitting devices1,3 or modulating the lasing efficiency 22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Control of such enhanced emission requires precise manipulation of phase structuring within the single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This manipulation, in turn, necessitates control of the phase transitions and thus dynamic management of the local temperature within the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' There are two major prerequisites for producing multi-phase structures in a dynamically controlled manner – (1) a mechanism for rapid and efficient heating at the nano- to micro-scales and (2) a mechanism for heat release from the heated location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In this context, the heating mechanism at small spatial scales can benefit from the thermoplasmonic effect, through which heat can be locally generated via absorption of incident light by a plasmonically resonant structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='24–26 This approach has been shown useful for efficient heat generation (up to a few thousands of K), followed by the rapid directional heat transfer to the material of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='26,27 Note that all- inorganic halide perovskites have remarkably small thermal conductivity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='42 W m-1K-1),28 yet possess high photo- and thermal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The latter underlines the possibility of maintaining spatial and temporal stability of the heat pattern and gradients across a single crystal, resulting in a stable multi-phase semiconducting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In this work, we demonstrate controlled multi-phase structuring of a single crystal of cesium lead bromine (CsPbBr3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We achieve this level of control by placing the perovskite crystal on a thermoplasmonic metasurface that consists of a 2D array of stacked titanium nitride (TiN) plasmonic nano-pads on top of silicon (Si) nano-pillars (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='29 When irradiated with visible light at a wavelength resonant with the TiN structure, the plasmonic nano-pad serves as an optically switchable heater, while the Si pillar provides a channel for heat dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This geometry produces sub-wavelength thermal gradients across the perovskite microplate, triggering on demand formation of stable phase domains within the original single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Results and discussion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Device concept CsPbBr3 perovskite undergoes two reversible phase transitions above the room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' These are the orthorhombic-to-tetragonal (361 K) and the tetragonal-to-cubic (\uf034\uf030\uf033\uf020\uf04b) phase transitions as determined with fast scanning calorimetry (FSC, 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (a) Schematic representation of a CsPbBr3 platelet mounted on a metasurface array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (b) A 72o tilted SEM image of the edge facet of the CsPbBr3 platelet on the TiN metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (c) Optical heating of halide perovskite crystal by the TiN/Si nanosctructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Color areas within the temperature gradient represent the \uf067 phase of the Pnma space group (blue), the \uf062 phase of the P4/mbm space group (green) and the \uf061 phase of the Pm3m space group (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (d) Finite-difference time-domain (FDTD) and finite element method (FEM) simulations of the axial temperature distribution across the TiN/Si and CsPbBr3 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Supplementary Figure S1 and Supplementary Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As the rate of the temperature sweep increases, the data clearly shows the lack of mirror symmetry between heating and cooling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This observation points to defect-induced spatial heterogeneity within the crystal and reflects an imbalance in the potential energy barriers associated with the conversion of structure from lower to higher symmetries and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This makes the FSC method highly sensitive to the crystalline imperfection content and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The presence of these phase transitions at high temperatures underlines the possibility to form a combination of different crystal phases in the material if steep and steady temperature gradients are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 1a schematically illustrates such a device concept that operates at ambient laboratory conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The CsPbBr3 perovskite platelet (10 \uf06dm x 14 \uf06dm x 1 \uf06dm) is placed on a metasurface that is comprised of a hexagonal 2D array of Si pillars with a subwavelength base (L<\uf06c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Each nanopillar is capped by a TiN plasmonic pad on top (Figures 1a and 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Upon illumination (continuous wave, 633 nm, 16 mW, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='6 \uf06dm spot size, NA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='7), the TiN pad functions as a photothermal heater, while the Si pillar transfers heat down to the bulk substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Silicon was chosen as the thermostat material because of its large thermal conductivity (148 Orthorhombic Tetragonal Cubic (c) (y phase) (β phase) (aphase) y 361 K 403K Cs Pb (a) X Br cwpump heat 361K T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='=T(h) CsPbBr 403K H NLI h c-Si (100) metasurfaceSi TiNCsPbBr (d) 650 (b) 600 a β FDTD/FEM Y 550 Temperature power-lawfit CsPbBr 500 450 400 200nm 350 TiN/Sivoxels 300 1000 500 0 500 1000 z(nm)4 W m-1K-1) and strong Raman response (Si-Si 521 cm- 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Moreover, its Raman activity is temperature sensitive, permitting its use as a probe for Raman- based thermometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The light-to-heat conversion is expected to be maximum at the plasmonic absorption resonance of the TiN structure, as characterized by the absorption power 0 abs P I \uf03d \uf073 , where abs \uf073 is the absorption cross section and I0 is the incident intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='24,25 The accessible temperature range at a thermal stationary state of the system will depend on several factors, namely the effective thermal conductivity of Si, the pillar’s lateral and axial dimensions, the permittivity \uf065 of the TiN pad and the incident flux I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The pillar geometry, defined by base lateral size L and height h, governs the heat dissipation efficiency and its effect has been discussed previously for composite TiN/Si rods,29–31 tubes and trenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='32 Taking into account the Fröhlich resonance condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' we can derive the temperature change at the top of the Si pillar as a function of structure height h and incident light intensity 0I as follows29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='31: 𝜀��� � �𝜆�� � �2𝜀Si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 𝛥𝑇��ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 𝐼�� � � � ���� ��� 𝜀��� �� � � �Si 𝐼� � �abs �Si � ��Si �� ℎ𝐼� ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (1) where \uf06c0\uf020is the wavelength at the plasmonic resonance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' \uf065TiN=\uf065’TiN+\uf020\uf065”TiN is the complex permittivity of a TiN heater,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Q=-\uf065’TiN/\uf020\uf065”TiN is a Q- factor for the plasmon resonance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' \uf06bSi is the temperature-dependent thermal conductivity of bulk Si and \uf062 is the geometry-dependent dimensionless thermal capacity of TiN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='25 For smaller pillar heights (<200 nm), the first term dominates and L T \uf044 is expected to show a linear dependence on 0I (Figure S2, see Supplementary Information) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For taller pillars, the contribution of the second term increases accordingly, resulting in a quadratic dependence of the temperature on 0I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Moreover, L T \uf044 is now dependent on the first temperature derivative of 𝜅Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For bulk Si, this derivative has a negative sign above room temperature, and, thus, L T \uf044 should monotonically increase with the incident intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For structures with height exceeding h>500 nm, significant deviation from experimental observations have been reported and explained in terms of thermal anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='32 Because the thermal conductivity of Si ( 1 1 Si 148 W m K \uf02d \uf02d \uf06b \uf03d ) significantly exceeds that both of air ( 1 1 air =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0263 W m K \uf02d \uf02d \uf06b ) and CsPbBr3 perovskite ( 3 1 1 CsPbBr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='42 W m K \uf02d \uf02d \uf06b \uf03d ), the pillar structure becomes the dominant channel for heat dissipation with its geometry being the key factor in determining the steady state temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hence, pillars of a specific height provide access to specific temperature ranges, while fine control within this range can be realized by varying the incident light intensity I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' When irradiated, an array of such nano- heaters can generate a two-dimensional temperature pattern formed by sub-wavelength hot spots (L<\uf06c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Induced thermal gradients along the axial direction in the perovskite allow particular phase domains to be formed, subject to the distance from the heating TiN pad (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 1d shows a combined finite- difference time-domain (FTDT) and finite element (FEM) method (ANSYS/Lumerical) simulation of the axial temperature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The simulation reveals the axial heat distribution within the layered system, comprised of a 1 \uf06dm Si pillar, a 50 nm TiN pad and a 1 \uf06dm CsPbBr3 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A pillar of this height is associated with a steady state temperature range of 320-520 K or a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='23 K/nm thermal gradient within the Si material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The maximum temperature at the plasmonic structure is chosen to be 630 K, a critical temperature point beyond which the CsPbBr3 optoelectronic properties drastically change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='33 The temperature gradient in the perovskite interior follows a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='54 | |z \uf02d dependence (R2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='997), which is mainly determined by crystal thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In these simulations, the surrounding medium is assumed to be air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Depending on the initial nano-pad temperature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' input light flux), the crystal interior can be comprised of a single \uf067 phase or a structure of two or all three phases, as shown for T0=630 K in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Confocal reflection images of a CsPbBr3 crystal using polarized light at (a) 303 K (orthorhombic phase), (b) 393 K (tetragonal phase), (c) 408 K (cubic phase) and back (d) 393 K (tetragonal phase), (e) 303 K (orthorhombic phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Optical visualization of phase transitions and crystal twinning The real time dynamics of domain formation and twinning in perovskite crystal on a microscale is shown in Supplementary Movie SM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In this experiment, the crystal was placed on a hot plate to perform temperature sweeps from 340 to 410 K and back, spanning the orthorhombic-to-tetragonal-to- cubic phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The heating and cooling rates were sufficiently slow to allow a uniform temperature to establish itself throughout the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Optical imaging and other spectroscopic experiments were performed with the aid of a sample piezo positioning feedback system, as described in Section 3 of Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This solution overcomes experimental obstacles such as the thermal expansion of the sample and setup elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It also corrects for beam defocusing by the Bragg-like grating formed through crystal twinning within the sample volume (see Methods and Supplementary Information, Figure S3 and Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 2 depicts confocal reflection images of the crystal surface at the selected steady state temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' At 303 K (Figure 2a), the crystal consists of parallel domains of the \uf067 phase that are oriented at a 45o angle (<110>) relative to the lab frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The transition to the tetragonal phase occurs around 393 K (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As the temperature is increased to 408 K, the stripes disappear completely, indicating the formation of a homogeneous cubic crystal (\uf061 phase, Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As the system cools down and crosses the cubic-to-tetragonal transition, crystal twinning triggers the formation of multiple tetragonal domains (Figure 2d) and a further temperature decrease brings the crystal back into the orthorhombic phase (Figure 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The data shows clear differences between the images of the crystal at the same temperature points, but opposite ends of the temperature cycle (Figures 2a and 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This difference in the patterns further confirms the results of the FSC experiments (Figure S2), indicating that the potential energy barriers are different when the phase transition proceeds along different directions of the temperature sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The symmetries of the original and final crystallographic phases associated with a transition are the key factors in the evolution of the crystal twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Upon careful examination, it is clear that the resultant orthorhombic phase reveals a 70 deviation angle relative to the previously orthogonal orientation of the domains in the tetragonal phase (Figure 2d and 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This is in excellent agreement with previous calculations by density functional theory (DFT) that yielded a \uf066~130 octahedral tilt for orthorhombic =303K (a) T=393K (b) heating 408K (c)1μm CsPbBr3 1um β-CsPbBr3 T=303K (e) T=393K (d)cooling 0-CsPbBra 83° 90° LLm 1um y-CsPbBr3 β-CsPbBr36 CsPbBr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='18 The rotation of the corner-sharing Br atoms of the [PbBr6]4- octahedron in the equatorial plane by \uf066\uf02f\uf032\uf020~ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='50 should result in a relative re- orientation of the domains to 900-\uf066\uf02f\uf032\uf020~ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='50, as demonstrated in Figure 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Temperature dependence of Raman and photoluminescence signatures Both the FSC and the optical imaging experiments reveal information about the phase transitions in the perovskite crystal, and both measurements point to the importance of lattice imperfections and distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' To examine their role at the microscopic level, we performed Raman and photoluminescence experiments, which are particularly sensitive to the electronic structure near defects and phase interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The Raman spectrum of CsPbBr3 perovskite features two main low energy vibrational modes, namely the 127 cm-1 TO (first-order transverse optical) and the 312 cm-1 2LO (second-order longitudinal optical) Pb-Br stretching phonon modes (Figure 3a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='34 It is important to note that the presence of the 312 cm-1 peak in the Raman spectrum is evidence for the more pristine CsPbBr3 structure relative to the presence of CsPb2Br5, with the latter being the result of exposure to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='35 The temperature dependence of the Raman spectra for both the TO and 2LO modes are different for various directions of the temperature sweep (Figures 3e and 3f) of a uniformly heated crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As the temperature is increased, the intensity of the TO phonon line (127 cm-1) undergoes two extrema corresponding to the orthorhombic-to-tetragonal (361 K) and tetragonal-to-cubic (403 K) phase transitions (Figure 3a and 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The \uf062-CsPbBr3 phase reveals an expected trend, namely the decrease of the Stokes intensity with temperature, caused by the bandgap widening and the depletion of carriers in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='18,19 Meanwhile, the temperature dependence of the Stokes bands ascribed to the \uf061 and \uf067 phases shows the opposite trend (Figure 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This observation can be explained by the interplay between the thermal volumetric expansion and the tilt of the [PbBr6]4- octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='18 It has been predicted that both mechanisms are capable of significant widening of the bandgap19, estimated to be <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 eV for \uf062-CsPbBr3 versus ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='36 eV for \uf067-CsPbBr3 and ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4 eV for \uf061- CsPbBr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' These bandgap variations offer possible explanations for the observed positive temperature trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For example, for the given experiments the Raman process in \uf062-phase is closer to the resonance for the used excitation photon energy (633 nm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='96 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This may lead to a signal increase when more \uf062- phase sites are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Another potential mechanism derives from the contribution of the shallow and deep states to the free carriers population at the conduction band is expected to increase with temperature,36 enabling to change the Raman polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='37 When the temperature is lowered, the Raman intensity of the TO mode decreases continuously and does not exhibit any extrema in this temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We speculate that such behavior can be understood from the dominant role of the crystallographic deformation of the [PbBr6]4- backbones while cooling down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Spatially resolved Raman intensity maps for each phonon mode at different temperatures are presented in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The images agree well with the confocal reflection images (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' They demonstrate a clear variation in the domain pattern as a function of the directionality of the temperature sweep across particular phase transitions - a result of the unequal potential energy barriers of the high-to-low and low-to-high symmetry conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The difference in crystal twinning also impacts the temperature trend of the TO and 2LO phonon lines, resulting in their characteristically different behavior (Figure 3e and 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For the TO phonon mode, electron-phonon scattering at the \uf047 point is more sensitive to twin domain formation due to the overall momentum restrictions for the one-phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This is opposite for the 2LO mode, for which there is a simpler path to fulfill momentum conservation due to the involvement of two phonons to scatter light inelastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' While the 2LO line clearly shows the \uf067→\uf062 transition (red curve, Figure 3f), at the same time it appears insensitive to the \uf062→\uf061 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The cooling curve exhibits similar behavior for both transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Whereas the multi-phonon mode can be utilized as a temperature probe for a defect-free crystal, the single-phonon TO mode is more sensitive to the orthorhombic-to-tetragonal and tetragonal-to- cubic phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Temperature-dependent Raman spectroscopy of a CsPbBr3 crystal at thermal equilibrium for TO phonon mode at 127 cm-1 (a, c) and LO two-phonon mode at 312 cm-1 (b, d) upon heating and cooling at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4 K/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (e,f) Cross-sections at peak center for TO and 2LO modes (dashed lines in (a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Photoluminescence (PL) microspectroscopy provides additional information on the carrier dynamics and the origin of the emission mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The latter has been investigated through power dependence and fluorescence lifetime studies and is discussed in detail in Supplementary Information, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Here, we focus primarily on the temperature trends of perovskite photoemission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' When the temperature is raised, the PL intensity drops dramatically (Figure 4a), reaching minimum at T\uf067→\uf062 at 361 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Further increase of the lattice temperature gives rise to a higher PL intensity for the tetragonal (\uf062) and cubic (\uf061) phases (red curve, Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The overall PL spectral shape reveals complex behavior through the sweep, showing splitting-like behavior at high temperatures (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' First, a blueshift of the mean of the spectral distribution (~16 meV) is observed (Figure 4d), indicating the bandgap expansion of the cubic phase at 423 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='19,37 Second, a red-shifted signature (~18 meV) is observed, which is suggested to originate from the competition between surface and interior contributions of the crystals (Figure 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='37,38 A radically different trend is observed when cooling is performed, with fluorescence showing a strong local maximum at the \uf067→\uf062\uf020\uf020transition (Figure 4b and Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A similar observation has been reported for methylammonium lead triiodide (CH3NH3PbI3 or MAPbI3), upon cooling from 160 K to 140 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='20 The nature of the PL enhancement across this phase transition can be understood as resulting from the funneling effect,20 when mobile carriers migrate to the “defect-free” low-bandgap tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The observed hysteresis agrees well with the confocal reflection and Raman studies, and can be understood in a similar manner - lattice reconstruction and the dependence of crystal twinning on the sign of the temperature change \uf044T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Since twinning requires the base of one domain to be matched to and shared with the side of another, its probability will strongly depend on the presence of inherent crystal imperfections and the geometry of the original and resultant phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This leads to significant differences in the overall pattern of the multi-domain assembly as a function of the sign of the temperature trend, and, in turn, the number of structural defects and phase interfaces being formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This phenomenon also explains the striking contrast in the PL quantum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It suggests that the presence of point heatingat0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4K/s 420 (a) 420 (b) Pb-Br mode at 127 cm Pb-Brmode at 312 cml β-& 400 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 (e) heating (f) Y-β 360 β heating per 340 30 888 320 320 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='2 ounts, 100150200250 260 310 360 Ramanshift(cm) Ramanshift(cm") coolingat0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4K/s420 (c) 0zt (d) 26 β-α 400 β-α 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4 380 cooling cooling 360 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 340 300330360390420 300330360390420 320 320 Temperature (K) Temperature (K) 100150200 250 260 310 360 Ramanshift(cm) Ramanshift(cm8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (a,b) False color PL maps for different sign of temperature sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (c) Cross-sections along the vertical dashed straight lines at the center of PL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (d) PL spectra for different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' defects and crystal twinning favor the \uf062→\uf067\uf020\uf020transition and hinder PL for the reverse direction of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Optical properties of multiple phase single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' After the optical characterization of pervoskite platelets held at a uniform temperature, using reflection, Raman and photoluminescene microspectroscopy, we next used these optical tools to study crystals subjected to a temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For this purpose, we employed the metasurface heating device discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1 to maintain stable temperature gradients in the crystal and control the distribution of phase domains in the axial (z) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 5a shows a scanning electron microscopy (SEM) image of a CsPbBr3 microplate placed on the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figures 5b1 to 5b3 (cyan, yellow and magenta) depict 72o tilted images of the corners marked with an arrow of the corresponding color (Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As is clear from the images, the structure is formed by two stacked crystal plates, most clearly observed through their exfoliation at one of the corners (Figure 5b1 and Supplementary Figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The metasurface is comprised of a 2D hexagonal array of TiN/Si voxels with a pillar height estimated to be approximately 900 nm as shown in Supplementary Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The top of the voxel is visualized in the inset of Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It is clear that, upon illumination, the TiN pads become damaged for intensities exceeding 3 MW/cm2 (green and blue contoured images in the inset of Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The confocal reflection image at 633 nm of the perovskite platelet is shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In this image, the uncovered TiN/Si voxels have been placed at the focal plane of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For such an arrangement, the voxels that are covered by the perovskite appear out of focus as light has to penetrate through the 1 \uf06dm-thick material of refractive index n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='37 This effect not only prevents efficient heating of the TiN pads, but also limits the efficient collection of the Raman signal from the voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The collection efficiency is instrumental, as the Raman response was utilized as a remote temperature probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In all further Energy(eV) cooling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='30 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 d (b) T=303K 410- T=358K (d) 403K T=403K 3 390 sity, x10 T=423K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0元 B sity 370 361K blueshift redshiftinten 350 intel 2 E 330- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 e 310 510515520525530535540 500 510 520 530 540 550 Wavelength(nm) Wavelength（nm)heating 5 a (a) heating 410 C 403K cooling 390 nsity,x1 e B 361K 3Q350 inte B++α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='330- P 310 510515520525530535540 300320340360380400420 Wavelength(nm) Temperature(K)9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' CsPbBr3 platelet on the thermoplasmonic TiN/Si metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (a) SEM image of the CsPbBr3 plate over the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The insets show TiN/Si voxels, marked with the red, green and blue squares, exposed to 633 nm cw illumination with the intensity of 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 MW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (b1)-(b3) SEM images (side views at the tilt angles of 72o (b1) and 48o (b2), (b3) from the sides marked with cyan, yellow and magenta arrows in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (с) A confocal reflection image at 633 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (d) False color PL spectra central frequency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (e) PL spectra taken at spots marked in Figure 5d with red, blue and green filled circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (f, g) Raman maps at 521 cm-1 (c-Si) and 127 cm-1 (Pb-Br mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (h) Raman spectra of Si pillar as a function of input light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The inset shows a cross section along a dashed white line and numerical deconvolution of the composite band into Lorentzian and Gaussian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (i) Temperature map measured based on Raman thermometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (j) Simulated cumulative Raman signal from phase-structured crystal of different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (k) Raman intensity vs the pumping intensity or gradient initial temperature T0 temperature for 127 cm-1 (green) and 312 cm-1 (red) of perovskite phonon modes and 521 cm-1 Si line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' experiments, the light was focused on the top of the TiN/Si voxels that are under the CsPbBr3 microplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 5d displays the results of confocal PL imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' PL spectra as a function of spectral position on the sample were collected using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='7 W cm-2 of 473 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It is important to note that such low fluxes did not introduce any meaningful temperature gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In addition, the excitation wavelength used is far away from the absorption resonance of the plasmonic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The false color PL map can be divided into three characteristic regions according to PL spectral shape and central frequency position (Figure 5e) - blue (522 nm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='375 eV), red (531 nm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='335 eV) and an intermediate green region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We observe a clear correlation between the PL spectrum and the sample thickness and/or stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Higher energy PL, centered around 522 nm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='375 eV), is observed in areas where two thin ~400 nm plates are stacked (blue spot in Figure 5d and Figure 5b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' However, the spectrum is red-shifted by 40 meV at the position where the sample appears to consist of a 1 \uf06dm-thick single plate (red spot in Figure 5d and (a)TiN:Simetasurface OMWI (C) 521cm(c-Si)() metasurface 127cm（Pb-Br）(g) 100nm C-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5MW 00nm CsPbBr,plate CsPbBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='plate 2m 2 μm LIT 100nm AT(K) RamanspectraofSi 20 40 60 80 100 120140 um PLspectralpositionmap540 5 780K (h) Calculatedintensity () (b1) P B60K 4 85 520 2 200nm 400nm 200nm 836K 800nm 600 nm JS 320K 1000nm 2 305K 500 500 510 520 530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 Pumpingintensity(MW/cm） 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 Energy(eV) Ramanshifts(cm) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='25 AT (K) 50 200(e) Ramanshiftbased 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 temperaturemap 600 (k) uim intensity 127 cm(Pb-Br) 312cm(Pb-Br) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 521 cm(c-Si) PL 450 40meV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 500510520530540550560 2m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 um Wavelength(nm) 300 Pumpingintensity(MW/cm210 Figure 5b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It has been suggested that the observed phenomenon is caused by the excitation of waveguide modes within the Fabry-Perot resonator through the absorption-emission-absorption mechanism 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' If true, monitoring of the PL spectral position offers a means to probe the distribution of the perovskite thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figures 5f and 5g show confocal Raman maps for the 521 cm-1 (c-Si peak of the pillar) and 127 cm-1 (TO Pb-Br phonon mode of CsPbBr3) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' It is evident that not all voxels under the platelet can be clearly differentiated in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This is caused by the damage while positioning the perovskite on the metasurface and/or by the poor contact at certain positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The enhanced Raman scattering of the TO mode at the crystal edges originates from structural inhomogeneities, where the density of surface states is higher (Figure 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For quantitative monitoring and visualization of the temperature at the voxel, we used Raman thermometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This method, thoroughly described elsewhere29–31 (see Supplementary Information, Section 8), utilizes the temperature dependent behavior of the c-Si Raman signal (521 cm- 1) as a remote probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Through the use of an Echelle grating, the spectral resolution of the imaging system reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1 cm-1 and enables temperature measurements with 5 K accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A detailed analysis of the open voxel temperature (blue voxel, Figure 5a) versus input flux is shown in Figure 5h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Note that the c-Si mode is asymmetrically broadened (inset, Figure 5h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This effect originates from the non-uniform heat distribution in the structure, resulting in the presence of contributions from both hot and cold portions of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='31 To further simplify the analysis, the spectrum was fitted with Lorentzian (hot medium contribution) and Gaussian (cold medium contribution) spectral line shapes, using a regularized least squares method (R2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The intensity map in Figure 5i clearly indicates that the contribution from hot domains deviates significantly from a linear incident intensity dependence for intensities exceeding 4 MW cm-2 (550 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We attribute this effect to temperature dependent changes in the TiN permittivity, which affects the plasmon resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In addition, the thermal conductivity of Si decreases when the temperature is raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='39 The signs of degradation of TiN pad appear at 750 K, where the Raman intensity peaks at about 5 MW cm-2 and then shifts back to the higher energy side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This is also confirmed by previous experiments using ellipsometry on TiN films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='29 Thus, in our experiments, the incident light flux enabled access to the 293 K to 473 K temperature range – sufficient to activate all necessary structural transitions in CsPbBr3 while preventing photo-damage of the plasmonic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 5h shows the resulting temperature map derived from the Raman shift using Equation S1 (see Figure S9) and measured at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 MW cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The local generation of hot spots produced thermal gradients throughout the perovskite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This effect resulted in the simultaneous formation of multiple phase domains, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' However, there are significant and fundamental differences between the temperature trends of the Raman signals when (1) heating of the whole crystal by a hot plate to achieve a uniform temperature profile, as opposed to (2) establishing a temperature gradient in the crystal with the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For the first case, the upward temperature trend discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='3 is shown by the red curve in Figures 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This profile shows clear extrema at phase transitions with an overall signal intensity decrease across the tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In the second case, when the crystal is locally heated by the plasmonic structures, the temperature gradient induces multiple phases in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For this case, the Raman response is the cumulative signal from all the phases in the collection volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The trend of the cumulative Raman signal RI versus the plasmonic pad temperature (0) m T should directly reflect the process of multi-phase structuring of the perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The trend can be modeled as discussed in Supplementary Information, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For simplicity, one can assume one-dimensional heat dissipation in a homogeneous perovskite crystal, in which the temperature profile obeys a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='54 ( ) (0) | | m m T z T z \uf02d \uf03d power law in the axial direction (Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The resulting Raman response can then written be as: 0 ( ) z RI I z dz \uf044 \uf03d \uf0f2 (2), where ( ) I z is the average Raman signal of the homogeneous media at a given z-plane (Figure S10) 11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (a) Linearly corrected Raman response versus the incident light flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' (b, c, d) represents simulated temperature points where single, double and triple phase structure occur as seen in Raman signatures from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The blue dashed line represents the subtracted linear contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' and z \uf044 is the total crystal thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 5j shows plots of IR vs I0 for different z \uf044 of perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As expected, for very thin crystals (<200 nm), the temperature trend of Raman signal closely follows the one previously observed for a thermally equilibrated crystal on a hot plate (green curve Figure 5j, red curve Figure 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For thicker crystals, multiple phases can contribute to the observed Raman signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The local maximum remains highly pronounced over the monotonically increasing Raman response, indicating the formation and growth of a two-phase structure (\uf067 and \uf062) in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This model agrees well with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Figure 5k shows the intensity evolution of the phonon modes (TO 127 cm-1, 2LO 312 cm-1) along with the c-Si line (521 cm-1) as a function of the incident light flux/pad temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As expected, the trends for the 127 cm-1 (Pb-Br) and 312 cm-1 (Pb-Br) modes are inherently different from the case of the thermally equilibrated system (red and green curves, Figures 5k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' For the TO mode, a clear presence of local maximum around the temperature of \uf067-to-\uf062 transition is observed, indicating the formation of a two-phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Upon further increase of the pad temperature (0) m T , another shallow bump at \uf044T ~ 140 K indicates triple phase formation (\uf061, \uf062 and \uf067).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' To visually emphasize these signatures, the linear contribution to the Raman-temperature trend has been subtracted in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The linear contribution has been determined from a simple linear fit over 0 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='5 MW cm-2 range (Figure 6a point b, Figure 6b), where the whole perovskite crystal remains in the single orthogonal \uf067 phase and the intensity of the TO Raman peak should linearly increase with the incident excitation flux and temperature (Figure 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Upon increasing the incident intensity, the formation of the tetragonal phase at the interface of the perovskite and △T (K) Raman enhancement 0 50 100 150 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='6 (a) Y,β α,β, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='4 b c d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='0 Q 0 1 2 3 4 5 Pumping intensity (MW/cm TiN CsPbBr TiN CsPbBrs TiN CsPbBr 550 550 550 (b) (c) (d) Temperature (K) 500 500 500 450 450 450 β,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' α,β, 400 400 400 350 350 350 300 300 300 0 500 1000 0 500 1000 0 500 1000 z (nm) z (nm) z (nm)12 TiN is expected to occur at point c (Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' At this point of the trend, the steep temperature gradient creates a spatially sharp defect area where the crystal is in a transitional form between the orthorhombic and tetragonal phase (Figure 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This scenario manifests itself as a shallow Raman intensity maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Further increase of the plasmonic pad temperature drives the \uf062-phase deeper into the crystal bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' At \uf044T~130 K another shallow maximum of the TO Raman peak indicates the formation of the \uf061-phase in close proximity to the TiN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Upon subsequent increase of the incident light flux, the \uf061 and \uf062 phases extend further into the crystal and significantly broaden (Figure 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This is expected to result in the smearing of the boundaries between the different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This effect is spatially asymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' different for the left (\uf061-\uf062) and the right (\uf062-\uf067) sides of \uf062 phase, following the highly nonlinear temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' At higher temperatures, the signal is an interplay of several contributions, in particular the phase layer thickness, the temperature, position along the gradient, and the sharpness of the phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We hypothesize the third maximum at \uf044T = 170 K is the result of such a cumulative effect and may be associated with the delocalized (disordered or randomly located) phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Among other instrumental contributors to the spatial phase formation are the crystal intrinsic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Their presence can trigger the spontaneous formation of different phases, resulting in a highly irregular structural front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' At higher TiN temperatures, the \uf061-\uf062 and \uf062-\uf067 boundaries broaden significantly and may capture more defects into these areas where the phases are highly mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' These experiments demonstrate that the Raman signal from perovskite subjected to a stable temperature gradient, as shown in Figure 5k and emphasized in Figure 6, reveals distinct behavior that is in contrast to a bulk crystal held at uniform temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The dependence of the Raman signal on the incident intensity shows clear signatures of particular phase formations, their extension into the bulk of the material, and, overall, the multi-phase structuring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Moreover, it shows that the detected Raman signal exhibits notable gain across \uf044T=150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Using Raman microspectroscopy as a probe, these results indicate that it is possible to generate on demand a distribution of single, double and triple phase structures in perovskite by simply controlling the incident light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Conclusions In this work, we have demonstrated the proof-of- principle multiphase structured single crystal CsPbBr3 halide perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' We have shown, the single, double and triple phase systems can be created in optically controlled fashion on the thermoplasmonic metasurface using the continuous wave illumination of modest intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Light-induced heat from plasmonic TiN nanopads forms strong temperature gradients within crystal bulk that are followed by sequence of corresponding phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Lattice distortions, defects and impurities operate like an optical nanoantenna, increasing the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Thus, multi-phase perovskite structures hold many interesting properties and open exciting possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' In such system, charge carriers migrate from lower symmetry lattice with large bandgap (orthorhombic and cubic) to higher symmetry, but lower bandgap crystal parts (tetragonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' There, highly concentrated and in close proximity to the boundaries, carriers efficiently recombine, leading to areas with significant enhancement of the optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' This multi-structured system promises to be highly beneficial to the development of next-generation ultracompact broadband light-emitting diodes showing high PL quantum yields above room temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Methods Synthesis of CsPbBr3 structures Perovskite microcrystals on glass substrates were synthesized by using a protocol similar to the previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='40 PbBr2 (110 mg) and CsBr (62 mg) were mixed and dissolved in 3 ml of anhydrous dimethyl sulfoxide (DMSO) inside a nitrogen-filled glovebox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Droplet of the prepared solution (volume 2 µl) was drop-casted on the substrate at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' After that, the substrate was sealed in a preheated up to 60 oC Petri dish containing 200 µl of liquid mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The solution was dried in the presence of azeotropic vapor for 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' As a result, the randomly oriented separate CsPbBr3 micro- crystals were formed on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 13 Synthesis, nanofabrication and characterization of a TiN/Si metasurface TiN thin films on c-Si (100) substrates were DC magnetron sputtered from a Ti target in the Ar/N2 environment with a volume proportion of 30:70 at elevated temperature of 350 oC and base pressure of 9 3 10\uf02d \uf0d7 mbar and power of 200 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Prior to the film growth, the c-Si substrate was sonicated in acetone for 15 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The thickness of the TiN films, equal to 50 5 \uf0b1 nm, was measured with a contact profilometer Alpha Step 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A 2D array of TiN/Si voxels were engraved with the help of focused ion beam (FIB) milling at a lower current of 1 pA by using Quanta 3D FEG (FEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Since the higher TiN/Si voxels are exposed to FIB for a longer time, their lateral size is reduced due to edge melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' To avoid this detrimental effect, we used different mask templates for short and long voxels so that their lateral size is the same regardless of height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The temperature-dependent permittivity of TiN thin films were measured with a spectroscopic ellipsometer (VASE, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Woollam) within the spectral range of 250-2500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The incident angle was 70°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The TiN sample was exposed to thermal annealing at the fixed temperature, whereas its permittivity was probed at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The temperature increment for each subsequent cycle was 100°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The temperature ranged from 25 oC to 600 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The samples were annealed at ambient air for 30 min using a heating stage (Linkam Scientific Model THMS600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The heating and cooling rates were 150 °C/min and 100 °C/min, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Fast Scanning Calorimetry The Fast Scanning Calorimetry (FSC) curves were registered on FlashDSC2+ (Mettler-Toledo, Greifensee, Switzerland) equipped with TC100MT intracooler with UFH1 sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The temperature calibration was performed using biphenyl ( o 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='2 C m T \uf03d ) and benzoic acid ( o 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='3 C m T \uf03d ) as standards to ±1 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A single perovskite crystal was placed in the center of the calorimetric sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' To improve signal- to-noise ratio the crystal size was chosen such as to almost match the active area of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Within the temperature range from 20 °C to 180 °C the sample was chemically stable, and the curves were repeatable, which allowed for averaging multiple scans to further improve signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Atomic force microscopy The multimode scanning probe microscope NTEGRA PRIMA (NT-MDT) was utilized for visualizing a topography of the CsPbBr3 microplate surface and the thermoplasmonic metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The AFM probes of the “VIT_P” series with resonant frequencies around 350 kHz were used in AFM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The CsPbBr3 microplate mounted on the metasurface fabricated by focused ion beam milling was measured in tapping mode with a free amplitude 0 A of 10-20 nm and a set-point value of 0 2 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Far- and near-field Raman spectroscopy and microscopy Raman spectra and maps were captured with a multi-purpose analytical instrument NTEGRA SPECTRA™ (NT-MDT) in inverted configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The confocal spectrometer was wavelength calibrated with a crystalline silicon (100) wafer by registering the first-order Raman band at 521 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A sensitivity of the spectrometer was as high as ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 3000 photon counts per 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1 s provided that we used a 100× objective (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='7), an exit slit (pinhole) of 100 \uf06dm and a linearly polarized light with the wavelength of 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='8 nm and the power at the sample of 16 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' No signal amplification regimes of a Newton EMCCD camera (ANDOR) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 128x128 pixel Raman maps were raster scanned with an exposure time per pixel of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1 s and were finally collected with the EMCCD camera cooled down to -95oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Raman spectra within the range of from -2000 to 2000 cm-1 were registered with a spectral resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='1 cm-1 using the Echelle grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Fluorescence Lifetime Imaging Microscopy To measure the PL decay time we used a system built-in the confocal optical spectrometer (NTEGRA SPECTRA) that includes a picosecond diode laser (BDL-SMN) generatig pulses of 473 nm wavelength, 30 ps pulse duration, and 80, 50, or 20 MHz repetition rate, a Simple-Tau 150 TCSPC FLIM module (Becker&Hickl), and a HPM-100-40 GaAsP hybrid 14 detector (Becker&Hickl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The detector has a detection efficiency of about 50% and is free of afterpulsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' FDTD/FEM calculation 3D simulation of optical absorption of a TiN/Si voxels consisting of stacked TiN and Si cylinders under cw illumination was performed by using an Ansys/Lumerical FDTD solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The height of the TiN pad was 50 nm, whereas the height of the Si pillar 900 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' To avoid anomalous electric fields near the TiN pad edge we used disks with rounded edges (10 nm rounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A mesh overlayer of 1 nm was utilized around the TiN pad and a rougher 10 nm mesh for the rest of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Perfectly matching layers were used as boundary conditions for three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The optical and thermal properties of Si and air were imported from the Ansys/Lumerical material database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The TiN pad was exposed to a 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='8 nm focused laser light (NA= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='7) with the intensity of 5 MW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The temperature profile was calculated through an Ansys/Lumerical FEM solver in the steady state regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The thermal conductivity of all constituents is assumed to be temperature- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The boundary condition of 300 K T \uf03d was set at the min 20000 nm z \uf03d \uf02d of the 20×20×5 \uf06dm3 simulation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Conflicts of interests There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Acknowledgement This work was supported by grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 19-12-00066- P of the RSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The PL decay time measurements were granted by the Kazan Federal University Strategic Academic Leadership Program (PRIORITY-2030).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' The authors acknowledge a technical support from our industrial partners: SCANSENS (GmbH, Germany) and NT-MDT BV (The Netherlands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Dey, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Feldmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hofkens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Luther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Pérez-Prieto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Manna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bodnarchuk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kovalenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Roeffaers, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Pradhan, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mohammed, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bakr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Yang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Müller-Buschbaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kamat, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Krahne, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Galian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Stranks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bals, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Biju, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Tisdale, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Yan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hoye, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Polavarapu, ACS Nano 2021, 15, 10775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Ahmadi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hu, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2017, 29, 1605254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Xiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Shan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zeng, ACS Nano 2021, 15, 17150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kovalenko, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Protesescu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bodnarchuk, Science 2017, 358, 745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Sun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Duan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Jin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kildishev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Qiu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Xiao, ACS Nano 2018, 12, 8847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Grätzel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mhaisalkar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Sum, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2014, 13, 476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Marçal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Benter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Irish, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Dzhigaev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Oksenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Rothman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Sanders, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hammarberg, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Sala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Björling, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Unger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mikkelsen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Joselevich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Timm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wallentin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2021, 5, L063001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zha, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Jie, Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Growth Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2020, 20, 4585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Rothmann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhu, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bach, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Spiccia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Etheridge, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Cheng, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2017, 8, 14547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Röhm, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Leonhard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Schulz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wagner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hoffmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Colsmann, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2019, 31, 1806661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Xiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Fang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Shao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Dai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Huang, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2020, 11, 2215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Bharadwaj, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Deutsch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Novotny, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Photonics 2009, 1, 438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Alaei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Circelli, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zeng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2017, 8, 3752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mannino, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Deretzis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Smecca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' La Magna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Alberti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Ceratti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Cahen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2020, 11, 2490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Ghaithan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Alahmed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Qaid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Hezam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Aldwayyan, ACS Omega 2020, 5, 7468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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Baffou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Cichos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Quidant, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 2020, 19, 946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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Kivshar, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Photonics 2021, 13, 643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kharintsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kharitonov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Chernykh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Alekseev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Filippov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Kazarian, Nanoscale 2022, 14, 12117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Danielson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 1963, 130, 1743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' P.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Sannikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zasedatelev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Lagoudakis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Zakhidov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Makarov, ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} +page_content=' Interfaces 2019, 11, 1040' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tE4T4oBgHgl3EQf4g2G/content/2301.05314v1.pdf'} diff --git a/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf b/aNAzT4oBgHgl3EQf2f7B/content/2301.01816v1.pdf new file mode 100644 index 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Vallejo-Pe˜na2 +1Theoretical Physics Department, CERN, +CH-1211 Geneva 23, Switzerland and +2ICRANet, Piazza della Repubblica 10, I–65122 Pescara, Italy +(Dated: January 27, 2023) +Abstract +The interaction between gravitational waves (GW) and electromagnetic waves (EMW) produces +quadrirefringence, a phenomenon consisting in a frequency and polarization dependency of the speed +of GW and EMW. Quadrirefringence can be due to the GW-EMW interaction in the source or during +the propagation from the source to the observer. +In the first case the astrophysical properties of the source can induce a unique characteristic imprint +on GW and EMW for each source, the binaries rainbow, which could be observed in systems with +EM counterparts, such as neutron stars or neutron star black hole binaries. Quadrirefringence could +be used for example to investigate the equation of state or neutron starts, while its effect on the +propagation from the source could be used to probe the large scale electromagnetic field using GW. +The interaction of the graviton with other fields will also induce similar effects, and will allow to +probe other fields using GW observations. +1 + +I. +INTRODUCTION +The detection of gravitational waves (GW) [1] by the Laser Interferometer Gravitational +Wave Observatory (LIGO) and Virgo has started the era of multi-messenger astronomy. Elec- +tromagnetic counterparts [2] are particularly useful to test the general relativity prediction that +GW and EMW propagate in the vacuum at the same speed [3–7]. In this paper we study the +effects of the interaction between GW and EMW, showing how the phenomenon of quadrirefrin- +gence naturally arises as a direct implication of the covariant formulation of electromagnetism +in curved space. +Quadrirefringence modifies the propagation of different polarizations of the GW and EMW, +making the effective propagation speed of each polarization, time and frequency dependent. +When quadrirefringence happens in the source, it leaves a characteristic imprint on GW and +EMW, which can be different for each binary system leading to the rainbow of binaries. +On cosmological scales quadrirefringence can instead be used to probe the large scale elec- +tromagnetic field. +II. +GRAVITOELECTROMAGNETISM LAGRANGIAN +The minimal coupling of gravity to electromagnetism is given by the Lagrangian density +L = LG + LEM = √−g +� +R + 1 +4FµνF µν� += √−g +� +R + 1 +4gµνgρσF µρF νσ� +, +(1) +where the Faraday’s tensor is defined as Fµν = ∂µAν − ∂νAµ in terms of the vector potential +Aµ, R is the Ricci scalar, and g is the determinant of the metric, and we are using units in +which 8πG = c = ǫ0 = 1, where G, c, ǫ0 are the Newton’s gravitational constant, the vacuum +speed of light and the vacuum permittivity. In the following we will treat both GW and EMW +as perturbations. +For a gravitational wave propagating along the x-axis, the leading order interaction La- +grangian density in the Einstein’s gauge is given by the cubic terms +LEM = 1 +2(F 02F 02−F 03F 03−F 12F 12+F 13F 13)h++(F 02F 03−F 12F 13)h× = Π+h++Π×h× , (2) +where the perturbed metric relevant for the computation of the effects of the graviton photon +2 + +interaction is +gµν = + + + + + + + +1 +0 +0 +0 +0 −1 +0 +0 +0 +0 +−1 + h+ +h× +0 +0 +h× +−1 − h+ + + + + + + + +. +(3) +From eq.(1) it is evident that the different polarizations of the GW and EMW are coupled +in different ways to each other, which is the origin of the quadrirefringence. +III. +GW PROPAGATION EQUATION +In the Einstein’s gauge the Lagrangian density for GW, including the leading order minimal +coupling to electromagnetism, is +Lh = h′2 +A − (∂hA)2 + LEM , +(4) +which gives the equations of motion +h′′ +A − ∇2hA = ΠA , +(5) +where ΠA was given in eq.(1). Not that the ΠA are different for each polarization mode, due +to the structure of the electromagnetic stress-energy-momentum tensor, which is typical of a +spin 1 gauge field with. Similar results are expected when considering other gauge fields, such +as the axion. In the Lagrangian approach ΠA is simply arising from the coupling ΠAhA. +IV. +EMW PROPAGATION EQUATION +The equations of motion for the EMW can be derived by varying the action with respect to +Aµ, giving the covariant form of the Maxwell’s equations +∇µF µν = +1 +√−g∂µ(√−gF µν) = 0 , +(6) +where the determinant of the perturbed metric is +√−g = (1 − h2 ++ − h2 +×)1/2 ≈ 1 − 1 +2(h2 ++ + h2 +×) . +(7) +This metric determinant is inducing the graviton photon coupling in the Maxwell’s equations, +which take the form +∂µF µν = Jν +eff = −∂µ +√−g +√−g F µν , +(8) +3 + +where we are denoting the effective gravitoelectromagnetic current as Jν +eff. The gravitoelec- +tromagnetic current arises naturally as a consequence of the photon graviton interaction. Note +that the effective current is due to an effective non minimal coupling of Aµ, i.e. it is not simply +obtained from a AµJµ term in the Lagrangian, since the current itself depends on the Faraday’s +tensor. +Imposing the Lorentz’s gauge condition ∂µAµ = 0, and taking the leading order interaction +terms in the determinant of the metric, we get the wave equation +□Aν = Jν +eff = (h+∂µh+ + h×∂µh×)F µν . +(9) +V. +QUADRIREFRINGENCE +The system of coupled differential equations +□hA = ΠA(h, Aρ) , +(10) +□Aν = Jν +eff(h, Aρ) , +(11) +describes the evolution and interaction of GW and EMW, and their interaction is associated +to the source terms which couple the two equations. In the above equations we have used an +abstract notation to make explicit the dependence on h and Aρ of the source terms given in +eq.(5) and eq.(9), to underline the coupling between the two equations. Quadrirefringence is +due to the fact that the source can be different for different polarizations of the GW or different +components of Aν. The effects of the polarization of EMW can be obtained by expressing the +electric and magnetic fields in terms of Aν using +E = −∇φ − ∂tA , +(12) +B = ∇ × A , +(13) +where Aν = (φ, A). +Since the GW propagating in the x direction have the form h(η − x) we get +Jν +eff = (h+h′ ++ + h×h′ +×)F 0ν + (h+∂xh+ + h×∂xh×)F 1ν . +(14) +Note that each component of the effective current is different form zero, implying that each +component of Aµ is affected by quadrirefringence. +If the effects of interaction are ignored the EMW and GW will propagate independently, +and the solutions of the wave equations will be of the standard oscillatory form, with each +4 + +polarization mode of the EMW and GW propagating at the speed of light, but if the effects +of interaction are take into account, each polarization of the GW and EMW will propagate +differently, since the source terms are different. +The strength of these effects depends on the size of the sources, i.e. ΠA and Jν +eff, which for +GW is related to the anisotropic part of the stress-energy-momentum tensor of the electromag- +netic field, and for EMW to the amplitude of the GW. +VI. +THE RAINBOW OF BRIGHT AND GREY SIRENS +So far we have not introduced any electromagnetic four current Jν +EM in the Lagrangians, i.e. +we have studied the interaction of GW and EMW in the absence of matter. +In a realistic astrophysical scenario Jν +EM will be definitely present in the source, and will be +associated to the specific dynamic of the astrophysical system producing the EMW and GW, +which could be a neutron stars (bright sirens) or neutron star black hole (grey siren) binary. +In this case the additional Lagrangian density and corresponding equations would take the +form +LJ = √−gAµJµ +EM , +(15) +□hA = ΠA(h, Aρ) , +(16) +□Aν = Jν +eff , (h, Aρ) + √−gJν +EM . +(17) +Due to the coupling between the equations, the current Jν +EM will also affect the GWs emitted +by the the binary. +Given the environmental dependence of Jν +EM(η, xi), i.e. its space and time evolution depend +on the specific dynamics of each astrophysical system, the effects of quadrirefringence can be +different for each binary system. Once more accurate measurements of bright sirens will be +available, it will be possible detect the object specific effects imprinted on GW and EMW, +predicted by the theory of general relativity and covariant electromagnetism. +This will allow to infer new information about the binary system dynamics, by combined +analysis of GW and EMW observations. +5 + +VII. +EFFECTIVE SPEEDS DESCRIPTION +It is possible to describe quadrirefringence using an effective approach [8, 9], in which the +effects of interaction are encoded in an appropriately defined polarization, frequency, and time +dependent effective speed of GW and EMW. While quadrirefringence can be studied without +adopting this effective description, by solving the system of coupled differential equations in +eq.(17), it is convenient to give this effective description, to gain more insight about the physical +implications. We will leave the details to a future work, but we provide below the effective +equations which can be derived +h′′ +A − 2c′ +T,A +cT,A +h′ +A − c2 +T,A∇2hA = 0 , +(18) +A′′ +ν − 2c′ +γ,ν +cγ,ν +Aν − c2 +γ,ν∇2Aν = 0 , +(19) +where cT,A(η, xi) and cγ,ν(η, xi) are respectively the GW and EMW space dependent effec- +tive speeds (SES). In Fourier space a similar set of equations can be derived in terms of the +momentum dependent effective speeds (MES) +˜h′′ +A − 2˜c′ +T,A +˜cT,A +˜h′ +A + ˜c2 +T,Ak2˜hA = 0 , +(20) +˜A′′ +ν − 2˜c′ +γ,ν +˜cγ,ν +˜Aν + k2˜c2 +γ,ν ˜Aν = 0 , +(21) +where a tilde denotes the Fourier modes, except for the MES ˜cT,A(η, k) and ˜cγ,ν(η, k), which +are not defined [8–11] as the Fourier transform of the SES. +Note that the GW-EMW interaction not only changes the effective speed of each polarization, +but it also induces an additional time/frequency/polarization dependent friction term, related +to the time derivative of the MES, which can have important observational implications [8] +for the gravitational and electromagnetic luminosity distances, which we will study in a future +work. +VIII. +CONCLUSIONS +We have shown that the interaction between GW and EMW waves induce the phenomenon +of quadrirefringence, the different propagation of the polarization modes of GW and EMW. +When this phenomenon happens in binary systems like bright or grey sirens it can induce +an object characteristic imprint on the emitted GW and EMW, the binaries rainbow, which +6 + +provides a new method to study the astrophysical properties of these systems, such as for +example the equation of state of neutron stars. On cosmological scales quadrirefringence can +be used to probe the electromagnetic field with gravitational waves. +We have studied the leading order effect of the classical calculation of the GW-EMW inter- +action, but it will also be important to consider quantum effects, and to study the regimes in +which they can provide important correction to the classical effect. +Note that while in this paper we have focused on the GW-EMW interaction effects, similar +effects are predicted by the model independent effective approach [8], and could arise also when +the graviton is coupled to any other field. These interaction effects will allow to use gravitational +waves observations to probe any other field the graviton is coupled to. +Acknowledgments +We thank Tessa Baker, Riccardo Sturani, Rogerio Rosenfeld, Nicola Tamanini, John Ellis, +and Mairi Sakellariadou for interesting discussions. +[1] LIGO Scientific, +Virgo, +B. P. Abbott et al., +Phys. Rev. Lett. 116, +061102 (2016), +arXiv:1602.03837. +[2] LIGO Scientific, +Virgo, +B. P. Abbott et al., +Phys. Rev. Lett. 119, +161101 (2017), +arXiv:1710.05832. +[3] T. Baker et al., Phys. Rev. Lett. 119, 251301 (2017), arXiv:1710.06394. +[4] P. Creminelli and F. Vernizzi, Phys. Rev. Lett. 119, 251302 (2017), arXiv:1710.05877. +[5] J. Sakstein and B. Jain, Phys. Rev. Lett. 119, 251303 (2017), arXiv:1710.05893. +[6] J. M. Ezquiaga and M. Zumalac´arregui, Phys. Rev. Lett. 119, 251304 (2017), arXiv:1710.05901. +[7] H. Wang et al., Astrophys. J. Lett. 851, L18 (2017), arXiv:1710.05805. +[8] A. E. Romano, (2022), arXiv:2211.05760. +[9] A. E. Romano, (2023), arXiv:2301.05679. +[10] A. E. Romano and S. A. Vallejo Pena, Phys. Lett. B 784, 367 (2018), arXiv:1806.01941. +[11] A. E. Romano, S. A. Vallejo-Pe˜na, and K. Turzy´nski, (2020), arXiv:2006.00969. +7 + diff --git a/aNFIT4oBgHgl3EQflStM/content/tmp_files/load_file.txt b/aNFIT4oBgHgl3EQflStM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff0a302efa93edc870e0132a4339058490202485 --- /dev/null +++ b/aNFIT4oBgHgl3EQflStM/content/tmp_files/load_file.txt @@ -0,0 +1,159 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf,len=158 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='11304v1 [gr-qc] 26 Jan 2023 Gravitoelectromagnetic quadrirefringence and the binaries rainbow Antonio Enea Romano 1,2, Sergio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Vallejo-Pe˜na2 1Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland and 2ICRANet, Piazza della Repubblica 10, I–65122 Pescara, Italy (Dated: January 27, 2023) Abstract The interaction between gravitational waves (GW) and electromagnetic waves (EMW) produces quadrirefringence, a phenomenon consisting in a frequency and polarization dependency of the speed of GW and EMW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Quadrirefringence can be due to the GW-EMW interaction in the source or during the propagation from the source to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In the first case the astrophysical properties of the source can induce a unique characteristic imprint on GW and EMW for each source, the binaries rainbow, which could be observed in systems with EM counterparts, such as neutron stars or neutron star black hole binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Quadrirefringence could be used for example to investigate the equation of state or neutron starts, while its effect on the propagation from the source could be used to probe the large scale electromagnetic field using GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' The interaction of the graviton with other fields will also induce similar effects, and will allow to probe other fields using GW observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' INTRODUCTION The detection of gravitational waves (GW) [1] by the Laser Interferometer Gravitational Wave Observatory (LIGO) and Virgo has started the era of multi-messenger astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Elec- tromagnetic counterparts [2] are particularly useful to test the general relativity prediction that GW and EMW propagate in the vacuum at the same speed [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In this paper we study the effects of the interaction between GW and EMW, showing how the phenomenon of quadrirefrin- gence naturally arises as a direct implication of the covariant formulation of electromagnetism in curved space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Quadrirefringence modifies the propagation of different polarizations of the GW and EMW, making the effective propagation speed of each polarization, time and frequency dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' When quadrirefringence happens in the source, it leaves a characteristic imprint on GW and EMW, which can be different for each binary system leading to the rainbow of binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' On cosmological scales quadrirefringence can instead be used to probe the large scale elec- tromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' GRAVITOELECTROMAGNETISM LAGRANGIAN The minimal coupling of gravity to electromagnetism is given by the Lagrangian density L = LG + LEM = √−g � R + 1 4FµνF µν� = √−g � R + 1 4gµνgρσF µρF νσ� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (1) where the Faraday’s tensor is defined as Fµν = ∂µAν − ∂νAµ in terms of the vector potential Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' R is the Ricci scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' and g is the determinant of the metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' and we are using units in which 8πG = c = ǫ0 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' where G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' ǫ0 are the Newton’s gravitational constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' the vacuum speed of light and the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In the following we will treat both GW and EMW as perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' For a gravitational wave propagating along the x-axis, the leading order interaction La- grangian density in the Einstein’s gauge is given by the cubic terms LEM = 1 2(F 02F 02−F 03F 03−F 12F 12+F 13F 13)h++(F 02F 03−F 12F 13)h× = Π+h++Π×h× , (2) where the perturbed metric relevant for the computation of the effects of the graviton photon 2 interaction is gµν = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 −1 0 0 0 0 −1 + h+ h× 0 0 h× −1 − h+ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (3) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (1) it is evident that the different polarizations of the GW and EMW are coupled in different ways to each other, which is the origin of the quadrirefringence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' GW PROPAGATION EQUATION In the Einstein’s gauge the Lagrangian density for GW, including the leading order minimal coupling to electromagnetism, is Lh = h′2 A − (∂hA)2 + LEM , (4) which gives the equations of motion h′′ A − ∇2hA = ΠA , (5) where ΠA was given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Not that the ΠA are different for each polarization mode, due to the structure of the electromagnetic stress-energy-momentum tensor, which is typical of a spin 1 gauge field with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Similar results are expected when considering other gauge fields, such as the axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In the Lagrangian approach ΠA is simply arising from the coupling ΠAhA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' EMW PROPAGATION EQUATION The equations of motion for the EMW can be derived by varying the action with respect to Aµ, giving the covariant form of the Maxwell’s equations ∇µF µν = 1 √−g∂µ(√−gF µν) = 0 , (6) where the determinant of the perturbed metric is √−g = (1 − h2 + − h2 ×)1/2 ≈ 1 − 1 2(h2 + + h2 ×) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (7) This metric determinant is inducing the graviton photon coupling in the Maxwell’s equations, which take the form ∂µF µν = Jν eff = −∂µ √−g √−g F µν , (8) 3 where we are denoting the effective gravitoelectromagnetic current as Jν eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' The gravitoelec- tromagnetic current arises naturally as a consequence of the photon graviton interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Note that the effective current is due to an effective non minimal coupling of Aµ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' it is not simply obtained from a AµJµ term in the Lagrangian, since the current itself depends on the Faraday’s tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Imposing the Lorentz’s gauge condition ∂µAµ = 0, and taking the leading order interaction terms in the determinant of the metric, we get the wave equation □Aν = Jν eff = (h+∂µh+ + h×∂µh×)F µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (9) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' QUADRIREFRINGENCE The system of coupled differential equations □hA = ΠA(h, Aρ) , (10) □Aν = Jν eff(h, Aρ) , (11) describes the evolution and interaction of GW and EMW, and their interaction is associated to the source terms which couple the two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In the above equations we have used an abstract notation to make explicit the dependence on h and Aρ of the source terms given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (5) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (9), to underline the coupling between the two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Quadrirefringence is due to the fact that the source can be different for different polarizations of the GW or different components of Aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' The effects of the polarization of EMW can be obtained by expressing the electric and magnetic fields in terms of Aν using E = −∇φ − ∂tA , (12) B = ∇ × A , (13) where Aν = (φ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Since the GW propagating in the x direction have the form h(η − x) we get Jν eff = (h+h′ + + h×h′ ×)F 0ν + (h+∂xh+ + h×∂xh×)F 1ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (14) Note that each component of the effective current is different form zero, implying that each component of Aµ is affected by quadrirefringence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' If the effects of interaction are ignored the EMW and GW will propagate independently, and the solutions of the wave equations will be of the standard oscillatory form, with each 4 polarization mode of the EMW and GW propagating at the speed of light, but if the effects of interaction are take into account, each polarization of the GW and EMW will propagate differently, since the source terms are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' The strength of these effects depends on the size of the sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' ΠA and Jν eff, which for GW is related to the anisotropic part of the stress-energy-momentum tensor of the electromag- netic field, and for EMW to the amplitude of the GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' THE RAINBOW OF BRIGHT AND GREY SIRENS So far we have not introduced any electromagnetic four current Jν EM in the Lagrangians, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' we have studied the interaction of GW and EMW in the absence of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In a realistic astrophysical scenario Jν EM will be definitely present in the source, and will be associated to the specific dynamic of the astrophysical system producing the EMW and GW, which could be a neutron stars (bright sirens) or neutron star black hole (grey siren) binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In this case the additional Lagrangian density and corresponding equations would take the form LJ = √−gAµJµ EM , (15) □hA = ΠA(h, Aρ) , (16) □Aν = Jν eff , (h, Aρ) + √−gJν EM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (17) Due to the coupling between the equations, the current Jν EM will also affect the GWs emitted by the the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Given the environmental dependence of Jν EM(η, xi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' its space and time evolution depend on the specific dynamics of each astrophysical system, the effects of quadrirefringence can be different for each binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Once more accurate measurements of bright sirens will be available, it will be possible detect the object specific effects imprinted on GW and EMW, predicted by the theory of general relativity and covariant electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' This will allow to infer new information about the binary system dynamics, by combined analysis of GW and EMW observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' 5 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' EFFECTIVE SPEEDS DESCRIPTION It is possible to describe quadrirefringence using an effective approach [8, 9], in which the effects of interaction are encoded in an appropriately defined polarization, frequency, and time dependent effective speed of GW and EMW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' While quadrirefringence can be studied without adopting this effective description, by solving the system of coupled differential equations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' (17), it is convenient to give this effective description, to gain more insight about the physical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' We will leave the details to a future work, but we provide below the effective equations which can be derived h′′ A − 2c′ T,A cT,A h′ A − c2 T,A∇2hA = 0 , (18) A′′ ν − 2c′ γ,ν cγ,ν Aν − c2 γ,ν∇2Aν = 0 , (19) where cT,A(η, xi) and cγ,ν(η, xi) are respectively the GW and EMW space dependent effec- tive speeds (SES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' In Fourier space a similar set of equations can be derived in terms of the momentum dependent effective speeds (MES) ˜h′′ A − 2˜c′ T,A ˜cT,A ˜h′ A + ˜c2 T,Ak2˜hA = 0 , (20) ˜A′′ ν − 2˜c′ γ,ν ˜cγ,ν ˜Aν + k2˜c2 γ,ν ˜Aν = 0 , (21) where a tilde denotes the Fourier modes, except for the MES ˜cT,A(η, k) and ˜cγ,ν(η, k), which are not defined [8–11] as the Fourier transform of the SES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Note that the GW-EMW interaction not only changes the effective speed of each polarization, but it also induces an additional time/frequency/polarization dependent friction term, related to the time derivative of the MES, which can have important observational implications [8] for the gravitational and electromagnetic luminosity distances, which we will study in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' CONCLUSIONS We have shown that the interaction between GW and EMW waves induce the phenomenon of quadrirefringence, the different propagation of the polarization modes of GW and EMW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' When this phenomenon happens in binary systems like bright or grey sirens it can induce an object characteristic imprint on the emitted GW and EMW, the binaries rainbow, which 6 provides a new method to study the astrophysical properties of these systems, such as for example the equation of state of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' On cosmological scales quadrirefringence can be used to probe the electromagnetic field with gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' We have studied the leading order effect of the classical calculation of the GW-EMW inter- action, but it will also be important to consider quantum effects, and to study the regimes in which they can provide important correction to the classical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Note that while in this paper we have focused on the GW-EMW interaction effects, similar effects are predicted by the model independent effective approach [8], and could arise also when the graviton is coupled to any other field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' These interaction effects will allow to use gravitational waves observations to probe any other field the graviton is coupled to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Acknowledgments We thank Tessa Baker, Riccardo Sturani, Rogerio Rosenfeld, Nicola Tamanini, John 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Vallejo-Pe˜na, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' Turzy´nski, (2020), arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content='00969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFIT4oBgHgl3EQflStM/content/2301.11304v1.pdf'} diff --git a/adAyT4oBgHgl3EQfW_eP/vector_store/index.faiss b/adAyT4oBgHgl3EQfW_eP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4890f137ee1c9683685d74a8e498212140c2cc70 --- /dev/null +++ b/adAyT4oBgHgl3EQfW_eP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Siavash Saeidpour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 Björn Corzilius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='2 Rostyslav Lesyuk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 Christian Klinke1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='6* 1 Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' University of Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Albert-Einstein-Straße 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 18059 Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Germany 2 Leibniz Institute for Catalysis (LIKAT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Albert-Einstein-Straße 29a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 18059 Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Germany 3 Institute of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' University of Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Albert-Einstein-Straße 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 18059 Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Germany 4 Department “Life,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Light & Matter”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' University of Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Albert-Einstein-Straße 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 18059 Rostock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Germany 5 Pidstryhach Institute for applied problems of mechanics and mathematics of NAS of Ukraine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Naukowa str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 3b, 79060 Lviv, Ukraine & Department of Photonics, Lviv Polytechnic National University, Bandery str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 12, 79000 Lviv, Ukraine 6 Department of Chemistry, Swansea University – Singleton Park, Swansea SA2 8PP, United Kingdom KEYWORDS colloidal synthesis, tin telluride, ligands, two-dimensional nanostructure, template- assisted growth 2 ABSTRACT We report the colloidal synthesis of 2D SnTe nanosheets through precursor hot- injection in a nonpolar solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' During the reaction, an important intermediate – Sn-template – is formed which defines the confined growth of SnTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' This "flake-like" structure gives the first evidence for the possible 2D morphology formation prior to the anion precursor injection (TOP- Te).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Additionally, we explore the role of each ligand in the reaction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Thus, we explain the formation and morphology evolution of 2D SnTe nanostructures from a mechanism perspective as well as the role of each ligand on the molecular scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The interplay of ligands provides the necessary conditions for the realization of stable low-dimensional SnTe nanomaterials with tunable size and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 3 Introduction Low-dimensional thermoelectric materials such as tin telluride (SnTe) can adopt advantages based on the quantum confinement effect, suggesting great potential for heat-electricity conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='1 As a IV-VI narrow bandgap semiconductor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='18 eV, bulk), SnTe exhibits an intrinsically high charge carrier concentration, which results in a relatively low Seebeck coefficient,2, 3 but optimization of the material through doping and alloying offers great promise for thermoelectric applications of this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='2 SnTe has also gained significant interest due to its exciting properties as a topological crystalline insulator, IR detection and radiation receivers material, as well as photovoltaic absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4-8 So far, attempts to obtain solution-based SnTe nanocrystals (NCs) mainly yielded zero-dimensional (0D) or one-dimensional (1D) nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='9-11 Recently we reported a synthesis protocol for two-dimensional (2D) colloidal SnTe nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='12 Now, we disclose the important role of ligands in the colloidal synthesis process information of the 2D morphology and the observed faceting of nanocrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The obtained nanomaterials are investigated by means of scanning electron microscopy (SEM), transmission electron microscope (TEM), high-resolution TEM, powder X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), energy-dispersive X-ray (EDX) analysis and Fourier transform infrared (FT-IR) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='13 Since SnTe has a cubic crystal structure with m3̅m symmetry and octahedral coordination geometry, the formation of anisotropic shapes has to be conducted by molecular templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' These templates are well-organized domains consisting of the cation and a specific combination of ligands/counterions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' When the tellurium precursor is injected into the reaction mixture, the tin/halogen arrangement of the template is replaced by the Sn-Te bond, and a bilayer oleic acid shell is formed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' These findings show the importance of co- ligands and emphasize the formation of templates in the syntheses of colloidal nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 4 Methods Chemicals and Materials: Tin (II) acetate (Sn(CH3CO2)2, anhydrous, ≥99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='99%, stored in a nitrogen filled glovebox), tellurium shots (Te, amorphous, 1-2 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='999 %;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' stored in a nitrogen filled glovebox), oleic acid (90%), trio-n-octylphosphine (TOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 97%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' stored in a nitrogen filled glovebox), diphenyl ether (≥99%), and 1-chlorotetradecane (1-CTD, 98%) were purchased from Sigma-Aldrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Ethanol and toluene were purchased from Honeywell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' All the chemicals were used as-received without additional purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Trio-n-ctylphosphine-Te (TOP-Te) was prepared and stored in a glovebox and all the syntheses were carried out applying standard air-free Schlenk-line techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Synthesis of 2D SnTe nanosheets: In a three necked flask equipped with a condenser, a septum and a thermocouple in a glass mantle, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mg of Sn (CH3CO2)2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='25 mmol), 2 mL of oleic acid (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='25 mmol), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mL of tri-n-octylphosphine (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='1 mmol) were mixed in 10 mL of diphenyl ether, stirred at 130 °C for 5 min and degassed under vacuum at 80 °C for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Afterwards, the reaction solution was heated to reaction temperature 210°C under nitrogen after adding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='15 mL of 1- chlorotetradecane (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mmol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 10 min later, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='8 mL of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='65 M tri-n-octylphosphine-tellurium (Te–P(octyl)3, TOP-Te) precursor solution was injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The solution turned from clear yellow to greyish yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The reaction was quenched after 1 min by removal of the heating mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The resultant nanostructures were purified by precipitation with toluene, centrifugation at 4000 rpm for 3 min (3 times), removal of the supernatant and re-suspension in toluene for further characterization or storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Synthesis of Sn templates: In a typical synthesis of Sn template, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mg of Sn (CH3CO2)2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='25 mmol), 2 mL of oleic acid (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='25 mmol), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mL of tri-n-octylphosphine (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='1 mmol) and 10 mL of diphenyl ether were mixed and dissolved in a 50 mL three-neck flask, stirred at 130 °C for 5 min 5 and degassed under vacuum at 80 °C for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Afterwards, the reaction solution was heated to reaction temperature 210°C under nitrogen after adding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='15 mL of 1-chlorotetradecane (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 mmol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 10 min later, the heating mantle was removed and when the temperature reached to 75°C, 15 mL of ethanol was injected into the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The resultant white powder was then purified by centrifugation with ethanol at 6000 rpm for 3 min (3 times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The product could then be re- suspended in ethanol for further characterization or storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Scanning electron microscope (SEM) and Energy Dispersive X-ray Spectroscopy (EDX): Standard SEM images and EDX elemental mapping were performed on a Zeiss EVO MA 10 microscope at an acceleration voltage of 10 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Samples were prepared by dropping 10 μL of the dilute NS solution onto a silicon wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Transmission Electron Microscopy (TEM): Standard TEM images were performed on a JEOL Jem-1011 microscope at an acceleration voltage of 100 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Samples were prepared by dropping 10 μL of the dilute NS solution onto carbon-covered copper grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Powder X-ray Diffraction (XRD): The crystal structure of the SnTe nanosheets was determined by XRD measurements, which were carried out on a Philips X’Perts PRO MPD diffractometer with monochromatic X-Ray radiation from a copper anode with a wavelength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='54 A (CuKα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The samples were prepared by dropping 10 μL of the dilute NS solution onto a silicon wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' X-ray Photoelectron Spectroscopy (XPS): measurements were performed on an ESCALAB 220iXL (Thermo Fisher Scientific) with monochromated Al Kα radiation (E = 1486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='6 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Samples are prepared on a stainless-steel holder with conductive double-sided adhesive carbon tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The electron binding energies were obtained with charge compensation using a flood electron source and referenced to the C 1s core level of carbon at 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='8 eV (C-C and C-H bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' For quantitative analysis the peaks were deconvoluted with Gaussian-Lorentzian curves using the 6 software Unifit 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' A background model composed of a Shirley background + polynomial background was used and the parameters are varied during the fit together with the peak parameters to find the optimal background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The peak areas were normalized by the transmission function of the spectrometer and the element specific sensitivity factor of Scofield1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Sensitivity-enhanced magic-angle spinning nuclear magnetic resonance (MAS NMR) by dynamic nuclear polarization (DNP): 1H-31P CPMAS NMR measurements were performed by suspending the dried nanomaterials in a 1,1,2,2-tetrachloroethane solution containing 15 mM TEKPol polarizing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 37 Experiments were carried out using a Bruker Biospin ASCENT 400DNP wide bore magnet operating at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 T (400 MHz 1H frequency) with a Bruker AVANCE III HD spectrometer and a Bruker 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='2 mm LTMAS DNP probe tuned to 1H and 31P in dual channel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Spectra were recorded at an MAS frequency of 15 kHz at a sample temperature of 100 K utilizing cross-polarization from hyperpolarized 1H under microwave irradiation from a Bruker/CPI 263 GHz gyrotron source operating at 130 mA beam current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The 1H-DNP enhancements for the frozen dispersions of SnTe-template and SnTe-nanosheets were 23 and 152, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Fourier transform infrared (FTIR): FTIR measurements were carried out by drying the nanomaterials and putting the powders on a diamond-ATR unit (PerkinElmer Lambda 1050+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The FTIR measurements are performed with a range from 500 to 4000 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Results and Discussion The coordinating ligands can modify the surface energy of exposed crystallographic facets (which assume an increasingly prominent role with increasing surface-to-volume ratio) or binding selectively to particular facets, thereby favoring specific morphologies or yielding altogether different polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='14 In our previous work, the type and content of ligands used in the colloidal 7 synthesis have been optimized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' however, a fundamental investigation and understanding of the mechanism would be very beneficial for the precise control of the formation and faceting processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' In this report, first we used a large amount of ethanol to prevent the reaction before the hot injection of the tellurium precursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As a result, a white powder, here called the Sn-template, was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Figure 1a shows a SEM image of the Sn-templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' We observe a flake-like structure giving the first evidence for the possible 2D morphology formation prior to the tri-n- octylphosphine (TOP)–Te injection, which is similar to the Zn-soft template formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='15 The selected area (red frame) in Figure 1a contains a large amount of Sn and carbon from the Sn-oleate complex, as well as a smaller amount of Cl which stems from the 1-chlorotetradecane (1-CTD) ligands (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Figures 1b and c depict a TEM image and a XRD pattern of SnTe nanosheets (NSs) after synthesis and purification steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The sample consist mainly of large nanosheets with lateral dimensions between 1 and 5 µm and a thickness of 57 nm calculated from XRD data using the Scherrer formula and 48 nm measured by atomic force microscopy (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The difference in thickness stems from the three-dimensional cubic side products which contribute to the XRD signals and have edge lengths bigger than 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The XRD pattern shows only 2 signals due to a strong texture effect attributed to the (200) and (400) reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' These signals can be attributed to the cubic crystal structure with the Fm3̅m space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 20 30 40 50 60 70 20 30 40 50 60 70 (200) (400) SnTe nanosheets (c) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') 2 theta (°) (200) (220) (222) (400) (420) Cubic, SnTe 00-046-1210 (a) (b) 2um 8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' (a) SEM image of the Sn templates with flake-like structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' (b) TEM image of the product containing mainly large nanosheets and minor fraction cubic particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' (c) XRD pattern of the SnTe nanosheets with a strong texture effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' In previous studies, we observed that halogenated compounds have an important influence on the shape of semiconductor nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='12, 13, 16, 17 The active species promoting the shape transformation and confined growth were halide ions produced in situ which influence both the nucleation and ripening of the nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='13 We also evidenced that the amount of 1- chlorotetradecane (1-CTD) is one of the key factors for obtaining relative uniform and well- defined 2D SnTe nanostripes while only agglomerates were produced if no haloalkanes were used during the synthesis of SnTe nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='12 The importance of 1-bromoheptane (Br-Hep) in the formation of CdSe nanosheets was earlier revealed, suggesting that two active species, Br-Hep and ionic Br , were present on the surface of the CdSe nanosheets and partially replace the carboxylate ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='13 As the new surface ligands, the formation of Cd-Br bonds can regulate the growth rate by modulation of the relative surface energies of those facets and consequently influence the shape of CdSe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='13 However, in the present work we find that 1-CTD influences the formation and final shape of SnTe nanosheets in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' X-ray photoelectron spectra (XPS) were recorded to analyze the composition and the chemical states of all elements in Sn-templates and SnTe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The survey spectra of both samples are shown in Figure S3a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As main elements C, O, and F as well as Sn, Te (for SnTe nanosheets), and Cl (for Sn-templates) with minor contributions can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The XPS quantification data are shown in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' High-resolution XPS scans of Sn 3d and Cl 2p of the Sn-templates, as well as Sn 3d and Te 3d of the SnTe nanosheets are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The Sn 3d5/2 peak of Sn-template can be found at a binding energy of 487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='2 eV (Figure 2a), which can 9 be assigned to Sn2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Figure 2b shows a rather small Cl 2p peak of Sn-templates at a binding energy of 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 eV which might be attributed to SnCl2 as minor Sn species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' We speculate that Cl does not exist as free Cl¯ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' This is in good accordance with the previous observations that Cl¯ ions, which are introduced to the synthesis by chloroalkanes, are attached to the surface of the nanostructures in the form of a halogen–metal complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='18 After the Te precursor (TOP-Te) was injected into the reaction, the product was further analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Figure 2c shows the XPS spectrum of Sn 3d for SnTe nanosheets (see Table S2 for details of the fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The peaks at 487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='0 and 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='5 eV correspond to the binding energies of Sn 3d5/2 and Sn 3d3/2 of SnTe (Sn2+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' However as shown for the template, the binding energy of Sn2+ attributed to tin oleate is located nearly at the same spectral position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The peaks located at 488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='8 and 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='3 eV indicate the oxidation state Sn4+ observed in the SnO2 crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='19 The weak peaks at 485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 and 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='0 eV are assigned to elemental Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Figure 2d shows the XPS spectrum of Te 3d (see Table S3 for details of the fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The peaks at 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 and 584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='9 eV are assigned to Te 3d5/2 and Te 3d3/2 in SnTe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The peaks at 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 and 582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='9 eV could be assigned to the Te 3d5/2 and Te 3d3/2 from elemental Te and the weak peaks at 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='1 and 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='4 eV could be assigned to the oxidized species of Te4+ in TeO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='20-22 The results express the clear propensity for oxidation of the SnTe nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='23 However, due to the formation of an oxide shell on SnTe nanocrystals, the synthesized material could be protected from being oxidized further and promote the stability of the nanocrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='11, 12 The data reveal also fluorine as very intense component of the surface, identified as contamination originated in the grease used for the glassware during the synthesis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' In fact, a majority of the F is bond to C as can be confirmed by the C 1s spectra shown in Figure S4 with strong peaks at 291 and 293 eV corresponding to C-F2 and C-F3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' XPS spectra: Sn 3d (a) and Cl 2p (b) of Sn template;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Sn 3d (c) and Te 3d (d) of SnTe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' In the context of understanding the bonding mode of the ligands with the nanomaterial surfaces, FTIR spectroscopy has been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As seen in Figure 3a and b, the bands at 2853 and 2922 cm-1 were attributed to the asymmetric CH2 stretch and the symmetric CH2 stretch modes of oleic acid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='24 It is worth noting that the band at 1710 cm-1, corresponding to stretching vibration of C=O in pure oleic acid (see Supporting Information, Figure S5), was absent in the spectrum of Sn-templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Instead, two new bands at 1457 and 1583 cm-1 appeared in the FTIR spectrum of the Sn-templates (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' They were attributed to the asymmetric (-COO¯) and symmetric (-COO¯) stretch vibration bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='24, 25 This indicates that there is no free oleic acid in the Sn template sample and a complexation between the carboxyl and Sn was formed, which is in good accordance with the XPS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As a result, oleic acid molecules are chemically adsorbed on the surface of Sn template through the chemical interaction between their -COO− groups and Sn atoms, meanwhile, the hydrophobic tails of oleic acid molecules face outwards (a Sn3d Sn-templates (b) CI2p Sn-templates Sn2+ 3ds/2 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') Sn2* 3d3/2 505 500 495 490 485 480 215 210 205 200 195 190 Bindingenergy (eV) Binding energy (eV) (c) Sn3d (d) SnTenanosheets Te3d SnTenanosheets 3d5/2 3d3/2 3d5/2 Sn2+ Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') 3d31/2 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') Te2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Sn4+ Te (0) Sn (0) 505 500 495 490 485 480 475 595 590 585 580 575 570 565 560 Bindingenergy (eV) Bindingenergy(eV) 11 (Figure 3c) and form a nonpolar shell which supports the single layer coated Sn-templates that can be dispersed in nonpolar carrier liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' After TOP-Te was injected into the reaction, the band at 1710 cm-1 appeared in the FTIR spectrum of SnTe nanosheets while the band at 1457 cm-1, the asymmetric (-COO¯) stretch vibration band, became very weak (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Thus, the FTIR signal changes also verify that the injection of TOP-Te to the synthesized oleic acid-coated Sn-template changes the kind of bonding between -COO¯ groups of the oleic acid and the Sn atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The band at 1710 cm-1 is the characteristic band of a secondary layer in bilayer oleic acid-coated nanomaterials which was already demonstrated by Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='24 The primary layer in the bilayer coated structure is chemically adsorbed on the surface of nanoparticles, and the secondary layer is physically adsorbed on the primary layer through the interpenetration of the tails of the primary and secondary surfactants at their interface (Figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='24, 26 Because the oleic acid of the secondary layer was only physically adsorbed on the primary layer, the 1710 cm-1 band due to the stretching vibration of C=O in oleic acid should appear in the FTIR spectrum of bilayer oleic acid- coated SnTe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The tri-n-octylphosphine (TOP)–Te complex was used as a tellurium precursor in the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' It has a higher cleavage rate compared to TOP-S and TOP-Se, which leads to the formation of more nuclei and faster exhaustion of the monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' This fast cleavage rate and a higher number of nuclei in the reaction showed an adverse effect on the anisotropic growth and resulted in 3D bulk structures instead of a 2D morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='27 Interestingly, the introduction of an appropriate amount of TOP is the guarantee for obtaining a two-dimensional structure along with the presence of the haloalkane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' For comparison, reference syntheses with different amounts of TOP were also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' No product could be obtained under these conditions without TOP while only agglomerates were produced if less or more amount of TOP compared to the standard procedure 12 was injected into the synthesis (Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' TOP plays an important role in slowing down particle growth, likely by blocking surface binding sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='28, 29 When TOP was used as a capping ligand in the synthesis of nanomaterials, the coordination with the TOP ligands could effectively stop the growth of the nanoparticles (even in the presence of unreacted precursors) which would influence the final shape of nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='30 In addition, TOP could act as a reducing agent as well, leading to producing SnO through an autocatalytic process at the SnTe NSs surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='31 Another significance of TOP that can ensure the acquisition of 2D structures is that TOP affects not only the reaction pathway for the halide transfer but also the dissolution of tin halides on SnTe nanocrystals which is similar to the structural development of CdSe tetrapods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='32 As shown in Figure S8, three resonances at ∼-12, ∼30 and ∼50 ppm in the 31P NMR spectrum of both the Sn template and SnTe nanosheets indicate the occurrence of ligand exchange and elimination when halide and TOP are present in the solution: alkylphosphonium halides (R4P+(halide)− where R = alkyl chain), formed from the reaction between alkylhalides (thermal decomposition product) and neat TOP, and halide-bound Sn is dissolved by forming (R3P)2Sn(halide)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='32 When the binding between Sn and Cl was replaced by Sn–Te bonds, a certain quantity of Cl− ions is released into the solution, resulting in more ligand exchange between TOP and halide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Therefore, stronger resonance at around -12 ppm (R3P) and 30 ppm (R4P+) are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' On the other hand, in reactions where oleic acid was used as a surfactant molecule, the formation of II-VI nanocrystals was accompanied by the formation of trialkylphosphine oxides and (OA)2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='33 In particular, the conversion of TOP-Te to its corresponding phosphine oxides is linked to the formation of anhydrides of oleic acid, suggesting that phosphonic and carboxylic acids are responsible for the cleavage of the phosphorus-chalcogen double bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='33 Thus, after the hot injection of TOP-Te, stronger resonance at around 50 ppm (corresponding to the conversion of TOP-Te to TOPO-Te) 13 is observed due to the cleavage of TOP-Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As H-OA is responsible for the cleavage of the P=Te bond, changing the concentration of these surfactants will likely change the TOP-Te cleavage kinetics in addition to the binding of surfactants to the nanocrystal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' This is especially important because the rate of this cleavage will influence particle nucleation and growth and offers an explanation of the influence of the amount of OA on the morphology of the ultimate product (Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Additionally, we investigated the influence of the other ligands and the synthesis temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The amount of oleic acid varies the lateral dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Low amounts of oleic acid yield small nanosheets while larger amounts caused big nanosheets (Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Since the Sn templates consist mainly of tin cations and a combination of oleate and oleic acid molecules, a higher amount of oleic acid means larger and better-organized templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' On the other hand, the co-ligand 1-CTD also influences the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' After a cleavage of the Cl¯ ion from the alkane rest, these small ions replace some of the oleate ligands in the template structure which alters the reactivity of the templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' An excessive amount of haloalkane molecules yield solely three dimensional (3D) structures while low amounts favor a large shape and size distribution (Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' It is important to note that the cleavage of this molecule does depend on the temperature and the time it spends in the reaction mixture, which means that the same amount of 1-CTD could produce different products when changing the reaction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' The temperature mainly determines the reactivity in a direct and indirect way as mentioned before through the cleavage rate of the 1-CTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' When the reaction temperature is below 170 °C no reaction takes place for this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Low reaction temperatures like 170 °C or 180 °C yield mainly 3D structures (Figure S11) while high temperatures like 250 °C produce products with a high size and shape distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='12 The results indicate that temperatures below 190 °C are not enough to separate the Cl¯ ions from the alkane 14 rest, which means that the synthesis is basically performed without this co-ligand and yields solely 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' At high temperatures, the cleavage rate of the 1-CTD is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Thus, the structure of the templates becomes more unstable resulting in templates with different shapes and reactivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' After hot injection of the TOP-Te precursor, the colorless solution transiently turned to yellow (the color of TOP-Te precursor) and the morphology evolution was investigated by taking aliquots during the reaction followed by TEM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' At the early stage, when the color of the solution starts to change (from yellow to dark yellow at 32 s after hot injection of TOP-Te), the SnTe nanoplatelets with hexagonal shape (≈2 μm length, ≈400 nm width) could be observed (Figure S12a, Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As the reaction proceeds, the nanoplatelets grow longer and the widths become larger (≈3 μm length, 600-700 nm width) (Figure S12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Based on evidence from XPS and FTIR measurements shown above, we propose a possible formation process of SnTe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' As depicted in Figure 3d, the two ligands of oleic acid and 1-CTD help Sn to form a flat structure, the Sn template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' After introduction of the Te precursor, the Cl¯ ions are replaced by Te to form Sn-Te bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Meanwhile, the SnTe nanostructure is coated with a well-organized primary oleic acid molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Then, the excess oleic acid was weakly adsorbed on the primary layer of the oleate-coated SnTe nanosheets to form a double layer shell through the steric intermolecular interaction between the subsequent molecule and the hydrophobic tail of oleate of the primary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 15 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' FTIR spectrum of Sn template (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' FTIR spectrum of final product SnTe nanosheets (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' scheme of the formation of Sn oleate (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' scheme of the proposed mechanism of the formation of SnTe nanosheets (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Conclusion In this work, we discussed the interaction of ligands with metal precursors at different stages of the synthesis and investigated the formation process of SnTe nanosheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' We see from the FTIR analysis that the formation of an OA-double layer is already evidenced at the stage of the Sn-white- complex (template).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' At the same time, the Cl-interaction with Sn is evidenced by XPS, suggesting that both OA and halogen ensure conditions for the 2D templating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Later on, Cl disappears and only OA and TOP stay on the surface leading to growth in three dimensions with different velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Thus, the collective interplay of different ligation agents is needed for obtaining the 2D morphology in the cubic system of SnTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Based on the experiments, we explained the formation and morphology evolution of 2D SnTe nanostructure from a mechanism perspective as well as the role of each ligand at the molecular scale which provides the building block for the realization of stable low-dimensional SnTe nanomaterials with tunable size and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' HO (a) Sn-template C(CH2)7CH=CH(CH2)7CH3 C(CH2)7CH-CH(CH2)CH 1583 1457 Transmittance (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=') (1) (b) SnTenanosheets 1710 3 400035003000 2500 2000 1500 1000 500 Wavenumber(cm-1) Sn RAAURAURAL 16 ASSOCIATED CONTENT Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' EDS analysis of Sn templates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' AFM images and measured height images of synthesized SnTe nanosheets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' XPS spectra and XPS quantification data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' fit parameters of the Sn 3d spectra and Te 3d spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' FT-IR spectrum of pure oleic acid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' TEM images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 31P NMR spectrum (PDF) AUTHOR INFORMATION Corresponding Author E-mail: christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='klinke@uni-rostock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content='de Funding Sources This work was supported by the China Scholarship Council and German Academic Exchange Service (DAAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1477 "Light-Matter Interactions at Interfaces", project number 441234705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' ACKNOWLEDGMENT We would like to thank Fabian Strunk for the help with the XPS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' We acknowledge the European Regional Development Fund of the European Union for funding the PL spectrometer (GHS-20-0035/P000376218) and X-ray diffractometer (GHS-20- 0036/P000379642) and the Deutsche Forschungsgemeinschaft (DFG) for funding an electron microscope Jeol NeoARM TEM (INST 264/161-1 FUGG) and an electron microscope ThermoFisher Talos L120C (INST 264/188-1 FUGG) and for supporting the collaborative research center LiMatI (SFB 1477).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 17 REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' N.' 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Solution-phase synthesis of monodispersed SnTe nanocrystallites at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Inorganic Chemistry Communications 2003, 6 (2), 181-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Zeng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Alivisatos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=', Mechanistic Study of Precursor Evolution in Colloidal Group II−VI Semiconductor Nanocrystal Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} +page_content=' Journal of the American Chemical Society 2007, 129 (2), 305-312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfivj4/content/2301.00404v1.pdf'} diff --git 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We describe all linear maps f : Nr → Nr satisfying [f(A), A] ∈ Z(Nr) for +every A ∈ Nr where Nr denotes the algebra of r × r strictly upper triangular matrices +with entries in a field F with center Z(Nr). +Contents +1. +Introduction +1 +2. +Preliminaries +2 +3. +Centralizing Maps on Nr +3 +References +5 +1. Introduction +Let R be a ring with center Z(R). We say that a map f : R → R is called commuting +(centralizing) if [f(x), x] = 0 (resp. [f(x), x] ∈ Z(R)) for all x ∈ R where [a, b] = ab − +ba denotes the standard commutator on R. Investigations into commuting mappings was +initiated by Posner [18] in 1957 when it was proven that a noncommutative prime ring +cannot exhibit a nonzero commuting derivation. This theorem has since been generalized +in numerous settings (see, for example, [2, 4, 5, 6, 12, 13, 19]). +The first general result in the study of commuting maps is due to Bre˘sar [5] when it was +shown that an additive commuting map f over a simple unital ring R must be of the so-called +standard form f(x) = λx+µ(x) for some λ ∈ Z(R) and additive µ : R → Z(R). Researchers +have since applied significant effort in examining the properties of commuting maps which +satisfy various conditions in numerous rings and algebras. The reader is referred to the +survey paper by Bre˘sar [7] for details on the early development of the theory of commuting +maps along with an overview of the classical results in the field. +Particular focus has been placed on studying the structure of commuting maps over +matrix rings and algebras, with notable success in the cases of upper and strictly upper +triangular matrices [1, 3, 8, 10]. In 2000, Beidar, Bre˘sar, and Chebotar [1] proved that a +commuting linear map defined on the algebra of r×r upper triangular matrices with entries +in a field F must also be of the standard form. More recently, it was shown by the author in +[3] that a commuting linear map defined on the algebra of strictly upper triangular matrices +with entries in a field of characteristic zero is almost of the standard form. +1 + +2 +JORDAN BOUNDS +Theorem 1.1 (Theorem 1 [3]). Let Nr = Nr(F) be the ring of r × r strictly upper +triangular matrices, r ≥ 4, over a field F of characteristic zero and suppose f : Nr → Nr +is a commuting linear map. Then there exist λ ∈ F and an additive map µ : Nr → Ω such +that +f(A) = λA + µ(A) +for all A ∈ Nr where Ω = {ae1,r−1 + be1,r + ce2,r : a, b, c ∈ F} and ei,j denotes the standard +matrix unit. +In this brief note we modify the methods used in [3] to generalize the conclusions of +Theorem 1.1, providing a complete characterization of all centralizing linear mappings on +Nr. More precisely, we prove the following theorem. +Theorem 1.2. Let f : Nr → Nr be a linear map such that [f(x), x] ∈ Z(Nr) for all +x ∈ Nr. Then there exist λ ∈ F and additive µ : Nr → Ω such that f(x) = λx + µ(x) for +all x ∈ Nr. +We begin with some preliminaries in Section 2, including elements from [3] which will +be necessary for our discussion. We then conclude with a proof of Theorem 1.2 in Section +3. +2. Preliminaries +Throughout the remainder of this paper we will assume r ≥ 4 is an integer and F is +a field of characteristic zero. Let ei,j denote the r × r matrix with ij-th entry 1 and all +other entries 0. We use Nr = Nr(F) (Mr = Mr(F)) to denote the ring of r × r strictly +upper triangular matrices with entries in F (resp. the ring of r × r matrices with entries +in F). It is well known that Nr is a nilpotent ring in which the identity Ar = Ir, the +r × r identity matrix in Mr, holds. Additionally, the center of Nr is known to be the set +Z(Nr) = {ae1,r : a ∈ F}. +From the definition it is clear that every commuting map is necessarily centralizing, +however, there exist maps on Nr which are centralizing and not commuting, as the following +example illustrates. +Example 2.1. Define g : Nr → Nr by g(A) = e1,r−1 + a1,re1,r + e2,r where A = (ai,j). +The map g satisfies +[g(A), A] = (ar−1,r − a1,2)e1,r, +hence g is clearly centralizing. However, the value ar−1,r − a1,2 is not identically 0, meaning +g is not commuting. +It should be noted that the subset Ω = {ae1,r−1 + be1,r + ce2,r : a, b, c ∈ F} of Nr given +in Theorem 1.2 is essential for the conclusion in [3]. The above example shows that this is +still the case in the more general setting of the present paper as the image of g is entirely +contained in Ω. +Finally, it will be convenient to utilize the following lemma which is an immediate +consequence of Theorem 3.2.4.2 of [11]. +Lemma 2.2. Let A = +r−1 +� +i=1 +ai,i+1ei,i+1 such that +r−1 +� +i=1 +ai,i+1 ̸= 0. Then the centralizer of A +in Nr is given by CA = {α1A + α2A2 + · · · + αr−1Ar−1}. + +CENTRALIZING MAPS ON THE RING OF STRICTLY UPPER TRIANGULAR MATRICES +3 +3. Centralizing Maps on Nr +We now prove our main results. Let f : Nr → Nr be a centralizing linear map. We +proceed via a similar method as in [3] by beginning with a basis for Nr and then extending +the result via the linearity of f. As such, the proofs in this section follow similar arguments +to those given in [3] with natural modifications made to suit the more general setting of +centralizing mappings. +Let J = +r� +i=1 +ei,i+1 and consider the sets S1 = {D−1JD : D = +r� +i=1 +diei,i with +r� +i=1 +di ̸= 0} +and S2 = {T −1 +i,j JTi,j : Ti,j = Ir − ei,j}. Note that the elements A ∈ S1 are all of the form +A = +r−1 +� +i=1 +aiei,i+1 where +r−1 +� +i=1 +ai ̸= 0. Setting S = S1 ∪ S2 we have that span(S) = Nr by +Lemma 5 in [3]. We first prove an analog of Lemma 2.2. +Lemma 3.1. Let A = +r−1 +� +i=1 +aiei,i+1 ∈ Nr such that +r−1 +� +i=1 +ai ̸= 0. Then there exist α1, . . . , αr−3 ∈ +F, and µA ∈ Ω such that f(A) = +r−3 +� +t=1 +αtAt + µA. +Proof. Suppose f(J) = (xi,j)1≤i n, every homomorphism ρ : Bn → +Bm either has cyclic image or else is reducible, i.e. there is a nonempty set of disjoint essential, +non-boundary-parallel curves on the m-punctured disk invariant under ρ(Bn). +As discussed in [CKM19], the reducibility conjecture in fact implies a stronger classification +of homomorphisms, predicting that all homomorphisms are recursively assembled from a small +list of basic ones using a certain set of operations. Our methods here can be used to show that +Conjecture 1.2 implies a complete classification of all holomorphic maps between configuration +spaces. +Theorem 1.3. If Conjecture 1.2 holds, then the statement of Theorem 3.1 holds for all n ≥ 5 with +no restriction on m. +Prior Results. Within the world of moduli spaces of complex-analytic objects, there have been +various efforts to understand the relationship between their behavior in the holomorphic and smooth +categories. The results here expand on the work [Lin04] of Lin mentioned above, who obtained +Theorem 3.1 in the case m = n, and also investigated the corresponding questions in the setting of +configuration spaces on CP1. Our proof is independent, making use of the developments of [CKM19] +that give a classification of maps between braid groups, as well as employing some powerful results +from Teichm¨uller theory. +Two other results, while not bearing directly on the results of this paper, were a source of +inspiration and merit mention. Antonakoudis–Aramayona–Souto [AAS18] give a classification of +holomorphic maps Mg → Mh for h ≤ 2g − 2, showing that the only non-constant map is the +identity. Letting Ah be the moduli space of h-dimensional, principally-polarized abelian varieties, +a recent result of Farb [Far21] shows that a nonconstant holomorphic map f : Mg → Ah exists for +h ≤ g iff h = g and f is the map sending a curve to its Jacobian. It is interesting to note that in +both these results, every smooth map is homotopic to a holomorphic map. This is in contrast to the +setting of this paper, where smooth maps between configuration spaces exist in relative abundance +(see above). +Proof Strategy. The main theme running through our arguments is the promotion of group- +theoretic rigidity statements to the level of holomorphic maps. In [Che20], the first author has +established a general rigidity theorem classifying homomorphisms f : PBn(X) → Λ, where X +is a Riemann surface, PBn(X) := π1(PConfn(X)) denotes the space of ordered configurations +of n points, and Λ is a torsion-free nonelementary hyperbolic group. When f is induced from +a holomorphic map F between complex manifolds, principles of complex analysis can be used to +show that F must have a specific form. In other settings, we make use of rigidity phenomena for +holomorphic maps of Riemann surfaces into the moduli space of Riemann surfaces, equivalently the +rigidity of holomorphically-varying families of Riemann surfaces over Riemann surfaces. We will use +two such theorems in our proof of Theorem 3.1, both concerning the monodromy of such families, the +homomorphism ρ : π1(B) → Mod(S) from the fundamental group of the base to the mapping class +group of the fiber. Specifically, we will use a result of Daskalopoulos-Wentworth [DW07] showing +that the monodromy of a non-isotrivial family of curves is necessarily “rich” in a certain technical +sense, and a result of Imayoshi–Shiga [IS88] stating that if two families have the same monodromy, + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +5 +then they are equal to each other. Inside Confn(C), there are many embedded Riemann surfaces: +given distinct points Y ∈ Confn−1(C), there is an associated embedding iY : C − Y → Confn(C) of +the finite-type surface C − Y into Confn(C). The holomorphic map Confn(C) → Confm(C) thus +equips each such surface with a family of Riemann surfaces (specifically, the fiber is C punctured +at m points). The monodromy of such families factors through the homomorphism Bn → Bm +induced by the map on the configuration spaces; we exploit the classification of [CKM19] along +with the criterion of Daskalopoulos-Wentworth to rule out many possibilities, and the rigidity result +of Imayoshi–Shiga gives control over the situations where a map is possible. The phenomenon of +affine twisting appears because the target space Confm(C) is not exactly the moduli space of m +points in C, but becomes a finite cover of this after erasing the action of the affine group Aff. +Acknowledgements. NS is supported by NSF Award No. DMS-2153879. LC is supported by +NSF Award No. 2005409. +2. background +2.1. Classification of homomorphisms between braid groups. The basic strategy of the +proof of the main results is to first understand all possible induced maps on the fundamental +groups (i.e. braid groups), and then to contrast this with known rigidity results about holomorphic +maps between the associated moduli spaces. In this section we carry out the first of these tasks. +Theorem 2.1 below is a corollary of the work of Chen–Kordek–Margalit [CKM19]. To state it, +we introduce the following terminology. A transvection of a group homomorphism f : G → H is a +homomorphism ft : G → H defined via +ft(g) = f(g)tℓ(g), +where ℓ : G → Z is a homomorphism and t ∈ H centralizes f(G). A homomorphism f : G → Bn +is said to be reducible if the image f(G) preserves a finite set of isotopy classes of essential non- +boundary-parallel curves on the n-punctured disk (viewing Bn as the mapping class group of the +punctured disk). Lastly, an element of Bn is said to have prefinite order if its image in Bn/Zn has +finite order. +Theorem 2.1. For n ≥ 5 and m ≤ 2n, let ρ : Bn → Bm be a homomorphism. Then exactly one +of the following conditions hold: +(1) n = m and ρ is the identity map up to transvection, +(2) ρ is either reducible and nontrivial, or else has infinite cyclic image generated by a pseudo- +Anosov, +(3) ρ has prefinite cyclic image. +Proof. Note first that the statement is equivalent to the assertion that at least one of the following +conditions hold: +(1’) n = m and ρ is the identity map up to transvection, +(2’) ρ is reducible (possibly trivial) +(3’) ρ has cyclic image. + +6 +LEI CHEN AND NICK SALTER +The latter assertion follows more readily from [CKM19, Theorem 1.1], but we will see in the +proof of Theorem 3.1 that the former organization corresponds to the classification of holomorphic +maps. +[CKM19, Theorem 1.1] asserts that for n ≥ 5, any homomorphism ρ : Bn → B2n is a transvection +of one of five “standard homomorphisms”, and [CKM19, Corollary 1.2] asserts that every ρ : Bn → +Bm for m < 2n is one of two of the five types possible when m = 2n. We first argue that if ρ +satisfies at least one of the conditions, then so does any transvection of ρ; then we will see that +each standard homomorphism satisfies one of the conditions. +There is nothing to check if ρ satisfies condition (1’). Suppose now that ρ is reducible. If ρ is +the trivial homomorphism, then any transvection of ρ has cyclic image (note that if t is pseudo- +Anosov, the transvection will not necessarily be reducible). If ρ is reducible and nontrivial, then +ρ has a nonempty “canonical reduction system” {Ci} of disjoint essential, non-boundary-parallel +curves (see e.g. [BLM83]). Given a transvection ρt, the element t commutes with every element of +ρ, from which it follows that t preserves {Ci} as well, so that ρt is likewise reducible. If ρ has cyclic +image, then ρ factors through the abelianization map ℓ : Bn → Z, and hence by construction any +transvection of ρ does as well. +We now see that each of the five “standard homomorphisms” described in [CKM19] satisfies +one of the above conditions. +Each of these is a routine verification; for brevity’s sake we will +assume familiarity with the five, as given in [CKM19, page 1]. The trivial homomorphism satisfies +conditions (2’) and (3’), the inclusion homomorphism satisfies (1’) if m = n and (2’) if m > n, and +the diagonal, flip-diagonal, and k-twist cabling maps are all reducible (condition (2’)). +□ +In anticipation of Theorem 3.2, we next consider the case of homomorphisms ρ : Bn → Mod(Sg) +(here Mod(Sg) denotes the mapping class group of a closed surface of genus g). Let H be the +hyperelliptic embedding defined in the introduction; H∗ is then the induced homomorphism on +orbifold fundamental groups. We have the following result of Chen–Mukerjea [CM20, Theorem +1.1]. +Theorem 2.2. For n ≥ 26 and g ≤ n − 2, let h : Bn → Mod(Sg) be a homomorphism. Then up to +transvection, h is either trivial or H∗. +2.2. Some results from complex analysis and Teichm¨uller theory. Before considering the +holomorphic theory of families of Riemann surfaces, we first mention a useful variant of the remov- +able singularity theorem that we will employ throughout the paper. +Proposition 2.3. Let X, Y be Riemann surfaces of finite type, each of negative Euler characteristic, +and let f : X → Y be holomorphic. Then f admits a holomorphic extension F : ¯X → ¯Y , where +¯X, ¯Y denote the compact Riemann surfaces associated to X, Y . +Proof. Since each of X, Y have negative Euler characteristic, they are uniformized by D and hence +admit complete hyperbolic metrics. By the Schwarz-Pick theorem, f is distance non-increasing in +these metrics. Let x ∈ ¯X − X be given, and let γ ⊂ X be a loop encircling x. The homotopy +class of γ admits representatives of arbitrarily short length, and hence the same is true for f(γ), +showing that f(γ) is either null-homotopic or else encircles a puncture in Y . Thus f admits a + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +7 +continuous extension F : ¯X → ¯Y , and by the usual removable singularity theorem, it follows that +F is holomorphic. +□ +We now turn our attention to families of Riemann surfaces. Let M′ +g,n be a finite orbifold cover +of Mg,n. Imayoshi–Shiga proved the following [IS88, Section 3]. +Theorem 2.4 (Imayoshi–Shiga). Let B be a Riemann surface of finite type. If f, h : B → M′ +g,n are +non-constant holomorphic maps and the monodromy maps f∗ = h∗ : π1(B) → Mod(Sg,n) coincide, +then f = h. +A homomorphism f : G → Mod(Sg,n) is said to be sufficiently large if the image contains two +pseudo-Anosov elements with distinct fixed point sets in PMF(Sg,n). We will not need to know the +precise meaning of these terms, only that a sufficiently large subgroup is not reducible, i.e. there is +no globally-invariant finite set of curves. Daskalopoulos-Wentworth proved the following [DW07, +Theorem 5.7]. +Theorem 2.5 (Daskalopoulos-Wentworth). Let B be a Riemann surface of finite type and f : B → +Mg,n be a non-constant holomorphic map. Then f∗ : π1(B) → Mod(Sg,n) is sufficiently large. +2.3. Relations between Confm(C), PConfm(C) and M0,m,1. Let M0,m,1 denote the moduli +space of m + 1 points on CP1, where one of the points is distinguished. We now discuss the natural +projection map πm : Confm(C) → M0,m,1. Let Aff be the affine group of C, which induces an +action on Confm(C) and on PConfm(C). +Lemma 2.6. The preimage π−1 +m (πm(X)) of a point X ∈ Confm(C) is the orbit Aff(X) ⊂ Confm(C). +Proof. The point πm(X) represents CP1 with distinct marked points (x0, {x1, ..., xm}), the first of +which is distinguished. We first apply a Mobi¨us transformation to send the first point into ∞. +Now πm(X) is represented as (∞, {x1, ..., xm}) where we have m distinct points x1, ..., xm ∈ C. +Two points (∞, {x1, ..., xm}) and (∞, {y1, ..., ym}) represent the same points in M0,m,1 if they are +holomorphically related, i.e. if and only if there is a biholomorphism f : C → C taking {x1 . . . , xm} +to {y1, . . . , ym}. The group of holomorphic automorphisms of C is Aff, and so (∞, {x1, ..., xm}) +and (∞, {y1, ..., ym}) represent the same points in M0,m,1 if and only if there is f ∈ Aff such that +f({x1, ..., xm}) = {y1, ..., ym}. +□ +Denote by PM0,m+1 the moduli space of m + 1 ordered distinct points in CP1. +Similar to +Lemma 2.6, we have the following (cf. [FM12, p. 247 ff.] for the assertion about orbifold funda- +mental groups). +Lemma 2.7. There is an isomorphism of complex orbifolds +PConfm(C)/ Aff ∼= PM0,m+1. +The quotient PConfm(C) → PConfm(C)/ Aff induces the natural map PBn → PBn/Zn on orbifold +fundamental groups. +A rotation group in Aff centered at c of order p is the cyclic subgroup generated by the element +g(x) = e2π/p(x − c) + c. + +8 +LEI CHEN AND NICK SALTER +Lemma 2.8. Any finite subgroup of Aff is a rotation group centered at some c ∈ C. +Proof. Let G < Aff be a finite subgroup. Any orientation-preserving finite group action on R2 is +cyclic by [vK19], which implies that G is generated by a single element f(x) = ax + b of order p. +This implies that the linear coefficient a must be a pth root of unity, and then the center can be +determined by the formula c = b/(1 − a). +□ +3. Holomorphic maps between configuration spaces on C: Theorem 3.1 +Our first main result, Theorem 3.1, gives a classification of holomorphic maps h : Confn(C) → +Confm(C) in the range m ≤ 2n. To state the result, we must first discuss one of the possible +archetypes. +Root maps. There are two maps +rp : Confk(C∗) → Confkp(C∗) +and +r′ +p : Confk(C∗) → Confkp+1(C), +where the first takes pth roots of the k distinct nonzero points, and the second takes the union of +the pth roots of the k distinct nonzero points and {0}. Such maps are called basic root maps. +A map Confn(C) → Confkp+ϵ(C) (with ϵ ∈ {0, 1}) is called a root map if it admits a factorization +Confn(C) → Confk(C∗) → Confkp+ϵ(C), +where the map Confn(C) → Confk(C∗) is a twist of a constant map by some holomorphic map +A : Confn(C) → C∗ and the latter map is a basic root map as above. By convention, we consider +the zero map Confk(C) → Conf1(C) ∼= C to be a root map of the second kind with p = 0. +The main result of this section is the following rigidity result about holomorphic maps. We note +that in the case m = n, Theorem 3.1 was established by Lin [Lin04, Theorem 1.4]. +Theorem 3.1. For n ≥ 5 and m ≤ 2n, if h : Confn(C) → Confm(C) is a non-constant holomorphic +map, then h is either an affine twist of the identity map or a root map. +We will also consider a variant of this result, where the target is the moduli space Mg. One +such holomorphic map is given by the hyperelliptic embedding H : Confn(C) → Mg for g = [ n−1 +2 ]: +H({x1, ..., xn}) = the algebraic curve {y2 = (x − x1)...(x − xn)} +Theorem 3.2. For n ≥ 26 and g ≤ n − 2, if h : Confn(C) → Mg is a non-constant holomorphic +map, then h is the hyperelliptic embedding. +Note that Theorem 3.2 does not mention affine twists, in contrast to Theorem 3.1. The reason +is that affine twisting gives equivalent points in the moduli space of the punctured sphere; the +hyperelliptic embedding does not depend on the actual location of the points but only on the holo- +morphic structure of the punctured sphere. Taking the orbifold structure on Mg into account, it is +possible to distinguish between two maps, one being a sort of twist of the other by the hyperelliptic +involution; see Section 3.5 for details. + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +9 +To prove Theorem 3.1, we divide into cases according to Theorem 2.1. Let +h : Confn(C) → Confn(C) +be a holomorphic map, and let h∗ be the map induced by h on fundamental group. Theorem 2.1 +asserts that there are three possibilities for h∗; we consider each in turn. +3.1. h∗ is the identity map up to transvection. +Proof. By hypothesis, h∗ : Bn → Bn has the form h∗(σ) = σζℓ(σ), where ζ ∈ Zn and ℓ : Bn → Z +is the signed word-length homomorphism. Note that Zn ≤ PBn, and so h∗(PBn) ≤ PBn. Thus h +admits a lift +˜h : PConfn(C) → PConfn(C), +and by hypothesis, ˜h∗ : PBn → PBn is also the identity map up to a transvection by an element +in the center of Bn. +Let π′ +n : PConfn(C) → PConfn(C)/ Aff be the natural quotient map. By Lemma 2.7, the space +PConfn(C)/ Aff is a finite orbifold cover of M0,n+1 and πorb +1 (PConfn(C)/ Aff) ∼= PBn/Zn, with +π′ +n,∗ the natural quotient PBn → PBn/Zn. +Let Y = (x1, ..., xn−1) be an n − 1-tuple of ordered distinct points in C. There is an embedding +iY : C − Y → PConfn(C) +such that iY (x) = (Y, x). From the above description of π′ +n,∗, we see that both π′ +n ◦ ˜h ◦ iY and +π′ +n ◦ iY induce the same map on the orbifold fundamental group. By Theorem 2.4, it follows that +π′ +n ◦ ˜h ◦ iY = π′ +n ◦ iY . +Therefore ˜h(Y, x) = g(Y, x)(Y, x) for some function g : PConfn(C) → Aff. +Claim 3.3. The map g : PConfn(C) → Aff is holomorphic and factors through Confn(C). +Proof. For X = (x1, . . . , xn) ∈ PConfn(C), we write +˜h(X) = (˜h1(X), . . . , ˜hn(X)) +with +˜hi(X) = g(X)(xi). +Writing g(X)(z) = az + b, we can solve for a, b using the fact that for any pair of distinct indices +i, j, the map g(X) takes xi to ˜hi(X) and xj to ˜hj(X). This yields the expressions +a = +˜hi(X) − ˜hj(X) +xi − xj +, +b = xj˜hi(X) − xi˜hj(X) +xi − xj +. +Observe that since ˜h is holomorphic, these expressions vary holomorphically with X, and that they +are independent of i and j by the assumption that ˜hi(X) = g(X)(xi) for all i. +□ +Thus in this case, we have shown that the original map h is the affine twist of the identity map +by the holomorphic map g : Confn(C) → Aff. + +10 +LEI CHEN AND NICK SALTER +3.2. The image of h∗ is reducible or else infinite cyclic pseudo-Anosov. Given a set Y = +{x1, ..., xn−1} of n − 1 distinct points in C, there is an embedding +iY : C − Y → Confn(C) +such that iY (x) = Y ∪ {x}. Composing iY with the natural projection πm : Confm(C) → M0,m,1, +we obtain a holomorphic map hY := πm ◦ h ◦ iY from the finite-type Riemann surface C − Y into +M0,m,1. +Suppose first that h∗ has cyclic image generated by a pseudo-Anosov. As the kernel of πm,∗ : +π1(Confm(C)) → πorb +1 (M0,m,1) is contained in the center Zn ≤ Bn = π1(Confm(C)) by [FM12, +p. 247 ff.], it follows that hY,∗ also has cyclic image generated by a pseudo-Anosov element. By +Theorem 2.5, this is not possible. Likewise, in the case when h∗ has reducible image, the induced +map hY ∗ also has nontrivial reducible image, which is again prohibited by Theorem 2.5. +3.3. h∗ has prefinite cyclic image. We will show that in this case h is a root map up to an affine +twist. +Claim 3.4. The map πm ◦ h is a constant map. +Proof. We continue to consider +hY : C − Y → M0,m,1 +as in Section 3.2. If hY were not constant, then it would determine a locally nontrivial family, which +would then have sufficiently large monodromy by Theorem 2.5, contrary to hypothesis. Therefore +πm ◦ h({x1, ..., xn}) = h{x1,...,xn−1}(xn), which does not depend on xn. This implies that it does not +depend on any coordinate, by symmetry. The claim follows. +□ +Denote the image of πm ◦ h by X ∈ M0,m,1, and let X0 = {x1, ..., xm} be a representative +of X in Confm(C). By Lemma 2.7, the image π−1 +m (X) is given as the orbit Aff(X0), and by the +orbit-stabilizer theorem, there is an isomorphism of complex manifolds, defining G := Stab(X0), +Aff(X0) ∼= Aff /G. +Thus h is given as a holomorphic map h : Confn(C) → Aff /G. By Lemma 2.8, G is a rotation +group of order p with center c and X0 is a G-invariant subset of C. By applying an affine twist, +we can assume that the center point of G is c = 0, so that G = µp, the pth roots of unity, and X0 +consists of all pth roots of some fixed subset Y0 = {y1, ..., yk} ⊂ C∗, possibly along with 0. +There is an isomorphism of complex manifolds +Aff /G ∼= C∗ × C +which sends the coset of z �→ az + b to (ap, b). Under this identification, let +A : Confn(C) → C∗ +be given as the first coordinate of the map h : Confn(C) → Aff /G, and likewise define B : +Confn(C) → C as the second coordinate. The map of configuration spaces +h : Confn(C) → Aff(X0) ⊂ Confkp+ϵ(C) + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +11 +is now seen to be an affine twist of a root map, factoring as shown below: +Confn(C) +(cY0)A +� Confk(C∗) +rp +� Confkp+ϵ +idB +� Confkp+ϵ . +□ +3.4. Proof of Theorem 1.3. Conjecture 1.2 asserts that every homomorphism ρ : Bn → Bm with +5 ≤ n < m is reducible, or else has cyclic image. The arguments of Section 3.2 and Section 3.3 +then apply to extend the classification to this setting. +3.5. Proof of Theorem 3.2. Suppose h : Confn(C) → Mg is holomorphic, with n, g satisfying +the bounds of Theorem 2.2. +We execute the same strategy as before, now following the cases +delineated by Theorem 2.2. If h∗ : Bn → Mod(Sg) is trivial, then we imitate the proof of Claim 3.4 +to see that h is constant. If the image of h∗ is cyclic, it is not sufficiently large, and so cannot arise +from a holomorphic map by Theorem 2.5. +It remains to consider the case where h∗ is a transvection of H∗ (the map induced by the +hyperelliptic embedding H : Confn(C) → Mg). Note that the image of H∗ is the hyperelliptic +mapping class group, which has centralizer Z/2Z, generated by the hyperelliptic involution ι. Thus, +there is exactly one nontrivial transvection of H∗, by ι. If h∗ : Bn → Mod(Sg) is given by H∗ , we +follow the argument of Section 3.1 to see that on each submanifold of the form C − Y ⊂ Confn(C), +the maps h and H coincide, hence coincide globally. If h∗ = Hι +∗, the same argument shows that +there is at most one holomorphic map in the homotopy class of h; it remains to give a construction. +As a map of sets, the underlying map Hι is the same as H, sending the configuration {x1, . . . , xn} +to the hyperelliptic curve y2 = (x − x1) . . . (x − xn). As a map of complex orbifolds, the two differ +in how the marking is specified on the universal covers: Hι arises from H by precomposition with +an affine twist by some holomorphic map ∆ : Confn(C) → C∗ that induces the abelianization +∆∗ : Bn → Z. +4. Rigidity of holomorphic maps between PConfn(C) and PConfn(CP1) +In this section, we will give a new proof of [Lin04, Theorem 2.5] (classifying holomorphic maps +F : PConfn(C) → C − {0, 1}) and use this to give a complete classification of holomorphic maps +between PConfn(C). +To state the results, we first define some basic ingredients: the maps sr, cr, RQ, and NI. The +maps sr and cr are holomorphic maps PConfn(C) → C − {0, 1}, both arising from the cross-ratio. +The “simple ratio” sr(i, j, k) : PConfn(C) → C − {0, 1} is given by +sr(i, j, k)(x1, ..., xn) = xk − xi +xj − xi +. +The second map is the cross-ratio cr(i, j, k, l) : PConfn(C) → C − {0, 1}, given by +cr(i, j, k, l)(x1, ..., xn) = xl − xi +xl − xk +xj − xk +xj − xi +. +Another interpretation of the maps sr(i, j, k) and cr(i, j, k, l) are the following. For (x1, ..., xn) ∈ +PConfn(C), there is a unique element in A ∈ Aff such that A(xi) = 0, A(xj) = 1. We define the + +12 +LEI CHEN AND NICK SALTER +value sr(i, j, k)(x1, ..., xn) = A(xk). Likewise, for (x1, ...xn) ∈ PConfn(C), there is a unique element +in A ∈ PSL(2, C) such that +A(xi) = 0, A(xj) = 1, A(xk) = ∞. +We define the value cr(i, j, k, l)(x1, ..., xn) = A(xl). +For later use, we record the following properties of cr and sr under permutation of indices; these +can be checked by direct inspection. +Lemma 4.1. There are identities +sr(j, i, k) = 1 − sr(i, j, k), +sr(j, k, i) = 1/sr(i, j, k), +and +cr(i, j, k, l) = cr(k, l, i, j), +cr(j, i, k, l) = 1 − cr(i, j, k, l), +cr(k, j, i, l) = 1/cr(i, j, k, l). +We will give a new proof of the following result of Lin [Lin04, Theorem 2.15], which has the +virtue of being somewhat shorter than the original. +Theorem 4.2. For n ≥ 3, any non-constant holomorphic map f : PConfn(C) → C − {0, 1} is +given by either sr(i, j, k) or cr(i, j, k, l). +Using this, we will classify holomorphic maps h : PConfn(C) → PConfm(C). We define two +basic ingredients. The first is the map RQn : PConfn(C) → PConfn−1(C), given by the following. +For (x1, ..., xn) ∈ PConfn(C), note that cr(x1, x2, x3, z), viewed as a M¨obius transformation, sends +x1, x2, x3 to 0, 1, ∞, respectively. Then define +RQn(x1, ..., xn) = (0, 1, cr(x1, x2, x3, x4), ..., cr(x1, x2, x3, xn)). +The terminology comes from the fact that the classical “resolving the quartic” map PConf4(C) → +PConf3(C) is affine equivalent to RQ4. +The second basic map we consider is the “normalized inversion” NIn : PConfn(C) → PConfn(C), +given by the formula +NIn(x1, . . . , xn) = +� +0, +1 +x2 − x1 +, . . . , +1 +xn − x1 +� +. +We can now state the main results of the section. +Theorem 4.3. Let m ≥ 2. Let h : PConfn(C) → PConfm(C) be a holomorphic map. Then up to +permutation of coordinates in the source and target and affine twisting, h is a composition of one +or more of the following: +• h = c a constant map, +• h = RQn, +• h = NIn, +• h = πm +n the forgetful map (x1, . . . , xn) �→ (x1, . . . , xm) (including m = n, the identity). +In particular, there is no holomorphic map PConfn(C) → PConfm(C) for 1 < n < m. +Theorem 4.4. Let m ≥ 3. Let h : PConfn(CP1) → PConfm(CP1) be a holomorphic map. Up to +a permutation on the domain and twisting by a holomorphic map A : PConfn(CP1) → PSL(2, C), +either F is a constant map or a forgetful map. + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +13 +4.1. Rigidity of PConf n(C) → C − {0, 1} and PConfn(CP1) → C − {0, 1}. We begin with +the following lemma. +Lemma 4.5. Any holomorphic map f : Aff → C − {0, 1} is constant. Likewise, any holomorphic +map f : PSL(2, C) → C − {0, 1} is a constant map. +Proof. Given f : Aff → C − {0, 1} holomorphic, there is an induced holomorphic map F between +the universal covers. As a complex manifold, Aff is isomorphic to C∗ × C and hence its universal +cover is C2, while the universal cover of C − {0, 1} is D; by Liouville’s theorem, it follows that F, +and hence f, must be constant. +Now suppose holomorphic f : PSL(2, C) → C − {0, 1} is given. The action by M¨obius transfor- +mation gives a fiber bundle +Aff → PSL(2, C) → CP1 +whose fibers are holomorphic submanifolds. By the previous paragraph, the restriction of f to each +fiber must be constant, and hence f factors through the base space CP1; the result now follows via +the maximum principle. +□ +We will also make use of the following result of [Che20, Theorem 1.4]: +Theorem 4.6. For n ≥ 3 and m ≥ 2, any homomorphism +f : PBn → Fm +factors through a forgetful map p∗ : PBn → PBm with m ∈ {3, 4}; in the case m = 4, there is a +further factorization through RQ4,∗ : PB4 → PB3. +Our other main tool will be the following lemma: +Lemma 4.7. For n ≥ 3, let +f : PConfn(C) → C − {0, 1} +be holomorphic. If f∗ : PBn → F2 factors through a forgetful map p∗ : PBn → PBm, then f factors +through a forgetful map p : PConfn(C) → PConfm(C). +Proof. Suppose that p∗ forgets the ith strand. Fixing a configuration of n − 1 points +Xi := x1, . . . , �xi, . . . , xn, +the restriction of f to the subspace C − Xi ⊂ PConfn(C) (where only the ith coordinate varies) +then lifts to the universal cover D of C − {0, 1}. However by the removability singularity theorem, +this extends to give a holomorphic map F : C → D, which must be constant by Liouville’s theorem, +as desired. +□ +Proof of Theorem 4.2. We proceed by induction on n. +Base case 1: n = 3. For τ ∈ C − {0, 1} fixed, define hτ : Aff → C − {0, 1} by +hτ(az + b) = h(b, a + b, aτ + b). +By Lemma 4.5, each hτ is constant, with value c(τ) ∈ C − {0, 1}. Since h is holomorphic, so too is +the induced map c : C − {0, 1} → C − {0, 1}. + +14 +LEI CHEN AND NICK SALTER +By the Great Picard Theorem, the points 0, 1, ∞ are at worst poles of c. Thus, c extends to a +holomorphic map ˆc : CP1 → CP1, a rational function so that ˆc−1({0, 1, ∞}) ⊂ {0, 1, ∞}. If this +containment is strict then ˆc has degree zero and hence c is constant; otherwise we can assume ˆc +restricts as the identity on the set {0, 1, ∞}, and that ˆc has no other singularities. +Writing ˆc(z) = p(z)/q(z), it follows that p(z) is divisible by z, there are no other zeros of p, that +q has no zeroes, and that ˆc(1) = 1. Thus ˆc(z) = z = sr(1, 2, 3)(0, 1, z), and as was discussed above, +this implies that h = sr(1, 2, 3) over the entire domain. +Base case 2: n = 4. By Theorem 4.6, h∗ : PB4 → F2 either factors through a forgetful map +PB4 → PB3 or else through (RQ4)∗. In the former case, we apply Lemma 4.7 to reduce to the +case n = 3, and so we assume that h∗ factors through (RQ4)∗. For τ ∈ C − {0, 1} fixed, define the +subset Xτ ⊂ PSL(2, C) via +Xτ := {A ∈ PSL(2, C)|A(0) ̸= ∞, A(1) ̸= ∞, A(∞) ̸= ∞, A(τ) ̸= ∞}. +By construction, Xτ((0, 1, ∞, τ)) = cr(1, 2, 3, 4)−1(τ). +Like in the case n = 3, we define the +holomorphic map hτ : Xτ → C − {0, 1} by +hτ(A) = h(A(0, 1, ∞, τ)). +We claim that hτ extends to a holomorphic map hτ : PSL(2, C) → C = {0, 1}. To see this, we +first claim that hτ lifts to holomorphic map Hτ : Xτ → D. This follows from the assumption +that h∗ factors through (RQ4)∗ = (0, 1, cr(1, 2, 3, 4))∗ and the characterization Xτ((0, 1, ∞, τ)) = +cr(1, 2, 3, 4)−1(τ), which shows that hτ is homotopic to a constant map. Thus Hτ is a bounded +holomorphic map. The space Xτ is the complement of hypersurfaces in the smooth complex variety +PSL(2, C). By the higher-dimensional removable singularity theorem [KW17, Theorem 4.7.2], it +follows that Hτ, and hence hτ, can be extended to PSL(2, C). +By Lemma 4.5, each map hτ : PSL(2, C) → C−{0, 1} is a constant map hτ = c(τ). We can now +apply the argument of the last two paragraphs in the case n = 3 to conclude that c : C − {0, 1} → +C − {0, 1} must be an automorphism, and hence h = cr(1, 2, 3, 4) up to an automorphism of +C − {0, 1} as claimed. +Inductive step: n ≥ 5. We proceed by induction on n, taking n = 4 as the base case. The +inductive step follows by Theorem 4.6 and Lemma 4.7. +□ +The following lemma gives a useful normalization of a map between configuration spaces. +Lemma 4.8 (Normalization). Every holomorphic map +h : PConfn(C) → PConfm(C) +is equivalent up to affine twisting to a unique holomorphic map +hs : PConfn(C) → PConfm(C) +such that the first two coordinates of hs(x1, ..., xn) are 0, 1. +Proof. Let p12 : PConfm(C) → PConf2(C) be the projection onto the first two coordinates. Then +p12 ◦ h is a holomorphic map. Let A : PConfn(C) → Aff be characterized by the condition that + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +15 +A(x1, ..., xn)(p12(x1, ..., xn)) = (0, 1). Via the identification of complex manifolds PConf2(C) ∼= Aff, +it follows that A is holomorphic. Defining hs as the affine twist hA, the claim follows. +□ +We now discuss the common values of cr(i, j, k, l) and sr(i, j, k). +Lemma 4.9. +(1) The function sr(1, 2, 3) − sr(i, j, k) has no zero in PConfn(C) if and only if (i, j, k) is one +of (1, 2, p), (1, p, 3), (p, 2, 3) for p ≥ 4. +(2) The function cr(1, 2, 3, 4) − sr(i, j, k) always has a zero in PConfn(C). +(3) The function cr(1, 2, 3, 4) − cr(i, j, k, l) has no zero in PConfn(C) if and only if (i, j, k, l) is +one of (p, 2, 3, 4), (1, p, 3, 4), (1, 2, p, 4), (1, 2, 3, p) for p ̸= 1, 2, 3, 4. +Proof. We will prove (3), the proofs of (1),(2) being similar. The function cr(1, 2, 3, 4)−cr(i, j, k, l) +has a zero if and only if the expression +x4 − x1 +x4 − x3 +x2 − x3 +x2 − x1 += xl − xi +xl − xk +xj − xk +xj − xi +. +(1) +has a solution. If {i, j, k, l} = {1, 2, 3, 4}, we can see by Lemma 4.1 that cr(i, j, k, l) is the image of +cr(1, 2, 3, 4) under some element of the dihedral group D3 acting on C−{0, 1} via z �→ 1−z, z �→ 1/z, +and it is straightforward to verify that no element of this group acts freely, implying that the +equation cr(1, 2, 3, 4) = cr(i, j, k, l) admits a solution in PConfn(C) in this case. +We therefore assume that index i is not in {1, 2, 3, 4}; without loss of generality, set i = 5. +If we view the equation as a function of x5 (fixing other points), it will have a unique solution +z5 = z5(x1, . . . , � +x5, . . . , xn) in CP1. If this is not a valid solution in PConfn(C), then z5 must either +be one of the xq for q ̸= 5 or else ∞. As z5 varies continuously with the parameters xi, i ̸= 5, if (1) +has no solutions, then there must be an identity z5 = xq for q ̸= 5 or else z5 = ∞. +In case z5 = ∞, (1) simplifies to the ostensible identity in C(x1, . . . , xn) +x4 − x1 +x4 − x3 +x2 − x3 +x2 − x1 += xj − xk +xl − xk +, +which is readily seen to not hold regardless of the values of j, k, l. If z5 = xq for some q ̸= 5, a +similar analysis shows that the only way (1) can be satisfied is if (p, j, k, l) = (1, 2, 3, 4). +□ +We now discuss similar results for PConfn(CP1). +Theorem 4.10. Every non-constant holomorphic map f : PConfn(CP1) → C − {0, 1} is of the +form cr(i, j, k, l). +Proof. Let f : PConfn(CP1) → C − {0, 1} be a holomorphic map. +Let En : PConfn(C) → +PConfn(CP1) be the natural embedding, which is a holomorphic map. By Theorem 4.2, f ◦ En is a +cross ratio map, either some sr or cr. Since En has dense image, the map f is uniquely determined +by f ◦ En. +If f ◦ En = sr(i, j, k), then letting xi approach ∞, the image under f ◦ En approaches 1. +Thus sr(i, j, k) cannot extend to PConfn(CP1) and so f ◦ En must be a four-term cross ratio map +cr(i, j, k, l), each of which extends to PConfn(CP1) via the same formula. +□ +Similarly, we have the following counterpart to Lemma 4.9. + +16 +LEI CHEN AND NICK SALTER +Theorem 4.11. The equation cr(1, 2; 3, 4) = cr(i, j, k, l) has no zero in PConfn(CP1) if and only +if (i, j, k, l) is one of (p, 2, 3, 4), (1, p, 3, 4), (1, 2, p, 4), (1, 2, 3, p) for p ̸= 1, 2, 3, 4. +4.2. Rigidity of PConf n(C) → PConfm(C). +Proof of Theorem 4.3. Let +h : PConfn(C) → PConfm(C) +be a holomorphic map. +We can assume the first two coordinates of h(x1, ..., xn) are (0, 1) by +Lemma 4.8. The remaining coordinates of h(x1, ..., xn) are functions +f3, ..., fm : PConfn(C) → C − {0, 1}. +Since the image lies in PConfm(C), the expressions fi − fj = 0 have no solutions in PConfn(C). +Case 1: f3 = sr(i, j, k). By Lemma 4.9, if f3 = sr(i, j, k), then all other fi’s are also simple +ratios sr. Up to a permutation of coordinates on the domain, we assume that f3 = sr(1, 2, 3). By +Lemma 4.9, f4 is either sr(1, 2, 4), sr(1, 4, 3) or sr(4, 2, 3). Applying Lemma 4.1, by applying an +affine twist and/or NIn, we can assume f4 = sr(1, 2, 4). We then claim that fk = sr(1, 2, k) for +all k ≥ 3, up to permutation on the domain. This follows from Lemma 4.9: the only tuple (i, j, k) +that differs from both (1, 2, 3) and (1, 2, 4) in a single entry is (1, 2, p) for some other p. Then we +can apply a permutation such that p = k. Applying the affine twist (x2 − x1)z + x1 then shows +that h is affine-equivalent to the forgetful map πm +n . +Case 2: f3 = cr(i, j, k, l). By Lemma 4.9, if f3 = cr(i, j, k, l), then all other fi’s are also four-term +cross ratio functions cr. Applying a permutation on the domain, we assume that f3 = cr(1, 2, 3, 4). +By Lemma 4.9, f4 is a cr function indexed by a tuple where exactly one entry of (1, 2, 3, 4) has +been replaced by 5. As in Case 1, we apply Lemma 4.1 so that possibly after an affine twist and/or +an application of NIn, we can assume that f4 = cr(1, 2, 3, 5). Arguing as in Case 1, it follows +that fk = cr(1, 2, 3, k + 1) for 3 ≤ k ≤ m up to permutation on the domain, and visibly then +h = πm +n ◦ RQn. +□ +4.3. Rigidity of PConfn(CP1) → PConfm(CP1) for m ≥ 3. +Proof. Let +h : PConfn(CP1) → PConfm(CP1) +be a holomorphic map. Similar to Lemma 4.8, we can assume that the first three coordinates of all +images of h are 0, 1, ∞ after applying a twist A : PConfn(CP1) → PSL(2, C). All the holomorphic +maps PConfn(CP1) → C − {0, 1} are given by cross ratio functions cr(i, j, k, l). +By a similar +argument as in Theorem 4.2, it can be shown that h is a forgetful map (in this setting, the maps +RQ, NI arise as twists and do not need to be considered separately). +□ +5. Pure configuration spaces in genus one +Here we consider the setting of holomorphic maps between pure configuration spaces of Riemann +surfaces where the target Y has genus one. +In outline, the proof is the same as that of the +previous section - we construct a normalization fs of f : PConfn(X) → PConfm(Y ), relative to +which the component functions are tightly constrained enough to be completely classifiable. In + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +17 +the previous section this classification of component functions (Theorem 4.2) relied on the group- +theoretic classification Theorem 4.6, which was proved in [Che20]. The results of [Che20] also treat +the case where g(X) ≥ 2 (this is recalled below in Theorem 5.9), but we must establish the case +g(X) = 1 ourselves. It is interesting to note that whereas all three results have a similar flavor +(asserting that maps from configuration spaces to hyperbolic groups factor through forgetful maps), +the precise nature of these forgetful maps reflect the different genus regimes (for g(X) = 0 we forget +all but three or four, for g(X) = 1 we forget all but two, and for g(X) ≥ 2 we forget all but one). +Theorem 5.1. Any homomorphism PBn(T 2) → Fm either factors through PB2(T 2) or has cyclic +image. +Using this, we will establish the main results of the section: +Theorem 5.2. Let X, Y be compact Riemann surfaces with g(X) = g(Y ) = 1. +Then up to +permutation of coordinates and twisting, any holomorphic map h : PConfn(X) → PConfm(Y ) is +induced by an isomorphism X → Y and a forgetful map. +Theorem 5.3. Let X, Y be compact Riemann surfaces with g(X) ≥ 2 and g(Y ) = 1. Then up to +twisting, any holomorphic map h : PConfn(X) → PConfm(Y ) is constant. +We observe that a twist A : PConfn(X) → Aut(Y ) is essentially the same thing as the case +m = 1 (see Equation (2) below), and so Theorem 5.3 really gives a reduction to the case m = 1. +It remains to give a classification of holomorphic h : PConfn(X) → Y . Certainly one possibility is +to factor through a forgetful map PConfn(X) → X, but there are more complicated examples, as +well. For instance, if X admits Y as one of its isogeny factors (i.e. the Jacobian Jac(X) admits +a finite cover isomorphic to a product Y × A), then it is possible to use the Abel-Jacobi map to +induce a map PConfn(X) → Y this way. We leave the problem of classifying m = 1 for future +work. +5.1. Theorem 5.1: from torus braid groups to free groups. We first discuss some facts +about the group PBn(T 2). Given a disk D2 embedded in T 2, we obtain an embedding of the pure +braid group i : PBn < PBn(T 2) as a subgroup. +We now introduce a generating set for PBn. Recall that PBn is the pure mapping class group of +the disk with n marked points; i.e., π0(Diff(Dn)), where Diff(Dn) is the group of diffeomorphisms +of D fixing n marked points pointwise. Consider the disk with n marked points Dn in Figure 1. +Figure 1. Dn. +Figure 2. +Figure 3. a124. + +18 +LEI CHEN AND NICK SALTER +Let L be a line segment below all the marked points x1, ..., xn. Let L1, ..., Ln be line segments +connecting x1, ..., xn to L as in Figure 1. Similarly, let U be a line segment above all marked points +and let U1, ..., Un be line segments connecting x1, ..., xn to U as shown in Figure 2. +For I ⊂ {1, ..., n}, let aI (resp. +a′ +I) be the boundary curve of the tubular neighborhood of +� +i∈I Li ∪ L (resp. � +i∈I Ui ∪ U). Let TI (resp. T ′ +I) be the Dehn twist about aI (resp. a′ +I). Figure 3 +gives an example of a curve representing a124. The following proposition about generating sets of +PBn is classical and can be found in [MM09, Theorem 2.3]. +Proposition 5.4. Both {Tij|1 ≤ i < j ≤ n} and {T ′ +ij|1 ≤ i < j ≤ n} are generating sets for PBn. +An embedding Dn �→ T 2 induces an injection PBn �→ PBn(T 2). We fix one such embedding, +and use this to identify TI and T ′ +I with elements of PBn(T 2). +Lemma 5.5. For n ≥ 2, any Dehn twist Tc about a simple closed curve c surrounding pi, pj is +conjugate in PBn(T 2) to Tij. +Proof. The Dehn twists Tc and Tij can be viewed as point-pushing maps about based loops c+ and +c+ +ij starting from pi around pj following c and cij respectively. The point-pushing subgroup based +at pi inside PBn(T 2) is a free group π1(T 2 −{x1, ..., ˆxi, ..., xn}). The loops c+ and c+ +ij are conjugate +in π1(T 2 − {x1, ..., ˆxi, ..., xn}), since as unbased loops they both encircle the puncture xj. +□ +We now prove the following statement; in anticipation of later use, we formulate it for a general +compact surface, not just tori (the Dehn twists Tij retain the meaning given above, as loops of the +ith point around the jth inside some topological embedding Dn �→ X). +Lemma 5.6. Let S be a compact surface and let en : PConfn(S) → PConfn−1(S)×S be the natural +embedding. The kernel of the induced map en∗ on the fundamental groups is normally generated by +{T1n, ..., Tn−1,n}. +Proof. Endow S with a complete Riemannian metric. Define W := PConfn−1(S)×S −PConfn(S), +noting that W is a union W = � Wi of n − 1 disjoint embedded copies of PConfn−1(S), according +to the unique i ∈ {1, . . . , n − 1} such that xn = xi. +Let N(W) be a tubular neighborhood; +likewise there is a decomposition N(W) = � N(W)i. +Let p : N(W) → W be the projection +taking (x1, . . . , xn) ∈ N(W)i to (x1, . . . , xn−1, xi). The map p is a homotopy equivalence because +it extends to a deformation retraction by radially contracting xn in to the associated xi, and the +set N(W)◦ +i := N(W)i − Wi has the homotopy type of a S1-bundle over Wi, with bundle map given +by p. +The use of van Kampen to obtain the result is complicated by the fact that W and N(W) are +disconnected. Accordingly, define Y0 := PConfn(S) and for i ≥ 1, +Yi = Yi−1 +� +N(W)◦ +i +N(W)i; +let qi ∈ N(W)◦ +i be a basepoint. + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +19 +Applying the van Kampen theorem to the above decomposition of Yi yields the following pushout +diagram: +π1(N(W)◦ +i , qi) +fi,∗ +� +pi,∗ +� +π1(Yi−1, qi) +ei,∗ +� +π1(N(W)i, qi) ∼= π1(Wi, qi) +� π1(Yi, qi) +Therefore the kernel of ei,∗ is normally generated by the image of ker(pi,∗) under fi,∗. Observe that +ker(pi,∗) is normally generated by the S1-fiber, which corresponds under fi,∗ to the Dehn twist Ti,n. +Inductively, we see that the kernel of the inclusion map π1(PConfn(S), qi) → π1(Yi, qi) is normally +generated by the Dehn twists Tj,n for j ≤ i; as Yn−1 = PConfn−1(S) × S, the result follows. +□ +Let Pi := π1(T 2 − {x1, ..., ˆxi, ..., xn}) < PBn(T 2) be the point-pushing subgroup based at the +point xi. +Lemma 5.7. The group PBn(T 2) is generated by elements in Pi for i ∈ {1, ..., n} +Proof. The subgroup Pn is the kernel of the homomorphism PBn(T 2) → PBn−1(T 2) induced by +forgetting the last point. The lemma is deduced by induction. +□ +We will make use of the result [Che20, Theorem 2.5]: +Theorem 5.8. Let G1, ..., Gn be groups and let Γ < G1 × ... × Gn be a finite index subgroup. Let +πi : Γ → Gi be the ith projection map and let Γi be the image of πi. Let Λ be a torsion-free, +non-elementary hyperbolic group. Then any homomorphism φ : Γ → Λ either factors through πi or +its image is a cyclic group. +We now start the proof of Theorem 5.1. +Proof of Theorem 5.1. Let ρ : PBn(T 2) → Fm be a homomorphism. +Case 1: n = 3. Suppose that one of ρ(T12), ρ(T13) or ρ(T23) is not trivial (we will consider the +situation where all three vanish below). We assume without loss of generality that ρ(T12) ̸= 1. +Let us consider the centralizer of T12. +The point push of the third point gives the embedding +P3 : π1(T 2 − {x1, x2}) → PB3(T 2). +Denote the based loop at x3 corresponding to T12 as c. +Then P3(c) = T −1 +12 T123. +The loop c = [a, b] is a commutator in π1(T 2 − {x1, x2}), where a, b +are standard generators for π1(T 2, x3) disjoint from T12, and so crucially, both P3(a) and P3(b) +commute with T12. As the centralizer of any nontrivial element of Fm is cyclic, it follows that +ρ([P3(a), P3(b)]) = ρ(T −1 +12 T123) = 1 and hence ρ(T123) = ρ(T12). By the same logic, either ρ(T23) = 1 +or ρ(T23) = ρ(T123), and likewise either ρ(T13) = 1 or else ρ(T13) = ρ(T123). +By the lantern relation, T12T23T13 = T123. If either ρ(T23) or ρ(T13) is trivial, this implies that +both are. The other possibility is that all three of ρ(T12), ρ(T23), ρ(T13) equal ρ(T123), which implies +that ρ(T123)2 = 1; as Fm is torsion-free, this implies ρ(T123) = 1, contrary to the assumption that +ρ(T12) ̸= 1. We conclude that ρ(T13) = ρ(T23) = 1, so by Lemma 5.6, it follows that ρ factors +through the product PB2(T 2) × T 2. By Theorem 5.8, we conclude that ρ either factors through +PB2(T 2) or else has cyclic image. Note that the argument of this paragraph covers the case where +all of ρ(T12), ρ(T13), ρ(T23) are trivial. + +20 +LEI CHEN AND NICK SALTER +Case 2: +n = 4. +As above, we can assume without loss of generality that ρ(T12) ̸= 1. +By +Theorem 4.6, the restriction ρ|PB4 either factors through a forgetful map to PB3 or RQ4. In the +first case, ρ(T14) = ρ(T24) = ρ(T34) = 1, which implies that ρ factors through e4∗ by Lemma 5.6. +Applying Theorem 5.8, either the image is cyclic or else we have reduced to the case n = 3. +Suppose then that ρ factors through RQ4. On the level of B4, the map RQ4 sends the standard +generators σ1, σ2 to themselves and σ3 to σ1. Thus ρ(T34) = ρ(σ2 +3) = ρ(σ2 +1) = ρ(T12). +Subcase 1: ρ(T23) does not commute with ρ(T12). We first claim that under this assumption, +ρ(T123) = ρ(T1234) = ρ(T234) = 1. +To see this, observe that each such element commutes with both ρ(T12) = ρ(T34) and ρ(T23) and +hence lies in the intersection of their centralizers, which is trivial. +γ +γ3 +γ4 +γ34 +¯c +Figure 4. The configuration of curves used in Subcase 1. +Referring to the left side of Figure 4, we define the elements S, S3, S4, S34 ∈ Mod(T 2, {x1, . . . , x4}) +as the Dehn twists about the curve γ with the corresponding subscript. Then there is a lantern +relation of the form +T34S4S3 = SS34 +We rearrange, expressing each side as an element of PB4(T 2): +T34S4S−1 = S−1 +3 S34 +Since S4S−1 commutes with T12 and T23, it follows that ρ(S4S−1) = 1 since ρ(T12) and ρ(T23) +do not commute by hypothesis. Thus ρ(S−1 +3 S34) = ρ(T34). There is a second lantern relation +T ′ +24T34T23 = T234, from which we conclude ρ(T ′ +24) = ρ(T34T23)−1. +Let c be the curve given as a regular neighborhood of the arc ¯c indicated on the right side of +Figure 4. By disjointness, Tc commutes with T ′ +24 and S−1 +3 S34. However ρ(S−1 +3 S34) = ρ(T34) = +ρ(T12) and ρ(T ′ +24) = ρ(T34T23)−1 = ρ(T12T −1 +23 ) don’t commute, which implies that ρ(Tc) = 1. By +Lemma 5.5, ρ(Tc) is conjugate to ρ(T13), implying that ρ(T13) = 1 as well. However, there is a +lantern relation T12T23T13 = T123, and since ρ(T123) = 1, this implies that ρ(T13) = ρ(T12T23)−1 ̸= 1, +a contradiction. + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +21 +Figure 5. Subcase 2: generators for P1 and P2. +Subcase 2: ρ(T23) commutes with ρ(T12). In this case, ρ(PB4) is in the centralizer of ρ(T12). +We now show that ρ(PB4(T 2)) is in also in the centralizer of ρ(T12), and hence ρ has cyclic image. +By Lemma 5.7, it suffices to show that ρ(Pi) is in the centralizer of ρ(T12) for i = 1, . . . , 4. The +subgroup P1 is the fundamental group of T 2 − {x2, x3, x4} and is generated by point-push maps +about the paths indicated in Figure 5. Expressing a point push as the product of Dehn twists +about the boundary components of a regular neighborhood, one observes that each such curve is +disjoint from either a12 or a34 with the exception of T23. By hypothesis, ρ(T23) commutes with +ρ(T12), and it follows that ρ(P1) commutes with ρ(T12) = ρ(T34) as claimed. A similar analysis of +the generating sets for P2, P3, P4 show that the same result holds, completing the argument. +Case 3: general n. This proceeds as in the first step of the case n = 4. If all ρ(Tij) = 1, +then the image is abelian and hence cyclic. +Otherwise, ρ(T12) ̸= 1 without loss of generality. +By Theorem 4.6, the restriction ρ|PBn factors through a forgetful map, so that (without loss of +generality) ρ(Ti,n) = 1 for 1 ≤ i ≤ n − 1. By Lemma 5.6, it follows that ρ factors through en,∗, and +applying Theorem 5.8 implies that either ρ has cyclic image or else we have reduced to the case +m = n − 1. +□ +5.2. Theorem 5.2: holomorphic maps between configuration spaces on elliptic curves. +To begin, we recall the classification of maps f : PBn(X) → Fm established (in greater generality) +in [Che20, Theorem 1.1]. +Theorem 5.9. Let X be a compact Riemann surface with g(X) ≥ 2, and let Fm be a free group of +rank m ≥ 2. Then every homomorphism f : PBn(X) → Fm either has cyclic image or else factors +through a forgetful map pi : PBn(X) → X. +Using this and Theorem 5.1, we establish the following analogue of Theorem 4.2. +Lemma 5.10. Let X, Y be compact Riemann surfaces, with g(Y ) = 1. Choose a base point O ∈ Y , +thereby endowing Y with a group structure. Let h : PConfn(X) → Y − {O} be a holomorphic map. +Then either h is constant or there is an isomorphism I : X ∼= Y , and h = I(xi − xj) for some i, j. +Proof. We proceed by induction, the case n = 1 being trivial (as every holomorphic map h : X → +Y − {O} is constant). Now let n ≥ 2 be given. By Theorem 5.1 and Theorem 5.9, either h∗ + +22 +LEI CHEN AND NICK SALTER +has cyclic image or h∗ factors through a forgetful map: either PBn(X) → PB2(X) in the case of +g(X) = 1 or PBn(X) → X in the case of g(X) ≥ 2. +Firstly, suppose that h∗ has cyclic image. Fix a configuration C = (x1, ..., xn−1) ∈ PConfn−1(X), +and for 1 ≤ k ≤ n, consider the natural embedding iC,k : X − C ⊂ PConfn(X) where all but +the kth coordinate is fixed at C; note this is a holomorphic map. By Proposition 2.3, h ◦ iC,k +extends to a holomorphic map H : X → Y . If H is not constant, then Hodge theory asserts that +H1(Y ; C) is spanned by holomorphic differentials and their conjugates. Since nonzero holomorphic +differentials pull back to nonzero holomorphic differentials along holomorphic maps, it follows that +H∗ : H1(Y ; C) → H1(X; C) is injective, and dually H∗ : H1(X; C) → H1(Y ; C) is surjective. This +contradicts the assumption that h ◦ iC,k has cyclic image. +Next, suppose that that g(X) ≥ 2 and h∗ factors through a forgetful map p : PBn(X) → X; +without loss of generality, assume that p is the projection onto the nth factor. As in the previous +paragraph, we fix a configuration C = (x1, ..., xn−1) ∈ PConfn−1(X) and consider h◦iC,k : X−C → +Y − {O}. By hypothesis, for 1 ≤ k ≤ n − 1, the induced map (h ◦ iC,k)∗ on fundamental group is +trivial and so there is a lift H : X − C → D. By the removable singularity theorem, this extends +to a holomorphic map H : X → D, but this must be constant by the maximum principle. We +conclude that h : PConfn(X) → Y − {O} factors through pn : PConfn(X) → X. The induced map +h : X → Y then has degree zero (since it misses O ∈ Y by hypothesis) and hence is constant, as +claimed. +Finally, suppose that g(X) = 1 and h∗ factors through a forgetful map PBn(X) → PB2(X); +for notational simplicity, we may therefore assume that n = 2. Let x1 ∈ X and ix1 : X − {x1} → +PConf2(X) be the natural embedding. Then the induced map h ◦ ix1 is also holomorphic. By +the removable singularity theorem (Proposition 2.3), h ◦ ix1 can be uniquely extended to a map +h ◦ ix1 : X → Y . Every holomorphic map between elliptic curves is either constant or else a covering +map. Since O has at most a single preimage, h ◦ ix1 is either an isomorphism or a constant map. +If h ◦ ix1 is a constant map, then this holds for all x1 ∈ X. Thus h factors through the forgetful +map to the first coordinate X, reducing further to the previous case n = 1, from which we conclude +that h is constant. +If h ◦ ix1 is an isomorphism, then we obtain a family of isomorphisms h ◦ ix1 : X → Y such +that h ◦ ix1(x1) = O. +Thus we obtain a global holomorphic map H : X × X → Y such that +H(x1, x1) = O. Choose a basepoint OX ∈ X, and let I = h ◦ iOX. Thus H(OX, x2) = I(x2). A +holomorphic map X → Y is uniquely determined by the map on the fundamental group and the +image of a single point. Thus H(x1, x2) = I(x2 − x1) for any other x1 since this map satisfies that +H(x1, x1) = O and is compatible on fundamental groups. +□ +Proof of Theorem 5.2. Let h : PConfn(X) → PConfm(Y ) be a holomorphic map, and fix a base- +point O ∈ Y . Let y1 ∈ Y denote the first coordinate of h(x1, . . . , xn); note that y1 is a holomorphic +function of x1, . . . , xn. Exploiting the group structure, define the normalization +hs(x1, ..., xn) = h(x1, ..., xn) − (y1, ..., y1). + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +23 +Thus, hs(x1, ..., xn) = (O, f2(x1, ..., xn), ..., fm(x1, ..., xn)) where fi : PConfn(X) → Y − {O} are +holomorphic maps. By Lemma 5.10, it follows that fp(x1, ..., xn) = Ip(xi−xj) for some isomorphism +Ip : X → Y and 1 ≤ i < j ≤ n. +To proceed, we consider the set of all isomorphisms I : X → Y . This set is a torsor for Aut(Y ), +the group of automorphisms of Y as a Riemann surface. There is a semi-direct product structure +Aut(Y ) ∼= (Y, O) ⋊ Aut(Y, O), +(2) +where (Y, O) indicates the elliptic curve structure on Y with O as the origin, and Aut(Y, O) indicates +the subgroup of Aut(Y ) fixing O, equivalently the subgroup of group automorphisms. The group +Aut(Y, O) is always finite, and always contains the negation map −id : x �→ −x. +Fix the identification I2 : X → Y associated with the first nontrivial coordinate f2 once and +for all; accordingly, we suppress I2 from the notation. For p ≥ 3, the torsor structure then gives +expressions Ip = (ϵp, αp) ◦ I2, where ϵp ∈ Y and αp ∈ Aut(Y, O). Succinctly, Ip(x) = αp(x) + ϵp, so +that we can write (after a permutation of coordinates in the domain) +fs(x1, . . . , xn) = (O, x2 − x1, α3(xi3) − α3(xj3) + ϵ3, . . . , αm(xim) − αm(xjm) + ϵm). +By hypothesis, the difference αp(xip) − αp(xjp) + ϵp − (x2 − x1) does not have O in its image, so +that by Lemma 5.10, there is an identity +αp(xip) − αp(xjp) + ϵp − (x2 − x1) = β(xk) − β(xl) + δ, +for distinct indices k, l and (β, δ) ∈ Aut(Y ), or equivalently +x1 + αp(xip) + β(xl) − (x2 + αp(xjp) + β(xk)) = δ − ϵp. +The right-hand side above is constant, and so the left-hand side must exhibit cancellation. If either +β or αp is not ±id, then this is not possible (e.g. lifting to the universal cover C of Y , the derivative +of the left-hand side would necessarily be everywhere nontrivial). Up to an exchange of indices, we +can assume that αp = β = id, so that +x1 + xip + xl − (x2 + xjp + xk) = δ − ϵp. +It follows that exactly one of the conditions ip = 2 or jp = 1 holds (they cannot both hold +simultaneously since then k = l). By a permutation of coordinates, we can assume j3 = 1 and +i3 = 3. At this point, our expression for fs has become +fs(x1, . . . , xn) = (O, x2 − x1, x3 − x1 + ϵ3, . . . , xim − xjm + ϵm), +subject to the condition that exactly one of the identities ip = 2 or jp = 1 holds for all p. By +comparing the second and pth entries, we conclude that ϵp = O for all p ≥ 3. +We claim that necessarily jp = 1 must hold for all p ≥ 4. Suppose to the contrary; then (without +loss of generality) +fs(x1, . . . , xn) = (O, x2 − x1, x3 − x1, x2 − xj4, . . . ). +By the previous analysis, comparing the third and fourth component forces j4 = 1, but then the +fourth component equals the second, a contradiction. Finally, we can re-normalize fs by translation +by x1, showing that f is given by a forgetful map as claimed. +□ + +24 +LEI CHEN AND NICK SALTER +5.3. Theorem 5.3: from higher genus to genus one. +Proof of Theorem 5.3. Write h = (f1, . . . , fm), with each fk : PConfn(X) → Y holomorphic. +Choose a basepoint O ∈ Y , and, as in Theorem 5.2, normalize h : PConfn(X) → PConfm(Y ) +by the twist +A(x1, . . . , xn)(y) = y − f1(x1, . . . , xn) +so that h = (O, f2, . . . , fm) with fk : PConfn(X) → Y − O holomorphic. By Lemma 5.10, each fk +must be constant. +□ +6. Pure configuration spaces in higher genus +In this section we consider the problem of classifying holomorphic maps h : PConfn(X) → +PConfm(Y ), where X, Y are Riemann surfaces of higher genus. We find that the situation is quite +rigid - Theorem 6.1 shows that either X = Y and the map is forgetful, or else h is induced from a +family of maps fi : X → Y which pairwise have disjoint graphs. In Theorem 6.3, we consider the +question of how many such fi can exist, which is essentially a variant of the de Franchis theorem +from classical algebraic geometry. +6.1. Classification of holomorphic maps. +Theorem 6.1. Let X and Y be compact Riemann surfaces, with g(X), g(Y ) ≥ 2. Suppose +h : PConfn(X) → PConfm(Y ) +is a nonconstant holomorphic map. +Then up to permutation of coordinates and the actions of +Aut(X) and Aut(Y ), either X = Y, n ≥ m, and h is a forgetful map PConfn(X) → PConfm(X), +or else h factors as the composition of a forgetful map PConfn(X) → X and a holomorphic map +X → PConfm(Y ). +Proof. Consider the composition pi ◦ h : PConfn(X) → Y , where pi : PConfm(Y ) → Y is the +projection onto the ith factor. +Claim 6.2. Each pi ◦ h is nonconstant and induces a surjection H1(PConfn(X); Q) → H1(Y ; Q). +Proof. We first show that each pi ◦h is nonconstant. If m = 1 and p1 is constant then h is constant, +contrary to hypothesis. If m > 1, then supposing that any pi ◦h is constant (say with value y0 ∈ Y ) +let j ̸= i be some other index, and consider pj ◦ h. If this is nonconstant, then let C ⊂ X be a +configuration of n−1 distinct points, and consider the inclusion iC,k : X−C ⊂ PConfn(X) as before. +Here k is chosen so as to make the composition pj◦h◦iC,k nonconstant. By the removable singularity +theorem (Proposition 2.3), this extends to give a nonconstant holomorphic map fC : X → Y and +by the de Franchis theorem, the space of such maps is discrete, and hence f := fC is independent +of C. Since f : X → Y is nonconstant, it has positive degree and hence is surjective. Thus, there +is some (x1, . . . , xn) ∈ PConfn(X) such that pjh(x1, . . . , xn) = y0 = pih(x1, . . . , xn), i.e. the map +fails to have codomain PConfn(Y ), contrary to assumption. +To see that (pi ◦ h)∗ is surjective, it suffices to see that f∗ : H1(X; Q) → H1(Y ; Q) is surjective, +where f is as in the above paragraph, but this is a general property of nonconstant holomorphic + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +25 +maps between compact Riemann surfaces, following from the existence of a transfer homomor- +phism f! : H1(Y ; Q) → H1(X; Q) with the property that f!f∗ = deg(f)IH1(X;Q) (see, e.g. [Tan10, +Definition 2.2] for a dual cohomological formulation). +□ +We consider the induced map (pi ◦ h)∗ : PBn(X) → π1(Y ). By [Che19, Lemma 2.5], either +(pi ◦ h)∗ factors as (pi ◦ h)∗ = f∗ ◦ pj,∗ for some f∗ : π1(X) → π1(Y ), where pj,∗ : PBn(X) → π1(X) +is induced by projecting onto the jth factor, or else (pi ◦ h)∗ has cyclic image, possibly trivial. +The latter case cannot happen, since such maps would not induce a surjection on H1, contrary to +Claim 6.2. +Defining [k] := {1, . . . , k}, we next claim that there is a function j : [m] → [n] and nonconstant +holomorphic maps αi : X → Y for which the diagram below commutes for all i: +PConfn(X) +h +� +pj(i) +� +PConfm(Y ) +pi +� +X +αi +� Y +(3) +The function j can be defined as follows: by the above, (pi ◦h)∗ : PBn(X) → π1(Y ) factors through +some projection pj,∗ : PBm(X) → π1(X); let j(i) be this j. We observe that h is determined by its +values on the collection of submanifolds (X − C)k. For k ̸= j(i), the restriction of the holomorphic +map pi ◦ h to (X − C)k is nullhomotopic by the preceding paragraph, and hence constant. Thus +pi ◦ h factors through pj(i) as required. +By construction, if j is constant, then h factors through some projection pj : PConfn(X) → X. +It remains to show that if j is nonconstant, then Y = X, n ≥ m, and h is a forgetful map up +to permutation of coordinates and the application of some α ∈ Aut(X) to each component of +PConfn(X). +We claim that if j is nonconstant, then αi1 = αi2 for all pairs of indices. Supposing to the +contrary, without loss of generality, we may take i = 1, j = 2, and j(1) = 1, j(2) = 2. Let x1 ∈ X +be given such that α1(x1) ̸= α2(x1), and let x2 ∈ X satisfy α2(x2) = α1(x1). Completing x1, x2 to +a point (x1, x2, . . . , xn) ∈ PConfn(X), we see that h(x1, . . . , xn) = (α1(x1), α2(x2), . . . , αn(xj(n))) +has a repeated entry α1(x1) = α2(x2), a contradiction. +To see that X = Y in this case, we show that the fixed map α : X → Y has degree 1. If not, let +x1 ̸= x2 ∈ X satisfy α(x1) = α(x2) = y; then h(x1, x2, . . . ) = (y, y, . . . ) has a repeated entry. Thus +X = Y , and by adjusting by α−1 if necessary, we may assume that α = id. Now it is clear that +n ≥ m, otherwise h(x1, . . . , xn) = (xj(1), . . . , xj(n)) would have a repeated entry. +□ +6.2. Bounds. To complement Theorem 6.1, we consider the problem of determining the maximal +m for which there is a nonconstant holomorphic map h : X → PConfm(Y ). This is closely related +to the effective de Franchis problem, which asks for bounds on the number of distinct holomorphic +maps fi : X → Y (known to be finite for g(X), g(Y ) ≥ 2 by the classical de Franchis theorem); +here we add the condition that the images of fi be pairwise-disjoint (pairwise have no coincidences, +in the terminology of [Tan10]). The general effective de Franchis problem is far from conclusively +resolved - Chamizo [Cha19] obtains an upper bound that is slightly larger than exponential in +g(X), while the largest known examples are linear in the genus (arising when X and/or Y has + +26 +LEI CHEN AND NICK SALTER +a large automorphism group which can be used to enlarge the number of morphism by pre/post +composition). +Here, we find that the condition that the morphisms pairwise have no coincidences imposes a +strong constraint, greatly reducing the upper bound, although in practice there is still a gap between +the upper bound of Theorem 6.3 and the largest known examples (arising when Y is equipped with +a group of free automorphisms). +Theorem 6.3. Let X, Y be compact Riemann surfaces each of genus at least 2, and let h : X → +PConfm(Y ) be a nonconstant holomorphic map. Then m ≤ 4g(X)g(Y ). +Proof. Each holomorphic map f : X → Y induces f∗ ∈ Hom(J(X), J(Y )), the induced map on +Jacobians; Martens observes [Mar78] that distinct morphisms f, g induce distinct maps f∗, g∗ so +long as g(Y ) ≥ 2. +Tanabe [Tan10, Definition 2.9], following ideas of Fuertes – Gonz´alez-Diez +[FGD93], Martens [Mar78] and ultimately Weil [Wei56], introduces a certain positive-definite inner +product ⟨·, ·⟩ on Hom(J(X), J(Y )). +According to [Tan10, Theorem 4.1], if f, g : X → Y are +holomorphic and have no coincidences (i.e. f(x) ̸= g(x) for all x ∈ X), then deg(f) = deg(g) and +cos(f∗, g∗) = g(Y )−1, where cos(v, w) := ⟨v, w⟩/∥v∥∥w∥ is defined as in any inner product space. If +h : X → PConfm(Y ) is given, the component functions h1, . . . , hm pairwise have no coincidences, +and thus determine a configuration of vectors h1,∗, . . . , hm,∗ ∈ Hom(J(X), J(Y )) where the angles +cos(hi,∗, hj,∗) = g(Y )−1 are pairwise fixed and equal. +As Hom(J(X), J(Y )) is a subgroup of Hom(H1(X; Z), H1(Y, Z)) ∼= Z4g(X)g(Y ), to prove the +claim, it suffices to show that such a configuration of vectors must be linearly independent. It +suffices to consider the associated unit vectors v1, . . . , vm. Let A be the matrix with Aij = ⟨vi, vj⟩; +linear-independence of {v1, . . . , vm} is equivalent to the nonsingularity of A. By hypothesis, +A = (1 − g(Y )−1)I + C, +where C is the matrix where every entry is given by g(Y )−1. The eigenvalues of C are 0 and +mg(Y )−1, and hence the eigenvalues of A are 1 − g(Y )−1 and 1 + (m − 1)g(Y )−1. As g(Y ) ≥ 2, +both of these are nonzero, which proves the claim. +□ +7. Distinct genus regime +In this section, we examine what happens when g(X) and g(Y ) belong to distinct genus regimes +(i.e. g = 0, 1, or g ≥ 2) - this is the setting in which there are very few holomorphic maps. The +results largely follow from basic principles, but we include the proofs for the sake of completeness. +Proposition 7.1. Let X, Y be Riemann surfaces of finite type with g(Y ) > g(X). Then every +holomorphic map h : PConfn(X) → PConfm(Y ) is constant. +Proof. Fix a configuration of distinct points C = {x1, . . . , xn−1} ⊂ X, and for 1 ≤ k ≤ n, consider +the inclusions (X − C)k �→ PConfn(X) as in the proof of Theorem 6.1. +Composing with the +projection pi : PConfm(Y ) → Y onto the ith factor, we obtain a holomorphic map fik : X −C → Y . +As g(Y ) > g(X), any such map must be constant; varying i, k, and C then shows that h itself is +constant. +□ + +HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES +27 +Proposition 7.2. Let Y be a compact Riemann surface of genus g(Y ) = 1. Then every holomorphic +map h : Confn(CP1) → Confm(Y ) is constant. +Proof. We consider the induced map on fundamental group +h∗ : Bn(CP1) → π1(Y ); +note that as the target is abelian, this factors through H1(Bn(CP1); Z). +It is well-known that +Bn(CP1) = Bn(S2) is a quotient of Bn by the word σ1σ2 . . . σn−1σn−1 . . . σ1. Thus, H1(Bn(CP1); Z) ∼= +Z/(2n − 2)Z. +As π1(Y ) ∼= Z2 is torsion-free, it follows that h∗ is the trivial map. +Applying +Lemma 5.6, we conclude that h extends to a holomorphic map +H : (CP1)n → Y. +Every such map is constant (e.g. H lifts to the universal cover C of Y and is therefore constant via +the maximum principle). +□ +We remark that in the setting of g(Y ) ≥ 2, the same argument (replacing Y with its Jacobian) +shows that any holomorphic map h : Confn(CP1) → Confm(Y ) has image contained in a fixed +linear system on Y , but this by itself is not enough to conclude that h itself is constant. +References +[AAS18] S. Antonakoudis, J. Aramayona, and J. Souto. Holomorphic maps between moduli spaces. +Ann. Inst. Fourier (Grenoble), 68(1):217–228, 2018. +[BLM83] J. Birman, A. Lubotzky, and J. McCarthy. Abelian and solvable subgroups of the map- +ping class groups. Duke Math. J., 50(4):1107–1120, 1983. +[Cha19] F. Chamizo. Morphisms and period matrices. Linear Algebra Appl., 582:103–113, 2019. +[Che19] L. Chen. +Surjective homomorphisms between surface braid groups. +Israel J. Math., +232(1):483–500, 2019. +[Che20] L. +Chen. +From +pure +braid +groups +to +hyperbolic +groups. +Preprint: +https://arxiv.org/abs/2007.15178, 2020. +[CKM19] L. Chen, K. Kordek, and D. Margalit. Homomorphisms between braid groups. Preprint: +https://arxiv.org/abs/1910.00712, 2019. +[CM20] L. Chen and A. Mukherjea. +From braid groups to mapping class groups. +Preprint: +https://arxiv.org/abs/2011.13020, 2020. +[DW07] G. Daskalopoulos and R. Wentworth. Harmonic maps and Teichm¨uller theory. In Hand- +book of Teichm¨uller theory. Vol. I, volume 11 of IRMA Lect. Math. Theor. Phys., pages +33–109. Eur. Math. Soc., Z¨urich, 2007. +[Far21] B. +Farb. +Global +rigidity +of +the +period +mapping. +Preprint: +https://arxiv.org/abs/2105.13178, 2021. +[Far22] B. Farb. Rigidity of moduli spaces and algebro-geometric constructions. To appear in +the conference proceedings of the Chern 110th birthday memorial, 2022. +[FGD93] Y. Fuertes and G. Gonz´alez D´ıez. On the number of coincidences of morphisms between +closed Riemann surfaces. Publ. Mat., 37(2):339–353, 1993. + +28 +LEI CHEN AND NICK SALTER +[FM12] B. Farb and D. Margalit. A primer on mapping class groups, volume 49 of Princeton +Mathematical Series. Princeton University Press, Princeton, NJ, 2012. +[IS88] Y. Imayoshi and H. Shiga. A finiteness theorem for holomorphic families of Riemann +surfaces. In Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), volume 11 +of Math. Sci. Res. Inst. Publ., pages 207–219. Springer, New York, 1988. +[KW17] J. Korevaar and J. Wiegerinck. Several complex variables. Korteweg-de Vries Institute +for Mathematics, 2017. +[Lin04] V. Lin. +Configuration spaces of C and CP1: +some analytic properties. +Preprint: +https://arxiv.org/abs/math/0403120, 2004. +[Mar78] H. Martens. Observations on morphisms of closed Riemann surfaces. Bull. London Math. +Soc., 10(2):209–212, 1978. +[MM09] D. Margalit and J. McCammond. Geometric presentations for the pure braid group. J. +Knot Theory Ramifications, 18(1):1–20, 2009. +[Tan10] M. Tanabe. Morphisms of closed Riemann surfaces and Lefschetz trace formula. Proc. +Amer. Math. Soc., 138(4):1295–1303, 2010. +[vK19] B. von Ker´ekj´art´o. +¨Uber die periodischen Transformationen der Kreisscheibe und der +Kugelfl¨ache. Math. Ann., 80(1):36–38, 1919. +[Wei56] A. Weil. On the theory of complex multiplication. In Proceedings of the international +symposium on algebraic number theory, Tokyo & Nikko, 1955, pages 9–22. Science Coun- +cil of Japan, Tokyo, 1956. +LC: Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD +20742 +Email address: chenlei@umd.edu +NS: Department of Mathematics, University of Notre Dame, Hurley Hall, Notre Dame, IN 46556 +Email address: nsalter@nd.edu + diff --git a/d9E_T4oBgHgl3EQf1Bzr/content/tmp_files/load_file.txt b/d9E_T4oBgHgl3EQf1Bzr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c428e62a64b37c05842d711b65704ab5f265998 --- /dev/null +++ b/d9E_T4oBgHgl3EQf1Bzr/content/tmp_files/load_file.txt @@ -0,0 +1,1412 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf,len=1411 +page_content='HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES LEI CHEN AND NICK SALTER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We give a complete classification of all holomorphic maps Confn(C) → Confm(C) provided that n ≥ 5 and m ≤ 2n extending the Tameness Theorem of Lin, which is the case m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious “effective de Franchis problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Introduction Let PConfn(X) denote the space of n ordered distinct points on a manifold X, and let Confn(X) denote the corresponding space of unordered tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If X has a complex structure, then PConfn(X) and Confn(X) inherit complex structures from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In this paper we take X, Y to be Riemann surfaces, and consider the family of problems of classifying holomorphic maps h : (P)Confn(X) → (P)Confm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As discussed in more detail below, a complete answer in full generality seems to be out of reach, involving unresolved questions related to the “effective de Franchis problem” of enumerating and bounding the set of all holomorphic maps X → Y as well as delicate group-theoretic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However, we are able to address a good portion of the general problem, especially in the ordered setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A general phenomenon recurring throughout our results is that of “twisting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Given a holo- morphic map h : (P)Confn(X) → (P)Confm(Y ), suppose a holomorphic map A : (P)Confn(X) → Aut(Y ) is given, where Aut(Y ) is the group of holomorphic automorphisms of Y , in our setting itself a complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then the twist hA of h is defined via the formula hA(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = A(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn)(h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note that the affine twist of a constant map need not be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In the case Y = C, the relevant automorphism group is the affine group Aff = {az +b | a ∈ C∗, b ∈ C}, in which case we call this an affine twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We also note that PConfn(X) admits a family of automorphisms given by permuting the coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' for ease of stating our main results, we will also consider this a kind of twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The following summarizes our main results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' see the indicated statements for precise details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1: For m ≥ 5 and n ≤ 2m, up to affine twisting, every holomorphic map h : Confn(C) → Confm(C) is either constant, the identity, or a “root map” (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Date: January 19, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='08333v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='GT] 19 Jan 2023 2 LEI CHEN AND NICK SALTER Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3: For m ≥ 2, up to a slight generalization of affine twisting, every holomorphic map h : PConfn(C) → PConfm(C) is either constant or a forgetful map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4: For n ≥ 3, up to twisting, every holomorphic map h : PConfn(CP1) → PConfm(CP1) is either constant or a forgetful map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2: Let X, Y be compact Riemann surfaces of genera g(X) = g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then X ∼= Y and up to twisting, every holomorphic map h : PConfn(X) → PConfm(Y ) is a forgetful map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3: Let X, Y be compact Riemann surfaces with g(X) ≥ 2 and g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to twisting, every holomorphic map h : PConfn(X) → PConfm(Y ) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1: Let X, Y be compact Riemann surfaces each of genus at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to twisting, either X ∼= Y and h is a forgetful map, or else h factors as the composition of a forgetful map p : PConfn(X) → X and a holomorphic map f : X → PConfm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We note that in the case m = n, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 was previously established by Lin [Lin04, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Farb [Far22, Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4] has asked for a classification of holomorphic maps PConfn(C) → PConfm(C), which is resolved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Observe that in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, a holomorphic map f : X → PConfm(Y ) is the same thing as an m-tuple f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , fm of holomorphic maps fi : X → Y with the properties that the graphs are pairwise disjoint in X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The study of the set Hol(X, Y ) of nonconstant holomorphic maps X → Y falls under the purview of the classical de Franchis theorem, which states that when g(Y ) ≥ 2, the set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Unlike its cousin the Hurwitz theorem (the case X = Y ), there is a very large gap between known upper bounds on |Hol(X, Y )| as a function of the genera g(X), g(Y ), and the lower bounds achieved by examples: by [Cha19], there is an upper bound that is slightly super- exponential in g(X), whereas, to the authors’ knowledge, there are no families of examples where the number of holomorphic maps grows faster than linearly with g(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However, we show that the extra constraint of having disjoint graphs is highly restrictive, which may be of independent interest: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3: Let X, Y be compact Riemann surfaces each of genus at least two, and let f : X → PConfm(Y ) be holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then m ≤ 4g(X)g(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Our final main result is of a slightly different flavor, considering instead the holomorphic rigid- ity properties of Confn(C) into the moduli space of curves Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' One such map is given by the hyperelliptic embedding H : Confn(C) → Mg for g = [ n−1 2 ]: H({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}) = the algebraic curve {y2 = (x − x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='(x − xn)} Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2: For n ≥ 26 and g ≤ n − 2, if h : Confn(C) → Mg is a non-constant holomorphic map of orbifolds, then h is the hyperelliptic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The holomorphic landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To put the results of this paper into better context, we describe here what is known about the totality of holomorphic maps between configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We organize the problem along two axes: first, whether the configurations are ordered or not, and secondly by the genera of the Riemann surfaces X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' An entry is left blank if we neither know of any interesting examples nor a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The “covering construction” mentioned below is the following: let p : Y → X be an unbranched covering of compact Riemann surfaces of HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 3 degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then for all n ≥ 1, there is a holomorphic map P : Confn(X) → Confdn(Y ), given by P({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn}) = p−1({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' PConf g(Y ) = 0 1 ≥ 2 g(X) = 0 Fully classified: All constant: All constant: Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 1 Fully classified: All constant: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 ≥ 2 Classified modulo Classified modulo case m = 1: understanding Hol(X, Y ) : Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 Conf g(Y ) = 0 1 ≥ 2 g(X) = 0 Partially classified: All constant: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 1 Abundant: covering construction ≥ 2 Abundant: covering construction We think that filling in the remaining entries of these tables is a worthwhile goal of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In particular, we would like to highlight the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For g(X), g(Y ) ≥ 2, does every holomorphic map h : Confn(X) → Confm(Y ) arise via the covering construction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity: holomorphic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' While it is hard to find holomorphic maps between configuration spaces of Riemann surfaces, there are many homotopy classes of continuous maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For instance, there are continuous maps Confn(C) → Confn+k(C) by adding k points “near infinity”, a “doubling map” Confn(C) → Conf2n(C) which replaces a configuration of points with two juxtaposed copies, and many more complicated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Even in the case when the induced map π1(Confn(C)) → π1(Confm(C)) factors through Z, there are many possibilities, encompassing all of the possible Nielsen-Thurston types of the associated mapping class of the m-punctured plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However, according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, none of the above can be induced by a holomorphic map, so long as we are in the range m ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Work of the first author, Kordek, and Margalit [CKM19] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1) explores this phenomenon and gives a complete classification of homotopy classes of continuous maps Confn(C) → Confm(C) in the range m ≤ 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' this is the source of the restriction m ≤ 2n in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This is accomplished by giving a classification of homomorphisms Bn → Bm, where Bn = π1(Confn(C)) is the braid group on n strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Following [CKM19], there is now a conjectural classification of all homomorphisms ρ : Bn → Bm for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To formulate it, we recall that one can also think of the braid group as Bn ∼= Mod(Dn), the mapping class group of the n-punctured disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 4 LEI CHEN AND NICK SALTER Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 (Reducibility conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 5 and m > n, every homomorphism ρ : Bn → Bm either has cyclic image or else is reducible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' there is a nonempty set of disjoint essential, non-boundary-parallel curves on the m-punctured disk invariant under ρ(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As discussed in [CKM19], the reducibility conjecture in fact implies a stronger classification of homomorphisms, predicting that all homomorphisms are recursively assembled from a small list of basic ones using a certain set of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Our methods here can be used to show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 implies a complete classification of all holomorphic maps between configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 holds, then the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 holds for all n ≥ 5 with no restriction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Prior Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Within the world of moduli spaces of complex-analytic objects, there have been various efforts to understand the relationship between their behavior in the holomorphic and smooth categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The results here expand on the work [Lin04] of Lin mentioned above, who obtained Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 in the case m = n, and also investigated the corresponding questions in the setting of configuration spaces on CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Our proof is independent, making use of the developments of [CKM19] that give a classification of maps between braid groups, as well as employing some powerful results from Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Two other results, while not bearing directly on the results of this paper, were a source of inspiration and merit mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Antonakoudis–Aramayona–Souto [AAS18] give a classification of holomorphic maps Mg → Mh for h ≤ 2g − 2, showing that the only non-constant map is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Letting Ah be the moduli space of h-dimensional, principally-polarized abelian varieties, a recent result of Farb [Far21] shows that a nonconstant holomorphic map f : Mg → Ah exists for h ≤ g iff h = g and f is the map sending a curve to its Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It is interesting to note that in both these results, every smooth map is homotopic to a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This is in contrast to the setting of this paper, where smooth maps between configuration spaces exist in relative abundance (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The main theme running through our arguments is the promotion of group- theoretic rigidity statements to the level of holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In [Che20], the first author has established a general rigidity theorem classifying homomorphisms f : PBn(X) → Λ, where X is a Riemann surface, PBn(X) := π1(PConfn(X)) denotes the space of ordered configurations of n points, and Λ is a torsion-free nonelementary hyperbolic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' When f is induced from a holomorphic map F between complex manifolds, principles of complex analysis can be used to show that F must have a specific form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In other settings, we make use of rigidity phenomena for holomorphic maps of Riemann surfaces into the moduli space of Riemann surfaces, equivalently the rigidity of holomorphically-varying families of Riemann surfaces over Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will use two such theorems in our proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, both concerning the monodromy of such families, the homomorphism ρ : π1(B) → Mod(S) from the fundamental group of the base to the mapping class group of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Specifically, we will use a result of Daskalopoulos-Wentworth [DW07] showing that the monodromy of a non-isotrivial family of curves is necessarily “rich” in a certain technical sense, and a result of Imayoshi–Shiga [IS88] stating that if two families have the same monodromy, HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 5 then they are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Inside Confn(C), there are many embedded Riemann surfaces: given distinct points Y ∈ Confn−1(C), there is an associated embedding iY : C − Y → Confn(C) of the finite-type surface C − Y into Confn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The holomorphic map Confn(C) → Confm(C) thus equips each such surface with a family of Riemann surfaces (specifically, the fiber is C punctured at m points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The monodromy of such families factors through the homomorphism Bn → Bm induced by the map on the configuration spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' we exploit the classification of [CKM19] along with the criterion of Daskalopoulos-Wentworth to rule out many possibilities, and the rigidity result of Imayoshi–Shiga gives control over the situations where a map is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The phenomenon of affine twisting appears because the target space Confm(C) is not exactly the moduli space of m points in C, but becomes a finite cover of this after erasing the action of the affine group Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' NS is supported by NSF Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' DMS-2153879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' LC is supported by NSF Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 2005409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Classification of homomorphisms between braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The basic strategy of the proof of the main results is to first understand all possible induced maps on the fundamental groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' braid groups), and then to contrast this with known rigidity results about holomorphic maps between the associated moduli spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In this section we carry out the first of these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 below is a corollary of the work of Chen–Kordek–Margalit [CKM19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To state it, we introduce the following terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A transvection of a group homomorphism f : G → H is a homomorphism ft : G → H defined via ft(g) = f(g)tℓ(g), where ℓ : G → Z is a homomorphism and t ∈ H centralizes f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A homomorphism f : G → Bn is said to be reducible if the image f(G) preserves a finite set of isotopy classes of essential non- boundary-parallel curves on the n-punctured disk (viewing Bn as the mapping class group of the punctured disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lastly, an element of Bn is said to have prefinite order if its image in Bn/Zn has finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 5 and m ≤ 2n, let ρ : Bn → Bm be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then exactly one of the following conditions hold: (1) n = m and ρ is the identity map up to transvection, (2) ρ is either reducible and nontrivial, or else has infinite cyclic image generated by a pseudo- Anosov, (3) ρ has prefinite cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note first that the statement is equivalent to the assertion that at least one of the following conditions hold: (1’) n = m and ρ is the identity map up to transvection, (2’) ρ is reducible (possibly trivial) (3’) ρ has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 6 LEI CHEN AND NICK SALTER The latter assertion follows more readily from [CKM19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1], but we will see in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 that the former organization corresponds to the classification of holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [CKM19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1] asserts that for n ≥ 5, any homomorphism ρ : Bn → B2n is a transvection of one of five “standard homomorphisms”, and [CKM19, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2] asserts that every ρ : Bn → Bm for m < 2n is one of two of the five types possible when m = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We first argue that if ρ satisfies at least one of the conditions, then so does any transvection of ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' then we will see that each standard homomorphism satisfies one of the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is nothing to check if ρ satisfies condition (1’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose now that ρ is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If ρ is the trivial homomorphism, then any transvection of ρ has cyclic image (note that if t is pseudo- Anosov, the transvection will not necessarily be reducible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If ρ is reducible and nontrivial, then ρ has a nonempty “canonical reduction system” {Ci} of disjoint essential, non-boundary-parallel curves (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [BLM83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Given a transvection ρt, the element t commutes with every element of ρ, from which it follows that t preserves {Ci} as well, so that ρt is likewise reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If ρ has cyclic image, then ρ factors through the abelianization map ℓ : Bn → Z, and hence by construction any transvection of ρ does as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We now see that each of the five “standard homomorphisms” described in [CKM19] satisfies one of the above conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Each of these is a routine verification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' for brevity’s sake we will assume familiarity with the five, as given in [CKM19, page 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The trivial homomorphism satisfies conditions (2’) and (3’), the inclusion homomorphism satisfies (1’) if m = n and (2’) if m > n, and the diagonal, flip-diagonal, and k-twist cabling maps are all reducible (condition (2’)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ In anticipation of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2, we next consider the case of homomorphisms ρ : Bn → Mod(Sg) (here Mod(Sg) denotes the mapping class group of a closed surface of genus g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let H be the hyperelliptic embedding defined in the introduction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' H∗ is then the induced homomorphism on orbifold fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We have the following result of Chen–Mukerjea [CM20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 26 and g ≤ n − 2, let h : Bn → Mod(Sg) be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to transvection, h is either trivial or H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Some results from complex analysis and Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Before considering the holomorphic theory of families of Riemann surfaces, we first mention a useful variant of the remov- able singularity theorem that we will employ throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be Riemann surfaces of finite type, each of negative Euler characteristic, and let f : X → Y be holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then f admits a holomorphic extension F : ¯X → ¯Y , where ¯X, ¯Y denote the compact Riemann surfaces associated to X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since each of X, Y have negative Euler characteristic, they are uniformized by D and hence admit complete hyperbolic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the Schwarz-Pick theorem, f is distance non-increasing in these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let x ∈ ¯X − X be given, and let γ ⊂ X be a loop encircling x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The homotopy class of γ admits representatives of arbitrarily short length, and hence the same is true for f(γ), showing that f(γ) is either null-homotopic or else encircles a puncture in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus f admits a HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 7 continuous extension F : ¯X → ¯Y , and by the usual removable singularity theorem, it follows that F is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We now turn our attention to families of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let M′ g,n be a finite orbifold cover of Mg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Imayoshi–Shiga proved the following [IS88, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4 (Imayoshi–Shiga).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let B be a Riemann surface of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If f, h : B → M′ g,n are non-constant holomorphic maps and the monodromy maps f∗ = h∗ : π1(B) → Mod(Sg,n) coincide, then f = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A homomorphism f : G → Mod(Sg,n) is said to be sufficiently large if the image contains two pseudo-Anosov elements with distinct fixed point sets in PMF(Sg,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will not need to know the precise meaning of these terms, only that a sufficiently large subgroup is not reducible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' there is no globally-invariant finite set of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Daskalopoulos-Wentworth proved the following [DW07, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5 (Daskalopoulos-Wentworth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let B be a Riemann surface of finite type and f : B → Mg,n be a non-constant holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then f∗ : π1(B) → Mod(Sg,n) is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Relations between Confm(C), PConfm(C) and M0,m,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let M0,m,1 denote the moduli space of m + 1 points on CP1, where one of the points is distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We now discuss the natural projection map πm : Confm(C) → M0,m,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let Aff be the affine group of C, which induces an action on Confm(C) and on PConfm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The preimage π−1 m (πm(X)) of a point X ∈ Confm(C) is the orbit Aff(X) ⊂ Confm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The point πm(X) represents CP1 with distinct marked points (x0, {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm}), the first of which is distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We first apply a Mobi¨us transformation to send the first point into ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Now πm(X) is represented as (∞, {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm}) where we have m distinct points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Two points (∞, {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm}) and (∞, {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ym}) represent the same points in M0,m,1 if they are holomorphically related, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' if and only if there is a biholomorphism f : C → C taking {x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xm} to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The group of holomorphic automorphisms of C is Aff, and so (∞, {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm}) and (∞, {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ym}) represent the same points in M0,m,1 if and only if there is f ∈ Aff such that f({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm}) = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Denote by PM0,m+1 the moduli space of m + 1 ordered distinct points in CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Similar to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, we have the following (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [FM12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 247 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='] for the assertion about orbifold funda- mental groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is an isomorphism of complex orbifolds PConfm(C)/ Aff ∼= PM0,m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The quotient PConfm(C) → PConfm(C)/ Aff induces the natural map PBn → PBn/Zn on orbifold fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A rotation group in Aff centered at c of order p is the cyclic subgroup generated by the element g(x) = e2π/p(x − c) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 8 LEI CHEN AND NICK SALTER Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Any finite subgroup of Aff is a rotation group centered at some c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let G < Aff be a finite subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Any orientation-preserving finite group action on R2 is cyclic by [vK19], which implies that G is generated by a single element f(x) = ax + b of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This implies that the linear coefficient a must be a pth root of unity, and then the center can be determined by the formula c = b/(1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Holomorphic maps between configuration spaces on C: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 Our first main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, gives a classification of holomorphic maps h : Confn(C) → Confm(C) in the range m ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To state the result, we must first discuss one of the possible archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Root maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There are two maps rp : Confk(C∗) → Confkp(C∗) and r′ p : Confk(C∗) → Confkp+1(C), where the first takes pth roots of the k distinct nonzero points, and the second takes the union of the pth roots of the k distinct nonzero points and {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Such maps are called basic root maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A map Confn(C) → Confkp+ϵ(C) (with ϵ ∈ {0, 1}) is called a root map if it admits a factorization Confn(C) → Confk(C∗) → Confkp+ϵ(C), where the map Confn(C) → Confk(C∗) is a twist of a constant map by some holomorphic map A : Confn(C) → C∗ and the latter map is a basic root map as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By convention, we consider the zero map Confk(C) → Conf1(C) ∼= C to be a root map of the second kind with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The main result of this section is the following rigidity result about holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We note that in the case m = n, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 was established by Lin [Lin04, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 5 and m ≤ 2n, if h : Confn(C) → Confm(C) is a non-constant holomorphic map, then h is either an affine twist of the identity map or a root map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will also consider a variant of this result, where the target is the moduli space Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' One such holomorphic map is given by the hyperelliptic embedding H : Confn(C) → Mg for g = [ n−1 2 ]: H({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}) = the algebraic curve {y2 = (x − x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='(x − xn)} Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 26 and g ≤ n − 2, if h : Confn(C) → Mg is a non-constant holomorphic map, then h is the hyperelliptic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 does not mention affine twists, in contrast to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The reason is that affine twisting gives equivalent points in the moduli space of the punctured sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' the hyperelliptic embedding does not depend on the actual location of the points but only on the holo- morphic structure of the punctured sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Taking the orbifold structure on Mg into account, it is possible to distinguish between two maps, one being a sort of twist of the other by the hyperelliptic involution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 9 To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, we divide into cases according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : Confn(C) → Confn(C) be a holomorphic map, and let h∗ be the map induced by h on fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 asserts that there are three possibilities for h∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' we consider each in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' h∗ is the identity map up to transvection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By hypothesis, h∗ : Bn → Bn has the form h∗(σ) = σζℓ(σ), where ζ ∈ Zn and ℓ : Bn → Z is the signed word-length homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note that Zn ≤ PBn, and so h∗(PBn) ≤ PBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus h admits a lift ˜h : PConfn(C) → PConfn(C), and by hypothesis, ˜h∗ : PBn → PBn is also the identity map up to a transvection by an element in the center of Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let π′ n : PConfn(C) → PConfn(C)/ Aff be the natural quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7, the space PConfn(C)/ Aff is a finite orbifold cover of M0,n+1 and πorb 1 (PConfn(C)/ Aff) ∼= PBn/Zn, with π′ n,∗ the natural quotient PBn → PBn/Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let Y = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn−1) be an n − 1-tuple of ordered distinct points in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is an embedding iY : C − Y → PConfn(C) such that iY (x) = (Y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' From the above description of π′ n,∗, we see that both π′ n ◦ ˜h ◦ iY and π′ n ◦ iY induce the same map on the orbifold fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4, it follows that π′ n ◦ ˜h ◦ iY = π′ n ◦ iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Therefore ˜h(Y, x) = g(Y, x)(Y, x) for some function g : PConfn(C) → Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The map g : PConfn(C) → Aff is holomorphic and factors through Confn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) ∈ PConfn(C), we write ˜h(X) = (˜h1(X), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , ˜hn(X)) with ˜hi(X) = g(X)(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Writing g(X)(z) = az + b, we can solve for a, b using the fact that for any pair of distinct indices i, j, the map g(X) takes xi to ˜hi(X) and xj to ˜hj(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This yields the expressions a = ˜hi(X) − ˜hj(X) xi − xj , b = xj˜hi(X) − xi˜hj(X) xi − xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Observe that since ˜h is holomorphic, these expressions vary holomorphically with X, and that they are independent of i and j by the assumption that ˜hi(X) = g(X)(xi) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Thus in this case, we have shown that the original map h is the affine twist of the identity map by the holomorphic map g : Confn(C) → Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 10 LEI CHEN AND NICK SALTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The image of h∗ is reducible or else infinite cyclic pseudo-Anosov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Given a set Y = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn−1} of n − 1 distinct points in C, there is an embedding iY : C − Y → Confn(C) such that iY (x) = Y ∪ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Composing iY with the natural projection πm : Confm(C) → M0,m,1, we obtain a holomorphic map hY := πm ◦ h ◦ iY from the finite-type Riemann surface C − Y into M0,m,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose first that h∗ has cyclic image generated by a pseudo-Anosov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As the kernel of πm,∗ : π1(Confm(C)) → πorb 1 (M0,m,1) is contained in the center Zn ≤ Bn = π1(Confm(C)) by [FM12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 247 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' ], it follows that hY,∗ also has cyclic image generated by a pseudo-Anosov element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5, this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Likewise, in the case when h∗ has reducible image, the induced map hY ∗ also has nontrivial reducible image, which is again prohibited by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' h∗ has prefinite cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will show that in this case h is a root map up to an affine twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The map πm ◦ h is a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We continue to consider hY : C − Y → M0,m,1 as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If hY were not constant, then it would determine a locally nontrivial family, which would then have sufficiently large monodromy by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5, contrary to hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Therefore πm ◦ h({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}) = h{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=',xn−1}(xn), which does not depend on xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This implies that it does not depend on any coordinate, by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Denote the image of πm ◦ h by X ∈ M0,m,1, and let X0 = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xm} be a representative of X in Confm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7, the image π−1 m (X) is given as the orbit Aff(X0), and by the orbit-stabilizer theorem, there is an isomorphism of complex manifolds, defining G := Stab(X0), Aff(X0) ∼= Aff /G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus h is given as a holomorphic map h : Confn(C) → Aff /G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8, G is a rotation group of order p with center c and X0 is a G-invariant subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By applying an affine twist, we can assume that the center point of G is c = 0, so that G = µp, the pth roots of unity, and X0 consists of all pth roots of some fixed subset Y0 = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', yk} ⊂ C∗, possibly along with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is an isomorphism of complex manifolds Aff /G ∼= C∗ × C which sends the coset of z �→ az + b to (ap, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Under this identification, let A : Confn(C) → C∗ be given as the first coordinate of the map h : Confn(C) → Aff /G, and likewise define B : Confn(C) → C as the second coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The map of configuration spaces h : Confn(C) → Aff(X0) ⊂ Confkp+ϵ(C) HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 11 is now seen to be an affine twist of a root map, factoring as shown below: Confn(C) (cY0)A � Confk(C∗) rp � Confkp+ϵ idB � Confkp+ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 asserts that every homomorphism ρ : Bn → Bm with 5 ≤ n < m is reducible, or else has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The arguments of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3 then apply to extend the classification to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose h : Confn(C) → Mg is holomorphic, with n, g satisfying the bounds of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We execute the same strategy as before, now following the cases delineated by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h∗ : Bn → Mod(Sg) is trivial, then we imitate the proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4 to see that h is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If the image of h∗ is cyclic, it is not sufficiently large, and so cannot arise from a holomorphic map by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It remains to consider the case where h∗ is a transvection of H∗ (the map induced by the hyperelliptic embedding H : Confn(C) → Mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note that the image of H∗ is the hyperelliptic mapping class group, which has centralizer Z/2Z, generated by the hyperelliptic involution ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus, there is exactly one nontrivial transvection of H∗, by ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h∗ : Bn → Mod(Sg) is given by H∗ , we follow the argument of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 to see that on each submanifold of the form C − Y ⊂ Confn(C), the maps h and H coincide, hence coincide globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h∗ = Hι ∗, the same argument shows that there is at most one holomorphic map in the homotopy class of h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' it remains to give a construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As a map of sets, the underlying map Hι is the same as H, sending the configuration {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn} to the hyperelliptic curve y2 = (x − x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' (x − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As a map of complex orbifolds, the two differ in how the marking is specified on the universal covers: Hι arises from H by precomposition with an affine twist by some holomorphic map ∆ : Confn(C) → C∗ that induces the abelianization ∆∗ : Bn → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity of holomorphic maps between PConfn(C) and PConfn(CP1) In this section, we will give a new proof of [Lin04, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5] (classifying holomorphic maps F : PConfn(C) → C − {0, 1}) and use this to give a complete classification of holomorphic maps between PConfn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To state the results, we first define some basic ingredients: the maps sr, cr, RQ, and NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The maps sr and cr are holomorphic maps PConfn(C) → C − {0, 1}, both arising from the cross-ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The “simple ratio” sr(i, j, k) : PConfn(C) → C − {0, 1} is given by sr(i, j, k)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = xk − xi xj − xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The second map is the cross-ratio cr(i, j, k, l) : PConfn(C) → C − {0, 1}, given by cr(i, j, k, l)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = xl − xi xl − xk xj − xk xj − xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Another interpretation of the maps sr(i, j, k) and cr(i, j, k, l) are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) ∈ PConfn(C), there is a unique element in A ∈ Aff such that A(xi) = 0, A(xj) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We define the 12 LEI CHEN AND NICK SALTER value sr(i, j, k)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = A(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Likewise, for (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='xn) ∈ PConfn(C), there is a unique element in A ∈ PSL(2, C) such that A(xi) = 0, A(xj) = 1, A(xk) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We define the value cr(i, j, k, l)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = A(xl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For later use, we record the following properties of cr and sr under permutation of indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' these can be checked by direct inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There are identities sr(j, i, k) = 1 − sr(i, j, k), sr(j, k, i) = 1/sr(i, j, k), and cr(i, j, k, l) = cr(k, l, i, j), cr(j, i, k, l) = 1 − cr(i, j, k, l), cr(k, j, i, l) = 1/cr(i, j, k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will give a new proof of the following result of Lin [Lin04, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='15], which has the virtue of being somewhat shorter than the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 3, any non-constant holomorphic map f : PConfn(C) → C − {0, 1} is given by either sr(i, j, k) or cr(i, j, k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Using this, we will classify holomorphic maps h : PConfn(C) → PConfm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We define two basic ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The first is the map RQn : PConfn(C) → PConfn−1(C), given by the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) ∈ PConfn(C), note that cr(x1, x2, x3, z), viewed as a M¨obius transformation, sends x1, x2, x3 to 0, 1, ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then define RQn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = (0, 1, cr(x1, x2, x3, x4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', cr(x1, x2, x3, xn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The terminology comes from the fact that the classical “resolving the quartic” map PConf4(C) → PConf3(C) is affine equivalent to RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The second basic map we consider is the “normalized inversion” NIn : PConfn(C) → PConfn(C), given by the formula NIn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = � 0, 1 x2 − x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , 1 xn − x1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We can now state the main results of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(C) → PConfm(C) be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to permutation of coordinates in the source and target and affine twisting, h is a composition of one or more of the following: h = c a constant map, h = RQn, h = NIn, h = πm n the forgetful map (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) �→ (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xm) (including m = n, the identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In particular, there is no holomorphic map PConfn(C) → PConfm(C) for 1 < n < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(CP1) → PConfm(CP1) be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Up to a permutation on the domain and twisting by a holomorphic map A : PConfn(CP1) → PSL(2, C), either F is a constant map or a forgetful map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity of PConf n(C) → C − {0, 1} and PConfn(CP1) → C − {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We begin with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Any holomorphic map f : Aff → C − {0, 1} is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Likewise, any holomorphic map f : PSL(2, C) → C − {0, 1} is a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Given f : Aff → C − {0, 1} holomorphic, there is an induced holomorphic map F between the universal covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As a complex manifold, Aff is isomorphic to C∗ × C and hence its universal cover is C2, while the universal cover of C − {0, 1} is D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' by Liouville’s theorem, it follows that F, and hence f, must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Now suppose holomorphic f : PSL(2, C) → C − {0, 1} is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The action by M¨obius transfor- mation gives a fiber bundle Aff → PSL(2, C) → CP1 whose fibers are holomorphic submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the previous paragraph, the restriction of f to each fiber must be constant, and hence f factors through the base space CP1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' the result now follows via the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We will also make use of the following result of [Che20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4]: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 3 and m ≥ 2, any homomorphism f : PBn → Fm factors through a forgetful map p∗ : PBn → PBm with m ∈ {3, 4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' in the case m = 4, there is a further factorization through RQ4,∗ : PB4 → PB3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Our other main tool will be the following lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 3, let f : PConfn(C) → C − {0, 1} be holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If f∗ : PBn → F2 factors through a forgetful map p∗ : PBn → PBm, then f factors through a forgetful map p : PConfn(C) → PConfm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose that p∗ forgets the ith strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Fixing a configuration of n − 1 points Xi := x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , �xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn, the restriction of f to the subspace C − Xi ⊂ PConfn(C) (where only the ith coordinate varies) then lifts to the universal cover D of C − {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However by the removability singularity theorem, this extends to give a holomorphic map F : C → D, which must be constant by Liouville’s theorem, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Base case 1: n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For τ ∈ C − {0, 1} fixed, define hτ : Aff → C − {0, 1} by hτ(az + b) = h(b, a + b, aτ + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5, each hτ is constant, with value c(τ) ∈ C − {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since h is holomorphic, so too is the induced map c : C − {0, 1} → C − {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 14 LEI CHEN AND NICK SALTER By the Great Picard Theorem, the points 0, 1, ∞ are at worst poles of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus, c extends to a holomorphic map ˆc : CP1 → CP1, a rational function so that ˆc−1({0, 1, ∞}) ⊂ {0, 1, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If this containment is strict then ˆc has degree zero and hence c is constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' otherwise we can assume ˆc restricts as the identity on the set {0, 1, ∞}, and that ˆc has no other singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Writing ˆc(z) = p(z)/q(z), it follows that p(z) is divisible by z, there are no other zeros of p, that q has no zeroes, and that ˆc(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus ˆc(z) = z = sr(1, 2, 3)(0, 1, z), and as was discussed above, this implies that h = sr(1, 2, 3) over the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Base case 2: n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, h∗ : PB4 → F2 either factors through a forgetful map PB4 → PB3 or else through (RQ4)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In the former case, we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7 to reduce to the case n = 3, and so we assume that h∗ factors through (RQ4)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For τ ∈ C − {0, 1} fixed, define the subset Xτ ⊂ PSL(2, C) via Xτ := {A ∈ PSL(2, C)|A(0) ̸= ∞, A(1) ̸= ∞, A(∞) ̸= ∞, A(τ) ̸= ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By construction, Xτ((0, 1, ∞, τ)) = cr(1, 2, 3, 4)−1(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Like in the case n = 3, we define the holomorphic map hτ : Xτ → C − {0, 1} by hτ(A) = h(A(0, 1, ∞, τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We claim that hτ extends to a holomorphic map hτ : PSL(2, C) → C = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To see this, we first claim that hτ lifts to holomorphic map Hτ : Xτ → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This follows from the assumption that h∗ factors through (RQ4)∗ = (0, 1, cr(1, 2, 3, 4))∗ and the characterization Xτ((0, 1, ∞, τ)) = cr(1, 2, 3, 4)−1(τ), which shows that hτ is homotopic to a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus Hτ is a bounded holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The space Xτ is the complement of hypersurfaces in the smooth complex variety PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the higher-dimensional removable singularity theorem [KW17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2], it follows that Hτ, and hence hτ, can be extended to PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5, each map hτ : PSL(2, C) → C−{0, 1} is a constant map hτ = c(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We can now apply the argument of the last two paragraphs in the case n = 3 to conclude that c : C − {0, 1} → C − {0, 1} must be an automorphism, and hence h = cr(1, 2, 3, 4) up to an automorphism of C − {0, 1} as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Inductive step: n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We proceed by induction on n, taking n = 4 as the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The inductive step follows by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ The following lemma gives a useful normalization of a map between configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8 (Normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Every holomorphic map h : PConfn(C) → PConfm(C) is equivalent up to affine twisting to a unique holomorphic map hs : PConfn(C) → PConfm(C) such that the first two coordinates of hs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) are 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let p12 : PConfm(C) → PConf2(C) be the projection onto the first two coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then p12 ◦ h is a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let A : PConfn(C) → Aff be characterized by the condition that HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 15 A(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn)(p12(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn)) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Via the identification of complex manifolds PConf2(C) ∼= Aff, it follows that A is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Defining hs as the affine twist hA, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We now discuss the common values of cr(i, j, k, l) and sr(i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' (1) The function sr(1, 2, 3) − sr(i, j, k) has no zero in PConfn(C) if and only if (i, j, k) is one of (1, 2, p), (1, p, 3), (p, 2, 3) for p ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' (2) The function cr(1, 2, 3, 4) − sr(i, j, k) always has a zero in PConfn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' (3) The function cr(1, 2, 3, 4) − cr(i, j, k, l) has no zero in PConfn(C) if and only if (i, j, k, l) is one of (p, 2, 3, 4), (1, p, 3, 4), (1, 2, p, 4), (1, 2, 3, p) for p ̸= 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We will prove (3), the proofs of (1),(2) being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The function cr(1, 2, 3, 4)−cr(i, j, k, l) has a zero if and only if the expression x4 − x1 x4 − x3 x2 − x3 x2 − x1 = xl − xi xl − xk xj − xk xj − xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' (1) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If {i, j, k, l} = {1, 2, 3, 4}, we can see by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 that cr(i, j, k, l) is the image of cr(1, 2, 3, 4) under some element of the dihedral group D3 acting on C−{0, 1} via z �→ 1−z, z �→ 1/z, and it is straightforward to verify that no element of this group acts freely, implying that the equation cr(1, 2, 3, 4) = cr(i, j, k, l) admits a solution in PConfn(C) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We therefore assume that index i is not in {1, 2, 3, 4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' without loss of generality, set i = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If we view the equation as a function of x5 (fixing other points), it will have a unique solution z5 = z5(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , � x5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) in CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If this is not a valid solution in PConfn(C), then z5 must either be one of the xq for q ̸= 5 or else ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As z5 varies continuously with the parameters xi, i ̸= 5, if (1) has no solutions, then there must be an identity z5 = xq for q ̸= 5 or else z5 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In case z5 = ∞, (1) simplifies to the ostensible identity in C(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) x4 − x1 x4 − x3 x2 − x3 x2 − x1 = xj − xk xl − xk , which is readily seen to not hold regardless of the values of j, k, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If z5 = xq for some q ̸= 5, a similar analysis shows that the only way (1) can be satisfied is if (p, j, k, l) = (1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We now discuss similar results for PConfn(CP1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Every non-constant holomorphic map f : PConfn(CP1) → C − {0, 1} is of the form cr(i, j, k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let f : PConfn(CP1) → C − {0, 1} be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let En : PConfn(C) → PConfn(CP1) be the natural embedding, which is a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2, f ◦ En is a cross ratio map, either some sr or cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since En has dense image, the map f is uniquely determined by f ◦ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If f ◦ En = sr(i, j, k), then letting xi approach ∞, the image under f ◦ En approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus sr(i, j, k) cannot extend to PConfn(CP1) and so f ◦ En must be a four-term cross ratio map cr(i, j, k, l), each of which extends to PConfn(CP1) via the same formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Similarly, we have the following counterpart to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 16 LEI CHEN AND NICK SALTER Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The equation cr(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 3, 4) = cr(i, j, k, l) has no zero in PConfn(CP1) if and only if (i, j, k, l) is one of (p, 2, 3, 4), (1, p, 3, 4), (1, 2, p, 4), (1, 2, 3, p) for p ̸= 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity of PConf n(C) → PConfm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(C) → PConfm(C) be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We can assume the first two coordinates of h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) are (0, 1) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The remaining coordinates of h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) are functions f3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', fm : PConfn(C) → C − {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since the image lies in PConfm(C), the expressions fi − fj = 0 have no solutions in PConfn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Case 1: f3 = sr(i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9, if f3 = sr(i, j, k), then all other fi’s are also simple ratios sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Up to a permutation of coordinates on the domain, we assume that f3 = sr(1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9, f4 is either sr(1, 2, 4), sr(1, 4, 3) or sr(4, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, by applying an affine twist and/or NIn, we can assume f4 = sr(1, 2, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We then claim that fk = sr(1, 2, k) for all k ≥ 3, up to permutation on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9: the only tuple (i, j, k) that differs from both (1, 2, 3) and (1, 2, 4) in a single entry is (1, 2, p) for some other p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then we can apply a permutation such that p = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Applying the affine twist (x2 − x1)z + x1 then shows that h is affine-equivalent to the forgetful map πm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Case 2: f3 = cr(i, j, k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9, if f3 = cr(i, j, k, l), then all other fi’s are also four-term cross ratio functions cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Applying a permutation on the domain, we assume that f3 = cr(1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9, f4 is a cr function indexed by a tuple where exactly one entry of (1, 2, 3, 4) has been replaced by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As in Case 1, we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 so that possibly after an affine twist and/or an application of NIn, we can assume that f4 = cr(1, 2, 3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Arguing as in Case 1, it follows that fk = cr(1, 2, 3, k + 1) for 3 ≤ k ≤ m up to permutation on the domain, and visibly then h = πm n ◦ RQn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity of PConfn(CP1) → PConfm(CP1) for m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(CP1) → PConfm(CP1) be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8, we can assume that the first three coordinates of all images of h are 0, 1, ∞ after applying a twist A : PConfn(CP1) → PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' All the holomorphic maps PConfn(CP1) → C − {0, 1} are given by cross ratio functions cr(i, j, k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By a similar argument as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2, it can be shown that h is a forgetful map (in this setting, the maps RQ, NI arise as twists and do not need to be considered separately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Pure configuration spaces in genus one Here we consider the setting of holomorphic maps between pure configuration spaces of Riemann surfaces where the target Y has genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In outline, the proof is the same as that of the previous section - we construct a normalization fs of f : PConfn(X) → PConfm(Y ), relative to which the component functions are tightly constrained enough to be completely classifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 17 the previous section this classification of component functions (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2) relied on the group- theoretic classification Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, which was proved in [Che20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The results of [Che20] also treat the case where g(X) ≥ 2 (this is recalled below in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9), but we must establish the case g(X) = 1 ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It is interesting to note that whereas all three results have a similar flavor (asserting that maps from configuration spaces to hyperbolic groups factor through forgetful maps), the precise nature of these forgetful maps reflect the different genus regimes (for g(X) = 0 we forget all but three or four, for g(X) = 1 we forget all but two, and for g(X) ≥ 2 we forget all but one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Any homomorphism PBn(T 2) → Fm either factors through PB2(T 2) or has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Using this, we will establish the main results of the section: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be compact Riemann surfaces with g(X) = g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to permutation of coordinates and twisting, any holomorphic map h : PConfn(X) → PConfm(Y ) is induced by an isomorphism X → Y and a forgetful map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be compact Riemann surfaces with g(X) ≥ 2 and g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to twisting, any holomorphic map h : PConfn(X) → PConfm(Y ) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We observe that a twist A : PConfn(X) → Aut(Y ) is essentially the same thing as the case m = 1 (see Equation (2) below), and so Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3 really gives a reduction to the case m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It remains to give a classification of holomorphic h : PConfn(X) → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Certainly one possibility is to factor through a forgetful map PConfn(X) → X, but there are more complicated examples, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For instance, if X admits Y as one of its isogeny factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' the Jacobian Jac(X) admits a finite cover isomorphic to a product Y × A), then it is possible to use the Abel-Jacobi map to induce a map PConfn(X) → Y this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We leave the problem of classifying m = 1 for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1: from torus braid groups to free groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We first discuss some facts about the group PBn(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Given a disk D2 embedded in T 2, we obtain an embedding of the pure braid group i : PBn < PBn(T 2) as a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We now introduce a generating set for PBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Recall that PBn is the pure mapping class group of the disk with n marked points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', π0(Diff(Dn)), where Diff(Dn) is the group of diffeomorphisms of D fixing n marked points pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Consider the disk with n marked points Dn in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' a124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 18 LEI CHEN AND NICK SALTER Let L be a line segment below all the marked points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', Ln be line segments connecting x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn to L as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Similarly, let U be a line segment above all marked points and let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', Un be line segments connecting x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn to U as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', n}, let aI (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' a′ I) be the boundary curve of the tubular neighborhood of � i∈I Li ∪ L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' � i∈I Ui ∪ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let TI (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' T ′ I) be the Dehn twist about aI (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' a′ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Figure 3 gives an example of a curve representing a124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The following proposition about generating sets of PBn is classical and can be found in [MM09, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Both {Tij|1 ≤ i < j ≤ n} and {T ′ ij|1 ≤ i < j ≤ n} are generating sets for PBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' An embedding Dn �→ T 2 induces an injection PBn �→ PBn(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We fix one such embedding, and use this to identify TI and T ′ I with elements of PBn(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For n ≥ 2, any Dehn twist Tc about a simple closed curve c surrounding pi, pj is conjugate in PBn(T 2) to Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The Dehn twists Tc and Tij can be viewed as point-pushing maps about based loops c+ and c+ ij starting from pi around pj following c and cij respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The point-pushing subgroup based at pi inside PBn(T 2) is a free group π1(T 2 −{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ˆxi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The loops c+ and c+ ij are conjugate in π1(T 2 − {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ˆxi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}), since as unbased loops they both encircle the puncture xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We now prove the following statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' in anticipation of later use, we formulate it for a general compact surface, not just tori (the Dehn twists Tij retain the meaning given above, as loops of the ith point around the jth inside some topological embedding Dn �→ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let S be a compact surface and let en : PConfn(S) → PConfn−1(S)×S be the natural embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The kernel of the induced map en∗ on the fundamental groups is normally generated by {T1n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', Tn−1,n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Endow S with a complete Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Define W := PConfn−1(S)×S −PConfn(S), noting that W is a union W = � Wi of n − 1 disjoint embedded copies of PConfn−1(S), according to the unique i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , n − 1} such that xn = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let N(W) be a tubular neighborhood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' likewise there is a decomposition N(W) = � N(W)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let p : N(W) → W be the projection taking (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) ∈ N(W)i to (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn−1, xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The map p is a homotopy equivalence because it extends to a deformation retraction by radially contracting xn in to the associated xi, and the set N(W)◦ i := N(W)i − Wi has the homotopy type of a S1-bundle over Wi, with bundle map given by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The use of van Kampen to obtain the result is complicated by the fact that W and N(W) are disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Accordingly, define Y0 := PConfn(S) and for i ≥ 1, Yi = Yi−1 � N(W)◦ i N(W)i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' let qi ∈ N(W)◦ i be a basepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 19 Applying the van Kampen theorem to the above decomposition of Yi yields the following pushout diagram: π1(N(W)◦ i , qi) fi,∗ � pi,∗ � π1(Yi−1, qi) ei,∗ � π1(N(W)i, qi) ∼= π1(Wi, qi) � π1(Yi, qi) Therefore the kernel of ei,∗ is normally generated by the image of ker(pi,∗) under fi,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Observe that ker(pi,∗) is normally generated by the S1-fiber, which corresponds under fi,∗ to the Dehn twist Ti,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Inductively, we see that the kernel of the inclusion map π1(PConfn(S), qi) → π1(Yi, qi) is normally generated by the Dehn twists Tj,n for j ≤ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' as Yn−1 = PConfn−1(S) × S, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Let Pi := π1(T 2 − {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', ˆxi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn}) < PBn(T 2) be the point-pushing subgroup based at the point xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The group PBn(T 2) is generated by elements in Pi for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', n} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The subgroup Pn is the kernel of the homomorphism PBn(T 2) → PBn−1(T 2) induced by forgetting the last point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The lemma is deduced by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We will make use of the result [Che20, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5]: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', Gn be groups and let Γ < G1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' × Gn be a finite index subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let πi : Γ → Gi be the ith projection map and let Γi be the image of πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let Λ be a torsion-free, non-elementary hyperbolic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then any homomorphism φ : Γ → Λ either factors through πi or its image is a cyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We now start the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let ρ : PBn(T 2) → Fm be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Case 1: n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose that one of ρ(T12), ρ(T13) or ρ(T23) is not trivial (we will consider the situation where all three vanish below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We assume without loss of generality that ρ(T12) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let us consider the centralizer of T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The point push of the third point gives the embedding P3 : π1(T 2 − {x1, x2}) → PB3(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Denote the based loop at x3 corresponding to T12 as c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then P3(c) = T −1 12 T123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The loop c = [a, b] is a commutator in π1(T 2 − {x1, x2}), where a, b are standard generators for π1(T 2, x3) disjoint from T12, and so crucially, both P3(a) and P3(b) commute with T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As the centralizer of any nontrivial element of Fm is cyclic, it follows that ρ([P3(a), P3(b)]) = ρ(T −1 12 T123) = 1 and hence ρ(T123) = ρ(T12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the same logic, either ρ(T23) = 1 or ρ(T23) = ρ(T123), and likewise either ρ(T13) = 1 or else ρ(T13) = ρ(T123).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the lantern relation, T12T23T13 = T123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If either ρ(T23) or ρ(T13) is trivial, this implies that both are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The other possibility is that all three of ρ(T12), ρ(T23), ρ(T13) equal ρ(T123), which implies that ρ(T123)2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' as Fm is torsion-free, this implies ρ(T123) = 1, contrary to the assumption that ρ(T12) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We conclude that ρ(T13) = ρ(T23) = 1, so by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, it follows that ρ factors through the product PB2(T 2) × T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8, we conclude that ρ either factors through PB2(T 2) or else has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Note that the argument of this paragraph covers the case where all of ρ(T12), ρ(T13), ρ(T23) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 20 LEI CHEN AND NICK SALTER Case 2: n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As above, we can assume without loss of generality that ρ(T12) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, the restriction ρ|PB4 either factors through a forgetful map to PB3 or RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In the first case, ρ(T14) = ρ(T24) = ρ(T34) = 1, which implies that ρ factors through e4∗ by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8, either the image is cyclic or else we have reduced to the case n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose then that ρ factors through RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' On the level of B4, the map RQ4 sends the standard generators σ1, σ2 to themselves and σ3 to σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus ρ(T34) = ρ(σ2 3) = ρ(σ2 1) = ρ(T12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Subcase 1: ρ(T23) does not commute with ρ(T12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We first claim that under this assumption, ρ(T123) = ρ(T1234) = ρ(T234) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To see this, observe that each such element commutes with both ρ(T12) = ρ(T34) and ρ(T23) and hence lies in the intersection of their centralizers, which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' γ γ3 γ4 γ34 ¯c Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The configuration of curves used in Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Referring to the left side of Figure 4, we define the elements S, S3, S4, S34 ∈ Mod(T 2, {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , x4}) as the Dehn twists about the curve γ with the corresponding subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then there is a lantern relation of the form T34S4S3 = SS34 We rearrange, expressing each side as an element of PB4(T 2): T34S4S−1 = S−1 3 S34 Since S4S−1 commutes with T12 and T23, it follows that ρ(S4S−1) = 1 since ρ(T12) and ρ(T23) do not commute by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus ρ(S−1 3 S34) = ρ(T34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is a second lantern relation T ′ 24T34T23 = T234, from which we conclude ρ(T ′ 24) = ρ(T34T23)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let c be the curve given as a regular neighborhood of the arc ¯c indicated on the right side of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By disjointness, Tc commutes with T ′ 24 and S−1 3 S34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However ρ(S−1 3 S34) = ρ(T34) = ρ(T12) and ρ(T ′ 24) = ρ(T34T23)−1 = ρ(T12T −1 23 ) don’t commute, which implies that ρ(Tc) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5, ρ(Tc) is conjugate to ρ(T13), implying that ρ(T13) = 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' However, there is a lantern relation T12T23T13 = T123, and since ρ(T123) = 1, this implies that ρ(T13) = ρ(T12T23)−1 ̸= 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 21 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Subcase 2: generators for P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Subcase 2: ρ(T23) commutes with ρ(T12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In this case, ρ(PB4) is in the centralizer of ρ(T12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We now show that ρ(PB4(T 2)) is in also in the centralizer of ρ(T12), and hence ρ has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='7, it suffices to show that ρ(Pi) is in the centralizer of ρ(T12) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The subgroup P1 is the fundamental group of T 2 − {x2, x3, x4} and is generated by point-push maps about the paths indicated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Expressing a point push as the product of Dehn twists about the boundary components of a regular neighborhood, one observes that each such curve is disjoint from either a12 or a34 with the exception of T23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By hypothesis, ρ(T23) commutes with ρ(T12), and it follows that ρ(P1) commutes with ρ(T12) = ρ(T34) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A similar analysis of the generating sets for P2, P3, P4 show that the same result holds, completing the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Case 3: general n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This proceeds as in the first step of the case n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If all ρ(Tij) = 1, then the image is abelian and hence cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Otherwise, ρ(T12) ̸= 1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, the restriction ρ|PBn factors through a forgetful map, so that (without loss of generality) ρ(Ti,n) = 1 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, it follows that ρ factors through en,∗, and applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='8 implies that either ρ has cyclic image or else we have reduced to the case m = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2: holomorphic maps between configuration spaces on elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To begin, we recall the classification of maps f : PBn(X) → Fm established (in greater generality) in [Che20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X be a compact Riemann surface with g(X) ≥ 2, and let Fm be a free group of rank m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then every homomorphism f : PBn(X) → Fm either has cyclic image or else factors through a forgetful map pi : PBn(X) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Using this and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, we establish the following analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be compact Riemann surfaces, with g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Choose a base point O ∈ Y , thereby endowing Y with a group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(X) → Y − {O} be a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then either h is constant or there is an isomorphism I : X ∼= Y , and h = I(xi − xj) for some i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We proceed by induction, the case n = 1 being trivial (as every holomorphic map h : X → Y − {O} is constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Now let n ≥ 2 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9, either h∗ 22 LEI CHEN AND NICK SALTER has cyclic image or h∗ factors through a forgetful map: either PBn(X) → PB2(X) in the case of g(X) = 1 or PBn(X) → X in the case of g(X) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Firstly, suppose that h∗ has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Fix a configuration C = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn−1) ∈ PConfn−1(X), and for 1 ≤ k ≤ n, consider the natural embedding iC,k : X − C ⊂ PConfn(X) where all but the kth coordinate is fixed at C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' note this is a holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3, h ◦ iC,k extends to a holomorphic map H : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If H is not constant, then Hodge theory asserts that H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' C) is spanned by holomorphic differentials and their conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since nonzero holomorphic differentials pull back to nonzero holomorphic differentials along holomorphic maps, it follows that H∗ : H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' C) → H1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' C) is injective, and dually H∗ : H1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' C) → H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' C) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This contradicts the assumption that h ◦ iC,k has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Next, suppose that that g(X) ≥ 2 and h∗ factors through a forgetful map p : PBn(X) → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' without loss of generality, assume that p is the projection onto the nth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As in the previous paragraph, we fix a configuration C = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn−1) ∈ PConfn−1(X) and consider h◦iC,k : X−C → Y − {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By hypothesis, for 1 ≤ k ≤ n − 1, the induced map (h ◦ iC,k)∗ on fundamental group is trivial and so there is a lift H : X − C → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the removable singularity theorem, this extends to a holomorphic map H : X → D, but this must be constant by the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We conclude that h : PConfn(X) → Y − {O} factors through pn : PConfn(X) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The induced map h : X → Y then has degree zero (since it misses O ∈ Y by hypothesis) and hence is constant, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Finally, suppose that g(X) = 1 and h∗ factors through a forgetful map PBn(X) → PB2(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' for notational simplicity, we may therefore assume that n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let x1 ∈ X and ix1 : X − {x1} → PConf2(X) be the natural embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then the induced map h ◦ ix1 is also holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the removable singularity theorem (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3), h ◦ ix1 can be uniquely extended to a map h ◦ ix1 : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Every holomorphic map between elliptic curves is either constant or else a covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since O has at most a single preimage, h ◦ ix1 is either an isomorphism or a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h ◦ ix1 is a constant map, then this holds for all x1 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus h factors through the forgetful map to the first coordinate X, reducing further to the previous case n = 1, from which we conclude that h is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h ◦ ix1 is an isomorphism, then we obtain a family of isomorphisms h ◦ ix1 : X → Y such that h ◦ ix1(x1) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus we obtain a global holomorphic map H : X × X → Y such that H(x1, x1) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Choose a basepoint OX ∈ X, and let I = h ◦ iOX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus H(OX, x2) = I(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' A holomorphic map X → Y is uniquely determined by the map on the fundamental group and the image of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus H(x1, x2) = I(x2 − x1) for any other x1 since this map satisfies that H(x1, x1) = O and is compatible on fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let h : PConfn(X) → PConfm(Y ) be a holomorphic map, and fix a base- point O ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let y1 ∈ Y denote the first coordinate of h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' note that y1 is a holomorphic function of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Exploiting the group structure, define the normalization hs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) − (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 23 Thus, hs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = (O, f2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', fm(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn)) where fi : PConfn(X) → Y − {O} are holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='10, it follows that fp(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=', xn) = Ip(xi−xj) for some isomorphism Ip : X → Y and 1 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To proceed, we consider the set of all isomorphisms I : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This set is a torsor for Aut(Y ), the group of automorphisms of Y as a Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' There is a semi-direct product structure Aut(Y ) ∼= (Y, O) ⋊ Aut(Y, O), (2) where (Y, O) indicates the elliptic curve structure on Y with O as the origin, and Aut(Y, O) indicates the subgroup of Aut(Y ) fixing O, equivalently the subgroup of group automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The group Aut(Y, O) is always finite, and always contains the negation map −id : x �→ −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Fix the identification I2 : X → Y associated with the first nontrivial coordinate f2 once and for all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' accordingly, we suppress I2 from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For p ≥ 3, the torsor structure then gives expressions Ip = (ϵp, αp) ◦ I2, where ϵp ∈ Y and αp ∈ Aut(Y, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Succinctly, Ip(x) = αp(x) + ϵp, so that we can write (after a permutation of coordinates in the domain) fs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = (O, x2 − x1, α3(xi3) − α3(xj3) + ϵ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , αm(xim) − αm(xjm) + ϵm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By hypothesis, the difference αp(xip) − αp(xjp) + ϵp − (x2 − x1) does not have O in its image, so that by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='10, there is an identity αp(xip) − αp(xjp) + ϵp − (x2 − x1) = β(xk) − β(xl) + δ, for distinct indices k, l and (β, δ) ∈ Aut(Y ), or equivalently x1 + αp(xip) + β(xl) − (x2 + αp(xjp) + β(xk)) = δ − ϵp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The right-hand side above is constant, and so the left-hand side must exhibit cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If either β or αp is not ±id, then this is not possible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' lifting to the universal cover C of Y , the derivative of the left-hand side would necessarily be everywhere nontrivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Up to an exchange of indices, we can assume that αp = β = id, so that x1 + xip + xl − (x2 + xjp + xk) = δ − ϵp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It follows that exactly one of the conditions ip = 2 or jp = 1 holds (they cannot both hold simultaneously since then k = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By a permutation of coordinates, we can assume j3 = 1 and i3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' At this point, our expression for fs has become fs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = (O, x2 − x1, x3 − x1 + ϵ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xim − xjm + ϵm), subject to the condition that exactly one of the identities ip = 2 or jp = 1 holds for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By comparing the second and pth entries, we conclude that ϵp = O for all p ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We claim that necessarily jp = 1 must hold for all p ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose to the contrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' then (without loss of generality) fs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = (O, x2 − x1, x3 − x1, x2 − xj4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the previous analysis, comparing the third and fourth component forces j4 = 1, but then the fourth component equals the second, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Finally, we can re-normalize fs by translation by x1, showing that f is given by a forgetful map as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 24 LEI CHEN AND NICK SALTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3: from higher genus to genus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Write h = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , fm), with each fk : PConfn(X) → Y holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Choose a basepoint O ∈ Y , and, as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2, normalize h : PConfn(X) → PConfm(Y ) by the twist A(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn)(y) = y − f1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) so that h = (O, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , fm) with fk : PConfn(X) → Y − O holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='10, each fk must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Pure configuration spaces in higher genus In this section we consider the problem of classifying holomorphic maps h : PConfn(X) → PConfm(Y ), where X, Y are Riemann surfaces of higher genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We find that the situation is quite rigid - Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1 shows that either X = Y and the map is forgetful, or else h is induced from a family of maps fi : X → Y which pairwise have disjoint graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3, we consider the question of how many such fi can exist, which is essentially a variant of the de Franchis theorem from classical algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Classification of holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X and Y be compact Riemann surfaces, with g(X), g(Y ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Suppose h : PConfn(X) → PConfm(Y ) is a nonconstant holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then up to permutation of coordinates and the actions of Aut(X) and Aut(Y ), either X = Y, n ≥ m, and h is a forgetful map PConfn(X) → PConfm(X), or else h factors as the composition of a forgetful map PConfn(X) → X and a holomorphic map X → PConfm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Consider the composition pi ◦ h : PConfn(X) → Y , where pi : PConfm(Y ) → Y is the projection onto the ith factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Each pi ◦ h is nonconstant and induces a surjection H1(PConfn(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q) → H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We first show that each pi ◦h is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If m = 1 and p1 is constant then h is constant, contrary to hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If m > 1, then supposing that any pi ◦h is constant (say with value y0 ∈ Y ) let j ̸= i be some other index, and consider pj ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If this is nonconstant, then let C ⊂ X be a configuration of n−1 distinct points, and consider the inclusion iC,k : X−C ⊂ PConfn(X) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Here k is chosen so as to make the composition pj◦h◦iC,k nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By the removable singularity theorem (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3), this extends to give a nonconstant holomorphic map fC : X → Y and by the de Franchis theorem, the space of such maps is discrete, and hence f := fC is independent of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Since f : X → Y is nonconstant, it has positive degree and hence is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus, there is some (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) ∈ PConfn(X) such that pjh(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = y0 = pih(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' the map fails to have codomain PConfn(Y ), contrary to assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To see that (pi ◦ h)∗ is surjective, it suffices to see that f∗ : H1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q) → H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q) is surjective, where f is as in the above paragraph, but this is a general property of nonconstant holomorphic HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 25 maps between compact Riemann surfaces, following from the existence of a transfer homomor- phism f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' : H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q) → H1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Q) with the property that f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='f∗ = deg(f)IH1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='Q) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [Tan10, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2] for a dual cohomological formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We consider the induced map (pi ◦ h)∗ : PBn(X) → π1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By [Che19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='5], either (pi ◦ h)∗ factors as (pi ◦ h)∗ = f∗ ◦ pj,∗ for some f∗ : π1(X) → π1(Y ), where pj,∗ : PBn(X) → π1(X) is induced by projecting onto the jth factor, or else (pi ◦ h)∗ has cyclic image, possibly trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The latter case cannot happen, since such maps would not induce a surjection on H1, contrary to Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Defining [k] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , k}, we next claim that there is a function j : [m] → [n] and nonconstant holomorphic maps αi : X → Y for which the diagram below commutes for all i: PConfn(X) h � pj(i) � PConfm(Y ) pi � X αi � Y (3) The function j can be defined as follows: by the above, (pi ◦h)∗ : PBn(X) → π1(Y ) factors through some projection pj,∗ : PBm(X) → π1(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' let j(i) be this j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We observe that h is determined by its values on the collection of submanifolds (X − C)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' For k ̸= j(i), the restriction of the holomorphic map pi ◦ h to (X − C)k is nullhomotopic by the preceding paragraph, and hence constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus pi ◦ h factors through pj(i) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By construction, if j is constant, then h factors through some projection pj : PConfn(X) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It remains to show that if j is nonconstant, then Y = X, n ≥ m, and h is a forgetful map up to permutation of coordinates and the application of some α ∈ Aut(X) to each component of PConfn(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We claim that if j is nonconstant, then αi1 = αi2 for all pairs of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Supposing to the contrary, without loss of generality, we may take i = 1, j = 2, and j(1) = 1, j(2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let x1 ∈ X be given such that α1(x1) ̸= α2(x1), and let x2 ∈ X satisfy α2(x2) = α1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Completing x1, x2 to a point (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) ∈ PConfn(X), we see that h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = (α1(x1), α2(x2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , αn(xj(n))) has a repeated entry α1(x1) = α2(x2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To see that X = Y in this case, we show that the fixed map α : X → Y has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If not, let x1 ̸= x2 ∈ X satisfy α(x1) = α(x2) = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' then h(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' ) = (y, y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' ) has a repeated entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus X = Y , and by adjusting by α−1 if necessary, we may assume that α = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Now it is clear that n ≥ m, otherwise h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn) = (xj(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xj(n)) would have a repeated entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To complement Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1, we consider the problem of determining the maximal m for which there is a nonconstant holomorphic map h : X → PConfm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' This is closely related to the effective de Franchis problem, which asks for bounds on the number of distinct holomorphic maps fi : X → Y (known to be finite for g(X), g(Y ) ≥ 2 by the classical de Franchis theorem);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' here we add the condition that the images of fi be pairwise-disjoint (pairwise have no coincidences, in the terminology of [Tan10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The general effective de Franchis problem is far from conclusively resolved - Chamizo [Cha19] obtains an upper bound that is slightly larger than exponential in g(X), while the largest known examples are linear in the genus (arising when X and/or Y has 26 LEI CHEN AND NICK SALTER a large automorphism group which can be used to enlarge the number of morphism by pre/post composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Here, we find that the condition that the morphisms pairwise have no coincidences imposes a strong constraint, greatly reducing the upper bound, although in practice there is still a gap between the upper bound of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3 and the largest known examples (arising when Y is equipped with a group of free automorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be compact Riemann surfaces each of genus at least 2, and let h : X → PConfm(Y ) be a nonconstant holomorphic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then m ≤ 4g(X)g(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Each holomorphic map f : X → Y induces f∗ ∈ Hom(J(X), J(Y )), the induced map on Jacobians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Martens observes [Mar78] that distinct morphisms f, g induce distinct maps f∗, g∗ so long as g(Y ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Tanabe [Tan10, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='9], following ideas of Fuertes – Gonz´alez-Diez [FGD93], Martens [Mar78] and ultimately Weil [Wei56], introduces a certain positive-definite inner product ⟨·, ·⟩ on Hom(J(X), J(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' According to [Tan10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1], if f, g : X → Y are holomorphic and have no coincidences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' f(x) ̸= g(x) for all x ∈ X), then deg(f) = deg(g) and cos(f∗, g∗) = g(Y )−1, where cos(v, w) := ⟨v, w⟩/∥v∥∥w∥ is defined as in any inner product space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' If h : X → PConfm(Y ) is given, the component functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , hm pairwise have no coincidences, and thus determine a configuration of vectors h1,∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , hm,∗ ∈ Hom(J(X), J(Y )) where the angles cos(hi,∗, hj,∗) = g(Y )−1 are pairwise fixed and equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As Hom(J(X), J(Y )) is a subgroup of Hom(H1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Z), H1(Y, Z)) ∼= Z4g(X)g(Y ), to prove the claim, it suffices to show that such a configuration of vectors must be linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It suffices to consider the associated unit vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let A be the matrix with Aij = ⟨vi, vj⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' linear-independence of {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , vm} is equivalent to the nonsingularity of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' By hypothesis, A = (1 − g(Y )−1)I + C, where C is the matrix where every entry is given by g(Y )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The eigenvalues of C are 0 and mg(Y )−1, and hence the eigenvalues of A are 1 − g(Y )−1 and 1 + (m − 1)g(Y )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As g(Y ) ≥ 2, both of these are nonzero, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Distinct genus regime In this section, we examine what happens when g(X) and g(Y ) belong to distinct genus regimes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' g = 0, 1, or g ≥ 2) - this is the setting in which there are very few holomorphic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' The results largely follow from basic principles, but we include the proofs for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let X, Y be Riemann surfaces of finite type with g(Y ) > g(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then every holomorphic map h : PConfn(X) → PConfm(Y ) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Fix a configuration of distinct points C = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' , xn−1} ⊂ X, and for 1 ≤ k ≤ n, consider the inclusions (X − C)k �→ PConfn(X) as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Composing with the projection pi : PConfm(Y ) → Y onto the ith factor, we obtain a holomorphic map fik : X −C → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As g(Y ) > g(X), any such map must be constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' varying i, k, and C then shows that h itself is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ HOLOMORPHIC MAPS BETWEEN CONFIGURATION SPACES OF RIEMANN SURFACES 27 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Let Y be a compact Riemann surface of genus g(Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Then every holomorphic map h : Confn(CP1) → Confm(Y ) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' We consider the induced map on fundamental group h∗ : Bn(CP1) → π1(Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' note that as the target is abelian, this factors through H1(Bn(CP1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' It is well-known that Bn(CP1) = Bn(S2) is a quotient of Bn by the word σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' σn−1σn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Thus, H1(Bn(CP1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Z) ∼= Z/(2n − 2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' As π1(Y ) ∼= Z2 is torsion-free, it follows that h∗ is the trivial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='6, we conclude that h extends to a holomorphic map H : (CP1)n → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Every such map is constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' H lifts to the universal cover C of Y and is therefore constant via the maximum principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' □ We remark that in the setting of g(Y ) ≥ 2, the same argument (replacing Y with its Jacobian) shows that any holomorphic map h : Confn(CP1) → Confm(Y ) has image contained in a fixed linear system on Y , but this by itself is not enough to conclude that h itself is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' References [AAS18] S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='15178, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [CKM19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Kordek, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Margalit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Homomorphisms between braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Preprint: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='org/abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='00712, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [CM20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Chen and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Mukherjea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' From braid groups to mapping class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Preprint: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='org/abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='13020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [DW07] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Daskalopoulos and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Wentworth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Harmonic maps and Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' In Hand- book of Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' I, volume 11 of IRMA Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Phys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Global rigidity of the period mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Preprint: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='org/abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='13178, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [Far22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Farb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Rigidity of moduli spaces and algebro-geometric constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' To appear in the conference proceedings of the Chern 110th birthday memorial, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' [FGD93] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Fuertes and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Gonz´alez D´ıez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' On the number of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' Science Coun- cil of Japan, Tokyo, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content=' LC: Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD 20742 Email address: chenlei@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='edu NS: Department of Mathematics, University of Notre Dame, Hurley Hall, Notre Dame, IN 46556 Email address: nsalter@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E_T4oBgHgl3EQf1Bzr/content/2301.08333v1.pdf'} diff --git a/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/2301.04542v1.pdf.txt b/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/2301.04542v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3196a9dfd6bf8ab14adfc83597721014a41484ae --- /dev/null +++ b/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/2301.04542v1.pdf.txt @@ -0,0 +1,2213 @@ +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Abstract. Classical good continuation for image curves is based on 2D position and orienta- +tion. It is supported by the columnar organization of cortex, by psychophysical experiments, +and by rich models of (differential) geometry. Here we extend good continuation to stereo. We +introduce a neurogeometric model, in which the parametrizations involve both spatial and ori- +entation disparities. Our model provides insight into the neurobiology, suggesting an implicit +organization for neural interactions and a well-defined 3D association field. Our model sheds +light on the computations underlying the correspondence problem, and illustrates how good +continuation in the world generalizes good continuation in the plane. +Keywords: Stereo vision, Sub-Riemannian geometry, 3D space of position-orientation, 3D +association field, Neurogeometry. +1. Introduction +Binocular vision is the ability of the visual system to provide information about the three- +dimensional environment starting from two-dimensional retinal images. Disparities are among +the main cues for depth perception and stereo vision but, in order to extract them, the brain needs +to determine which features in the right eye correspond to those in the left eye, and which do +not. This generates a coupling problem, which is usually referred to as the stereo correspondence +problem. +Our goal in this paper is to develop a perceptual organization approach to stereo, +extending good continuation for planar curves to that for 3D spatial curves. +Orientation good continuation in the plane (retinotopic coordinates) is one of the foundational +principles of Gestalt perceptual organization. It enjoys an extensive history [88]. It is supported +by psychophysical investigations (e.g., [33, 30, 28, 31, 55]), which reveal connections to contour +statistics; it is supported by physiology (orientation selectivity), which reveals the role for long- +range horizontal connections [12]; and it is supported by computational modeling ([9, 78]), which +reveals a key role for geometry. Historically, good continuation in depth is much less well devel- +oped than good continuation in the plane, despite having comparable origins. Quoting Koffka +[53, p. 161-162] : +...a perspective drawing, even when viewed monocularly, does not give the same +vivid impression of depth as the same drawing if viewed through a stereoscope +with binocular parallax ... for in the stereoscope the tri-dimensional force of the +parallax co-operates with the other tri-dimensional forces of organization; instead +of conflict between forces, stereoscopic vision introduces mutual reinforcement. +Our specific goal in this paper is to develop good continuation in depth analogously to the models +of contour organization in two dimensions. Psychophysical investigations suggest this should be +feasible ([50, 23, 24, 87]); our focus will be more mathematical. +Specifically, in Koffka’s words, we seek to develop a computational model of "mutual reinforce- +ment." Although only one dimension higher than contours in the plane, contours extending in +1Department of Mathematics, University of Bologna, Italy. +2CAMS, CNRS - EHESS, Paris, France. +3Departments of Computer Science and Biomedical Engineering, Yale University, New Haven, CT, United +States. +1 +arXiv:2301.04542v1 [q-bio.NC] 11 Jan 2023 + +2 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +depth raise subtle new issues; this is why a computational model can be instructive. First among +the issues is the choice of coordinates which, of course, requires a mathematical framework for +specifying them. In the plane, position and orientation are natural; smoothness is captured by +curvature or the relationship between nearby orientations along a contour. For stereo, there is +monocular structure in the left eye and in the right. Spatial disparity is a standard variable +relating them, and it is well known that primate visual systems represent this variable directly +[71]. Spatial disparity is clearly a potential coordinate. However, other physiological aspects +are less clear. The columnar architecture so powerful for contour organization in the plane is +monocular; it may well be a different story for contours in depth where, at least in V1, there are +ocular dominance bands (see next Section). Nevertheless, orientationally-selective cells provide +the input for stereo so, at a minimum, both positional disparity and orientation – one orientation +for the right eye and (possibly) another for the left – should be involved. While it is traditional +to assume only "like" orientations are matched [40, 60, 13, 16, 66], our sensitivity to orientation +disparity questions this, making orientation disparity another putative variable. We shall show +that orientations do play a deep role in stereo, but that it is not necessarily efficent to represent +them as a disparity. Furthermore, this settles a classical debate in stereo psychophysics about ori- +entation: since its physiological realization could be confounded with disparity gradients [63, 15], +orientation may be redundant. This is not the case, since it is the orientation of the "gradient" +that matters. Thus we can describe the main technical goal of this paper: to provide a represen- +tation of the geometry of spatial disparity and orientation in support of using good continuation +in a manner that both incorporates the biological "givens" and provides a rigorous foundation +for the stereo correspondence problem. As has been the case with curve organization, we further +believe that our modeling will illuminate the underlying connectomics of stereo, at least at the +earlier stages, even if the columnar organization is not clear. This would be important if, as is +the case with orientation columns in the mouse [74], the neural architectural support for stereo +is laid out only implicitly in the connections (rather than in explicit columns). +Hubel and Wiesel introduced disparity-tuned neurons [41]. They observed that single units +could be driven from both eyes and that it was possible to plot separate receptive fields (RF) +for each eye. We emphasize these monocular RFs are tuned to orientation; see [20] for more on +the physiology. The architecture and the neural connections of the visual cortex underlying the +establishment of stereoscopic correspondence in binocular vision have recently been studied in +[68], and a review of neural models can be found in [72]. +The classical model for expressing the left/right-eye receptive field combination is the binoc- +ular energy model (BEM), first introduced in [5]. It encodes disparities through the receptive +profiles of simple cells, raising the possibility of both position and phase disparities [43]. How- +ever Read and Cumming [73], building upon [4], showed that phase disparity neurons tend to +be strongly activated by false correspondence pairs. Therefore, it is widely concluded, the most +relevant disparity in the receptive fields is the position alone. This, however, neglects the orien- +tation difference between the two eyes [65], neglecting the orientation disparity. Although there +are attempts to extend the energy model to incorporate binocular differences in receptive-field +orientation [14], they are limited. The geometrical model we will present incorporates orientation +differences directly. +Many other mathematical models for stereo vision based on neural models have been devel- +oped. +Many have observed (e.g., [60]) that orientations should match between the two eyes, +although small differences are allowed. This, of course, assumes the structure is frontal-parallel. +Subsequently, Jones and Malik [44] used a set of linear filters tuned to different orientations (and +scales) but their algorithm was not built on aneurophysiological basis. Subsequently, Zucker et +al. [3, 57, 89] built a more biologically-inspired model that addressed the connections between +neurons. Their differential-geometry model employed position, orientations and curvatures in 2D +retinal planes, modeling binocular neurons with orientations given by tangent vectors of Frenet +geometry. Our results here are related, although the geometry is deeper. (We develop this be- +low.) A more recent work, based on differential geometry and precisely Riemannian geometry, + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +3 +is developed in [64]. Before specifying these results, however, we introduce the specific type of +geometry that we shall be using. It follows directly from the columnar organization often seen in +predators and primates. +1.1. Columnar architectures and sub-Riemannian geometry. We propose a sub- Rie- +mannian model for the cortical-inspired geometry underlying stereo vision based on the encod- +ing of positional disparities and orientation differences in the information coming from the two +eyes. +We build on neuromathematical models, starting from the work of Hoffmann [37] and +Koenderink-van Doorn [52], with particular emphasis on the neurogeometry of monocular simple +cells ([18, 69, 70, 76, 77, 78]). +To motivate our mathematical approach, it is instructive to build on an abstraction of visual +cortex. We start with monocular information, segregated into ocular dominance bands [56] in +layer 4; these neurons have processes that extend into the superficial layers. We cartoon this in +Fig. 1.1, which shows an array of orientation hypercolumns arranged over retinotopic position. +It is colored by dominant eye inputs; the binocularly-driven cells tend to be closer to the ocular +dominance boundaries, while the monocular cells are toward the centers. A zoom emphasizes the +orientation distribution along a few of the columns near each position; horizontal connections (not +shown) effect the interactions between these units. This raises the basic question in this paper: +what is the nature of the interaction among groups of cells representing different orientations at +nearby positions and innervated by inputs from the left and right eyes? The physiology suggests +(Fig. 1.1(right)) the answer lies in the interactions among both monocular and binocular cells; +our model specifies this interaction, starting from the monocular ones. +(a) +(b) +(c) +Figure 1.1. Cartoon of visual cortex, V1, superficial layers. (A) Macroscopic +organization: A number of (abstracted) orientation hypercolumns, colored by +left-eye (green)/right-eye (purple) dominant inputs. The color grading empha- +sizes that at the center of the ocular dominance bands the cells are strongly +monocular, while at the boundaries they become binocularly-driven. +(B) A +zoom in to a few orientation columns showing left and right monocular cells +at the border of ocular dominance bands. Cells in these nearby columns will +provide the anatomical substrate for our model. (C) More recent work shows +that both monocular and binocular inputs matter to these cells (redrawn from +[83], using data from ferret). This more advanced wiring suggests the connection +structures in our model. +1.2. Informal Setup and Overview. Since much of the paper is technical, we here specify, +informally, the main ingredients of the model and the results. We first list several of the key +points, then illustrate them directly. +• Stereo geometry enjoys a mathematical structure that is a formal extension of plane +curve geometry. In the plane, points belonging to a curve are described by an orientation + +0HMonocular +inputs +Binocular inputs4 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +at a position, and these are naturally represented as elements (orientation, position) +of columns. In our model, these become abstract fibres. The collection of fibres across +position is a fibre bundle. Elements of the (monocular) fibre can be thought of as neurons. +• For stereo, we shall need fibres that are a "product" of the left and right-eye monocular +columns. +The natural coordinates on the stereo fiber bundle are position, positional +disparity and orientations from the left and right eyes respectively, which describe fiber +over each position. +• The columnar organization of the stereo system, beyond what is shown in the Figure +1.1, is completely unclear. While visual area MT is suggestive of columns for direction +of motion ([61, 22]) and perhaps V4 for slant ([36]), there is no direct evidence of which +we are aware in V1 for spatial or orientation disparity columns. This is the reason why +models can be insightful. +• Curvature provides a kind of "glue" to enable transitions from points on fibres to nearby +points on nearby fibres. These transitions specify "integral curves" through the stereo +fibre bundle. +• The integral curve viewpoint provides a direction of information flow (information diffuses +through the bundle) thereby suggesting underlying circuits. +• The integral curves formalize association field models. Their parameters describe the +spray of curves that is well in accordance with 3D curves as studied in psychophysical +experiments in [32, 34, 51]. +• Our formal theory resolves several conjectures in the literature [48, 49, 58]. +• Our formal theory provides a new framework for specifying the correspondence problem, +by illustrating how good continuation in the 3-D world generalizes good continuation in +the 2-D plane. This is the point where consistent binocular-binocular interactions are +most important. +• Our formal theory has direct implications for understanding torsional eye movements. +It suggests, in particular, that the rotational component is not simply a consequence +of development, but that it helps to undo inappropriate orientation disparity changes +induce by eye movements. This provides a novel role for Listing’s Law, and is treated in +a companion paper (in preparation). +We now illustrate these ideas (Fig. 1.2). Consider a three-dimensional stimulus as a space +curve γ : R −→ R3, with a unitary tangent at the point of fixation. Since the tangent is the +derivative of a curve, the binocular cells naturally encode the unitary tangent direction ˙γ to the +spatial 3D stimulus γ. This space tangent projects to a tangent orientation in the left eye1, and +perhaps the same or a different orientation in the right eye. A nearby space tangent projects +to another pair of monocular tangents, illustrated as activity in neighboring columns. Note how +connections between the binocular neurons support consistency along the space curve. It is this +consistency relationship that we capture with our model of the stereo association field. +Since space curves live in 3D, two angles are required to specify its space tangent at a point. +In other words, monocular tangent angles span a circle in the plane; space tangent angles span a +2-sphere in 3D. In terms of the projections into the left-eye and the right-eye, the space tangent +can be described by the parameters n = (θ, ϕ) of S2 (Fig. 1.3). Thus, we can suitably describe +the space of stereo cells – the full set of space tangents at any position in the 3D world – as the +manifold of positions and orientations R3 ⋊ S2. Moving from one position in space to another, +and changing the tangent orientation to the one at the new position, amounts to what is called +a group action on the appropriate manifold. We informally introduce these notions in the next +subsection; a more extensive invitation to these ideas is in Appendix A. +1.2.1. Sub-Riemannian Geometry. We live in a 3D world in which distances are familiar; that is, +a space of points with a Euclidean distance function defined between any pair of them. Apart from +1We are here being loose with language. By a tangent orientation in the left eye, we mean the orientation of +a left-eye innervated column in V1 + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +5 +(a) +(b) +Figure 1.2. (A) Stereo projection of the highlighted tangent vector to the stim- +ulus γ ∈ R3 in the left-eye innervated and right-eye innervated monocular orien- +tation columns. (Each short line denotes a neuron by its orientation preference.) +Joint activity across the eyes, which denotes the space tangent, is illustrated by +the binocular neuron (circle). Note the two similar but distinct monocular ori- +entations. Connections from the actively stimulated monocular neurons to the +binocular neuron are shown as dashed lines. (B) Stereo projection of a consecu- +tive pair of tangents to the stimulus γ ∈ R3 in the left and right retinal columns. +Each space tangent projects to a different pair of monocular columns because +of the spatial disparity. Consistency in the responses of these four columns cor- +responds to consistency between the space tangents attached nearby positions +along γ. This consistency is realized through the binocular neural connection +(solid line). +practical considerations we can move in any direction we would like. Cars, however, have much +more restricted movement capabilities. They can move forward or backward, but not sideways. +To move in a different direction, cars must turn their wheels. +Here is the basic analogy: in +cortical space information can move to a new retinotopic position in a tangent direction, or it can +move up or down a column (orientation fibre) to change direction. Moving in this fashion, from +an orientation at a position to another orientation at a nearby position, is clearly more limited +than arbitrary movements in Euclidean space. Euclidean geometry, as above, is an example of +a Riemannian geometry; the limitations involved in moving through a cortical columnar space + +r3 +Y +r1 +r2r3 +Y +r1 +r2R +YR +X6 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Figure 1.3. The full geometry of stereo. +Note how the stereo correspon- +dence problem allows to establish the relationship between the 3D tangent point +(P, θ, φ) and the projections pL and pR, the disparity and the orientations θL +and θR. +specify a sub-Riemannian geometry. Just as cars can move along a roads that are mostly smooth, +excitatory neurons mainly connect to similarly "like" (in orientation) excitatory neurons. This +chain of neurons indicates a path through sub-Riemannian space; the fan of such paths is the +cortical connectivity which can be considered the neural correlate of association fields. Again, +for more information please consult Appendix A. +Moving now out to the world, we must be able to move between all points. Repeating the +above metaphor more technically, we equip R3 ⋊ S2 with a group action of the three-dimensional +Euclidean group of rigid motions SE(3). +Notice, importantly, that this group is now acting +on the product space of positions and orientations. A bit more is required, though, since the +geometry of the stereo vision is not solved only with these punctual and directional arguments. + +3 +2 +R +R +y.GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +7 +As we showed in Fig. 1.2 there is the need to take into account the relationships between nearby +tangents; in geometric language this involves a suitable type of connections. It is therefore natural +to look at integral curves of the sub-Riemannian structure, which encode in their coefficients the +fundamental concept of 3D curvature and torsion. An example of this is shown in Fig 1.4. Notice +how the 3D association field envelopes a space curve, in the same way that a 2D association field +envelopes a planar curve. This figure illustrates, in a basic way, the fundamental result in this +paper. +Figure 1.4. Main result of the paper. The three-dimensional space curve γ is +enveloped by the 3D the association field centered at a point. Formally, this +association field is a fan of integral curves in the sub-Riemmanian geometry +computed entirely within the columnar architecture. (It is specifically described +by equation (36) with varying c1 and c2 in R, but that will take some work to +develop.) +1.3. Overview of Paper. The paper is organized as follows: in Section 2, we describe the +geometrical and neuro-mathematical background underlying the problem of stereo vision. +In +particular, we review the standard stereo triangulation technique to relate the coordinate system +of one retina with the other, and put them together in order to reconstruct the three-dimensional +space. Then, we briefly review the classical neurogeometry of monocular simple cells selective +for orientation and the underlying connections. The generalization of co-circularity for stereo is +also introduced. In Section 3, starting from binocular receptive profiles, we introduce the neuro- +mathematical model for binocular cells. First we present the cortical fiber bundle of binocular +cells. It follows the differential interpretation of the binocular profiles in terms of the neurogeom- +etry of the simple cells, and we show how this is well in accordance with the results of the stereo +triangulation. Then, we give a mathematical definition of the manifold R3 ⋊ S2 with the sub- +Riemannian structure. Finally, we study the integral curves and the suitable change of variables +that allow us to switch our analysis from cortical to external space. In Section 4 we proceed to the +validation of our geometry with respect to psychophysical experiments. We combine information +about the psychophysics of 3D perception and formal conjectures; it is here that we formulate a +3D association field analogous to the 2D association field. At the end, we show an example of + +C今 +r28 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +a lifting of a stimulus and how our integral curves properly connect corresponding points. This +illustrates the use of our model as a basis for solving the correspondence problem. +2. Stereo vision and neuro-mathematical background +2.1. Stereo geometry. In this subsection we briefly recall the geometrical configuration under- +lying 3D vision, to define the variables that we use in the rest of the paper, mainly referring to +[29, Ch. 6]. For a complete historical background see [38, 39]. +2.1.1. Stereo variables. We consider the global reference system (O, i, j, k) in R3, with O = +(0, 0, 0), and coordinates (r1, r2, r3). +We introduce the optical centers CL = (−c, 0, 0) and +CR = (c, 0, 0), with c real positive element, and we define two reference systems: (CL, iL, jL), +(CR, iR, jR), the reference systems of the retinal planes RL and RR with coordinates respectively +(xL, y), (xR, y). In the global system we suppose the retinal planes to be parallel and to have +equation r3 = f, with f denoting the focal length. This geometrical set-up is shown in Figure +2.1. +Figure 2.1. Reconstruction of the 3D space point Q through points QL the +retinal plane RL and QR in , RR. +Remark 2.1. If we know the coordinate of a point Q = (r1, r2, r3)T in R3, then it is easy to +project it in the two planes via perspective projection, having c the coordinate of the optical centers +and f focal length. This computation defines two projective maps ΠL and ΠR, respectively, for +the left and right retinal planes: +(1) +ΠL : +R3 +−→ +R2 +ΠR : +R3 −→ +R2 +� +� +r1 +r2 +r3 +� +� +�→ +� +f(r1+c) +r3 +fr2 +r3 +� +, +� +� +r1 +r2 +r3 +� +� �→ +� +f(r1−c) +r3 +fr2 +r3 +� +. + +r3 +r2 +r1 +C +R +YR +RR +Q +Q +R +RL +XR +XLGOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +9 +Proof. A point on the left retinal plane of local coordinates (xL, y)T has global coordinates +QL = (−c + xL, y, f)T , and it corresponds to a point Q = (r1, r2, r3)T in the Euclidean R3 +such that CL, QL and Q are aligned. This means that the vectors QL − CL = (xL, y, f)T and +Q − CL = (r1 + c, r2, r3)T are parallel, obtaining the following relationships: +(2) +xL = f r1 + c +r3 +, +y = f r2 +r3 +. +Analogously, considering QR and CR, we get: +(3) +xR = f r1 − c +r3 +, +y = f r2 +r3 +. +□ +In a standard way, the horizontal disparity is defined as the differences between retinal coor- +dinates +(4) +d := xL − xR +2 +, +up to a scalar factor. Moreover, it is also possible to define the coordinate x as the average of +the two retinal coordinates x := xL+xR +2 +, leading to the following change of variables: +(5) +� +� +� +� +� +x = fr1 +r3 +y = fr2 +r3 +d = fc +r3 +←→ +� +� +� +� +� +r1 = xc +d +r2 = yc +d +r3 = fc +d +, +where the set of coordinates (x, y, d) is known as cyclopean coordinates [46]. +2.1.2. Tangent estimation. Corresponding points in the retinal planes allow to project back into +R3. An analogous reasoning can be done for the tangent structure: if we have tangent vectors of +corresponding curves in the retinal planes, it is possible to project back and recover an estimate +of the 3D tangent vector. Let us recall here this result; a detailed explanation can be found in +[29]. +Remark 2.2. Let γL and γR be corresponding left and right retinal curves; i.e., perspective +projections of a curve γ ∈ R3 through optical centers CL and CR with focal length f. Knowing +the left and right retinal tangent structures, it is possible to recover the direction of the tangent +vector ˙γ. +Proof. Starting from a curve γ ∈ R3, we project it in the two retinal planes obtaining γL = ΠL(γ) +and γR = ΠR(γ) from eq. (1). The retinal tangent vectors are obtained through the Jacobian +matrix 2 of the left and right retinal projections ˙γL,R(t) = (JΠL,R)γ(t) ˙γ(t): +(6) +˙γR(t) = +� f(γ3 ˙γ1+(c−γ1) ˙γ3) +γ3(t)2 +f(γ3 ˙γ2−γ2 ˙γ3) +γ2 +3 +� +, ˙γL(t) = +� f(γ3 ˙γ1−(c+γ1) ˙γ3) +γ3(t)2 +f(γ3 ˙γ2−γ2 ˙γ3) +γ2 +3 +� +. +2The Jacobian matrix (JΠ)p evaluated at point p represents how to project displacement vectors (in the sense of +derivatives or velocities or directions). In details, if ˙γ(t) is the displacement vector in R3, then the matrix product +(JΠ)γ(t) ˙γ(t) is another displacement vector, but in R2. In other words, the Jacobian matrix is the differential of +Π at every point where Π is differentiable; common notation includes JΠ or DΠ. + +10 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Extending the tangent vectors and the points into R3, we get ˜tL = (˙γL1, ˙γL2, 0)T , and ˜mL = (γL1− +c, γL2, f)T , and UtL = (PL)−1 ˜mL × (P −1 +L )˜tL, with the projection matrix PL = +� +� +1 +0 +−c/f +0 +1 +0 +0 +0 +1 +� +�. +The same reasoning holds for the right structure, with projection matrix PR = +� +� +1 +0 +c/f +0 +1 +0 +0 +0 +1 +� +� . +Then UtR × UtL is a vector parallel to the tangent vector ˙γ: +(7) +UtR × UtL = +� +� +� +� +� +f 42c(˙γ2γ3 − ˙γ3γ2) +γ4 +3 +� +�� +� +λ(t) +˙γ1, f 42c(˙γ2γ3 − ˙γ3γ2) +γ4 +3 +˙γ2, f 42c(˙γ2γ3 − ˙γ3γ2) +γ4 +3 +˙γ3 +� +� +� +� +� +T += λ(t) (˙γ1(t), ˙γ2(t), ˙γ3(t))T += λ(t)˙γ(t). +□ +2.2. Elements of neuro-mathematics. We now provide background on the geometric mod- +eling of the monocular system, and good continuation in the plane. Our goal is to illustrate +the role of sub-Riemannian geometry in the monocular system, which will serve as the basis for +generalization in the stereo system starting from the neuro-mathematical model of Citti and Sarti +[18]. +2.2.1. Classical neurogeometry of simple cells. We model the activation map of a cortical neuron’s +receptive field (RF) by its receptive profile (RP) ϕ. A classical example is the receptive profiles +of simple cells in V1, centered at position (x, y) and orientation θ, modeled (e.g in [7, 21, 45]) as +a bank of Gabor filters ϕ{x,y,θ}, which act on a visual stimulus. +Formally, it is possible to abstract the primary visual cortex as R2 ×S1, or position-orientation +space, thereby naturally encoding the Hubel/Wiesel hypercolumnar structure [40]. An example +of this structure is displayed in image (A) of Figure 2.4 from [9]. +Following the neuro-mathematical model of Citti and Sarti [18], the set of simple cells RPs +can be obtained via translations of vector (x, y)T and rotation of angle θ from a unique "mother" +profile ϕ0(ξ, η) +(8) +ϕ0(ξ, η) = exp +�2πiξ +λ +� +exp +� +−ξ2 + η2 +2σ2 +� +, +a Gabor function with real (even) and imaginary (odd) parts (Figure 2.2). Translations and +rotations can be expressed as: +(9) +T(x,y,θ)(ξ, η) = +�cos θ +− sin θ +sin θ +cos θ +� �ξ +η +� ++ +�x +y +� +, +where T(x,y,θ) denotes the action of the group of rotations and translations SE(2) on R2. This +group operation associates to every point (ξ, η) a new point (˜x, ˜y), according to the law (˜x, ˜y) = +T(x,y,θ)(ξ, η). Hence a general RP can be expressed as +(10) +ϕ(x,y,θ)(ξ, η) = ϕ0(T −1 +(x,y,θ)(ξ, η)), +and this represents the action of the group SE(2) on the set of receptive profiles. +The retinal plane R is identified with the R2 plane, whose coordinates are (x, y). When a +visual stimulus I : R −→ R+ of intensity I(x, y) activates the retinal layer, the neurons centered + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +11 +(a) +(b) +Figure 2.2. Even (A) and odd (B) part of Gabor function: the surface of +the two-dimensional filters, their common bi-dimensional representation and a +mono-dimensional section. +at every point (x, y) produce an output O(x, y, θ), which can be modeled as the integral of the +signal I with the set of Gabor filters: +(11) +O(x, y, θ) = +� +R +ϕ{x,y,θ}(ξ, η)I(ξ, η)dξdη, +where the function I represents the retinal image. +For (x, y) fixed, we will denote ¯θ the point of maximal response: +(12) +max +θ |O(x, y, θ)| = |O(x, y, ¯θ)|. +We will then say that the point (x, y) is lifted to the point (x, y, ¯θ). This is extremely important +conceptually to understand our geometry: it illustrates how an image point, evaluated against +an simple cell RP, is lifted to a "cortical" point by introducing the orientation explicitly. If all +the points of the image are lifted in the same way, the level lines of the 2D image I are lifted to +new curves in the 3D cortical space (x, y, ¯θ). +We shall now introduce a set of directions for moving on the cortical space (x, y, ¯θ), in the sense +of vector fields. This is important because it will be necessary to move within this space, across +both positions and orientations. Biologically, such movements would be the flow of information +from one cell in a column to another cell in a nearby column. +To begin, in the right hand side of the equation (11) the integral of the signal with the real and +imaginary part of the Gabor filter is expressed. The two families of cells have different shapes, +hence they detect (or play a role in detecting) different features. Since the odd-symmetry cells sug- +gest boundary detection, we concentrate on them, but this is mainly a convenience for computa- +tion. The output of a simple cell can then be locally approximated as O(x, y, θ) = −X3,p(Iσ)(x, y), +where p = (x, y, θ) ∈ SE(2), Iσ is a smoothed version of I, obtained by convolving it with a +Gaussian kernel, and +(13) +X3,p = − sin θ∂x + cos θ∂y, +is the directional derivative in the direction ⃗X3,p = (− sin θ, cos θ, 0)T . From now on, we will +denote (by a slight abuse of notation) ω⋆ := ⃗X3,p to remind the reader familiar with the language +of 1-forms the correspondence of these quantities, and the relation with the Hodge star operator. +3 +3The purpose of introducing this notation is also to motivate an implication of the mathematical model in [18]; +see Appendix B.2.1 for explanation. + +12 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Now, think of vector fields as defining a coordinate system at each point in cortical space. +Then, in addition to above, the vector fields orthogonal to X3,p are: +(14) +X1,p = cos θ∂x + sin θ∂y, +X2,p = ∂θ +and they define a 2-dimensional admissible tangent bundle4 to R2 × S1. One can define a scalar +product on this space by imposing the orthonormality of X1,p and X2,p: this determines a sub- +Riemannian structure on R2 × S1. +(a) +(b) +(c) +Figure 2.3. (A) Examples of the compatibilities around the central point of +the image, derived from planar co-circularity. Brightness encodes compatibility +values. Figure adapted from [9]. (B) Starting from the central initial oriented +point, the solid line indicates a configuration between the patches where the +association exists while the dashed line indicates a configuration where it does +not. Figure adapted from [30]. (C) Association field of Field, Hayes and Hess. +Figure adapted from [30]. +The visual signal propagates, in an anisotropic way, along cortical connectivity and connects +more strongly cells with comparable orientations. This propagation establishes the connection +between the geometry just developed and 2-dimensional contour integration. This is a formal- +ization of the Gestalt law of good continuation [53, 54]. It first arose in a simpler form, namely +co-circularity in the plane [67], to describe the consistency and the compatibility of neighboring +oriented points, in accordance with specific values of curvature. An example of these compatibil- +ities can be found in Figure 2.3, image (A). It is complemented by psychophysical experiments, +e.g. [42, 84, 86]. In particular, Field et al. in [30] describe the association rules for 2-dimensional +contour integration, introducing the concept of association fields. A representation of these con- +nections can be found in Figure 2.3, images (B) and (C). Note that this is equivalent to the union +(over curvature) in [67]. Neurophysiological studies [10, 12, 35, 59, 82] suggest that the cortical +correlate of the association field is the long-range horizontal connectivity among cells of similar +(but not necessarily identical) orientation preference. +Based on these findings, Citti and Sarti in [18] modeled cortical propagation as propagation +along integral curves of the vector fields X1 and X2, namely curves γ : [0, T] ⊂ R −→ R2 × S1 +described by the following differential equation: +(15) +˙γ(t) = ⃗X1,γ(t) + k ⃗X2,γ(t), +t ∈ [0, T], +obtained by varying the parameter k ∈ R. (k acts analogously as curvature.) An example of +these curves is in Figure 2.4(B). Their 2D projection is a close approximation of the association +fields (Figure 2.4(B)). +A related model has been proposed by Duits et al. [26]. They study the geodesics of the sub- +Riemannian structure to take into account all appropriate end-conditions of association fields. +4as defined in Appendix A.3 + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +13 +(a) +(b) +(c) +Figure 2.4. (A) Orientation columns of cells in (x, y, θ) coordinates. Long- +range horizontal connections between cells relate an orientation signal at posi- +tion (x, y, θ) to another orientation at (x′, y′, θ′). Figure adapted from [9]. (B) +Horizontal integral curves in R2 × S1 generated by the sub-Riemannian model +[18]. (C) Projection of the fan of the integral curves in the (x, y) plane. Figure +adapted from [18]. +2.2.2. Generalizing co-circularity for stereo. The concept of co-circularity in R2 has been de- +veloped by observing that a bidimensional curve γ can be locally approximated at 0 via the +osculating circle. Zucker et al. in [3, 57, 58] generalize this concept with the Frenet differential +geometry of a three dimensional curve. +While in the two-dimensional case the approximation of the curve using the Frenet 2D basis +causes the curvature to appear in the coefficient of the Taylor series development (1st order), in +the three-dimensional case the coefficients involve both the curvature and torsion. So, in [3] the +authors propose heuristically to generalize the osculating circle for space curves with an osculating +helix, with a preference for r3-helices to improve stability in terms of camera calibration. In this +way the orientation disparity is encoded in the behavior of the helix in the 3D space: there is no +difference in orientation in the retinal planes if the helix is confined to be in the fronto-parallel +plane (the helix becomes a circle), otherwise moving along the 3D curves the retinal projections +have different orientations. +In [57, 58] they observe that, by introducing the curvature variable as a feature in the two +monocular structures, and assuming correspondence, it is possible to reconstruct the 3D Frenet +geometry of the curve, starting from the two-dimensional Frenet geometry, up to the torsion +parameter. In particular, they prove: +Proposition 2.1. Given two perspective views of a 3D space curve with full calibration, the +normal N and curvature k at a curve space point are uniquely determined from the positions, +tangents, and curvatures of its projections in two images. Thus the Frenet frame {T, N, B} and +curvature k at the space point can be uniquely determined. +Hence, using the knowledge of the Frenet basis together with the fundamental addition of the +curvature variable, Zucker et al. introduced the concept of transport. This allowed moving the +3D Frenet frame in a consistent way with the corresponding 2D Frenet structures of the left and +right retinal planes, to establish stereo correspondence between pairs of (left and right) pairs of +tangents. See Figure 2.5 image (B). +Remark 2.3. The model that we propose in this paper is related to, but differs from, what has +just been stated. In particular, to remain directly compatible with the previous neuro-geometric +model, we will work only with the monocular variables of position and orientation. Rather than +using curvature directly, we shall assume that these variables are encoded within the connections; + +Long range connections to compatible cells +X +Rearrangement of orientation +hypercolumns by retinotopic position14 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +(a) +(b) +Figure 2.5. (A) Geometrical setup of Proposition 2.1. The spiral curve 3D +projects in the left and right retinal planes together with the Frenet structure. +(B) Stereo correspondence between pairs of (left-right) pairs of tangents. Both +figures are taken from [58]. +mathematically they appear as parameters. A theoretical result of our model is that the heuristic +assumption regarding the r3-helix can now be established rigorously. +Let us also mention the paper [1], where the curvature was considered as independent variable +and helices have been obtained in the 2D space. +3. The neuromathematical model for stereo vision +3.1. Binocular profiles. Binocular neurons receive inputs from both the left and right eyes. To +facilitate calculations, we assume these inputs are first combined in simple cells in the primary +visual cortex, a widely studied approach ([5, 20, 47, 62]). It provides a first approximation in +which binocular RPs are described as the product of monocular RPs; see Figure 3.1, image (A). +This is of course a simplification – see [80], for instance – but it is compatible with existing neural +findings. +This binocular model allows us to define disparity and frontoparallel coordinates as +(16) +� +d = xL−xR +2 +x = xR+xL +2 +, +perfectly in accordance with the introduction of cyclopean coordinates in (4). In this way (x, y, d) +correspond to the neural correlate of (r1, r2, r3), via the change of variables (5). +3.2. The cortical fiber bundle of binocular cells. The hypercolumnar structure of monoc- +ular simple cells (orientation selective) has been described as a jet fiber bundle in the works of +Petitot and Tondut [70], among many others. We concentrate on the fiber bundle R2 × S1, with +fiber S1; see e.g. [9] among many others. +In our setting, the binocular structure is based on monocular ones; recall the example illus- +trations from the Introduction. In particular, for each cell on the left eye there is an entire fiber +of cells on the right, and vice versa, for each cell on the right there is an entire fiber of cells on +the left. This implies that the binocular space is equipped with a symmetry that involves the left +and right structures, allowing us to use the cyclopean coordinates (x, y, d) defined in (16). +Hence, we define the cyclopean retina R, identified with R2, endowed with coordinates (x, y). +The structure of the fiber is F = R×S1 ×S1, with coordinates (d, θL, θR) ∈ F. The total space is + +T +B +X +fi +Z +P1 +Ci +XI +Pr +f. +W +Zr +5 +XTransport in F +Y +x +pairi +- +y +pairi +VT +xrGOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +15 +Figure 3.1. Comparisons between binocular interaction RPs and the product +of left and right eye RPs, where left and right RPs are shown in Figure 2.2. +Binocular interaction RPs (Raw data) of a cell is shown on the left. Contour +plots for the product of left and right eye RPs (L×R) are shown in the right +along with 1-dimensional profiles of the left (L) and right (R) eye RPs. Figure +adapted from [5]. +defined in a trivial way, E = R×F = R2×R×S1×S1, and the projection π : E −→ R is the trivial +projection π(x, y, d, θL, θR) = (x, y). The preimage of the projection E(x,y) := π−1({(x, y)}), for +every (x, y) ∈ R, is isomorphic to the fiber F, and the local trivialization property is naturally +satisfied. +A schematic representation can be found in Figure 3.2. The base has been depicted as 1- +dimensional, considering the restriction R|x of the cyclopean retina R on the coordinate x. +The left image displays only the disparity component of the fiber F, encoding the relationships +between left and right retinal coordinates. The right image shows the presence of the left and +right monodimensional orientational fibers. +3.3. Binocular energy model. To simplify calculations, as stated in the Introduction, we follow +the classical binocular energy model [5] for binocular RPs. The basic idea is a binocular neuron +receives input from each eye; if the sum OL + OR of the inputs from the left and right eye is +positive, the firing rate of the binocular neuron is proportional to the square of the sum, and it +vanishes, if the sum of the inputs is negative: +(17) +OB = (Pos(OL + OR))2, +with Pos(x) = max{x, 0}, OB the binocular output. + +Raw data (B) +AD +B +AD +A +kd058m22.03r.03.58 +kd449m20.02r.02.59 +B +B +x +D +Prediction (LxR) +E +D +D +AD +LxR +LxR +口 +6(deg) +5 (deg) +6 +(deg) +5 +(deg)16 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Figure 3.2. Left: schematic representation of the fiber bundle in two dimen- +sion, with relationships between left and right retinal coordinates. Right: repre- +sentation of the selection of a whole fiber of left and right simple cells, for every +x and for every d. +If OL + OR > 0, then the output of the binocular simple cell can be explicitly written as +OB = O2 +L + O2 +R + 2OLOR. The first two terms represent responses due to monocular stimulation +while the third term 2OLOR can be interpreted as the binocular interaction term. +The activity of a cell is then measured from the output and will be strongest at points that +have a higher probability of matching each other. The maximum value over d of this quantity is +the extracted disparity. +It is worth noting that neurophysiological computations of binocular profiles displayed in Fig- +ure 3.1 assume the mono-dimensionality of the monocular receptive profile, ignoring information +about orientation of monocular simple cells. However, this information will be needed to encode +different types of orientation disparity. +Remark 3.1 (Orientation matters). In 2001, the authors of [14] conducted investigations on the +response of binocular neurons to orientation disparity, by extending the energy model of Anzai, +Ohzawa and Freeman to incorporate binocular differences in receptive-field orientation. More +recently, the difference between orientations in the receptive fields of the eyes has been confirmed +[81]. +The binocular energy model is a type of minimal model. It serves as a starting point, allowing +the combination of monocular inputs. But is not sufficient to solve the stereo-matching problem. +Remark 3.2 (Connections). It is argued in [68, 75] that, in addition to the neural mechanisms +that couple characteristics (such as signals, stimuli, or particular features) relating the left and +right monocular structures, there must be a system of connections between binocular cells, which +characterizes the processing mechanism of stereo vision; see also Samonds et al. in [75] in par- +ticular. +3.4. Differential interpretation of binocular RPs. It is possible to write the interaction +term OLOR coming from (17), in terms of the left and right receptive profiles: +(18) +OLOR = +� +ϕθL,xL,y(˜xL, ˜yL)IL(˜xL, ˜yL)d˜xLd˜yL +� +ϕθR,xR,y(˜xR, ˜yR)IR(˜xR, ˜yR)d˜xRd˜yR += +� � +ϕθL,xL,y(˜xL, ˜yL)ϕθR,xR,y(˜xR, ˜yR)IL(˜xL, ˜yL)IR(˜xR, ˜yR)d˜xRd˜yRd˜xLd˜yL. + +d +d +xR = X-d +xL = x+d +xR = x-d +xL = x+d +x +xGOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +17 +If we fix (˜xR, ˜yR, ˜xL, ˜yL), we derive the expression of the binocular profiles ϕL,R = ϕθR,xR,yϕθL,xL,y +as the product of monocular left and right profiles. This is in accordance with the measured pro- +files of Figure 3.1. +Proposition 3.1. The binocular interaction term can be associated with the cross product of the +left and right directions defined through (13), namely ω⋆ +L and ω⋆ +R of monocular simple cells: +(19) +OLOR = ω⋆ +L × ω⋆ +R. +Proof. The idea is that the binocular output is the combined result of the left and right actions of +monocular cells, thus identifying a direction in the space of cyclopean coordinates. The detailed +proof of this proposition can be found in Appendix B. +□ +To better understand the geometrical idea behind Proposition 3.1, we recall that the retinal +coordinates can be expressed in terms of cyclopean coordinates (4) as xR = x−d and xL = x+d, +and so we can write ω⋆ +L and ω⋆ +R in the 3D space of coordinates (x, y, d) as: +(20) +ω⋆ +R =(− sin θR, cos θR, sin θR)T +ω⋆ +L = +(− sin θL, cos θL, − sin θL)T . +We define ωbin := ω⋆ +L × ω⋆ +R as the natural direction characterizing the binocular structure: +(21) +ωbin = +� +� +sin(θR + θL) +2 sin θR sin θL +sin(θR − θL) +� +� . +Remark 3.3. The vector ωbin of equation (21) can be interpreted as the intersection of the +orthogonal spaces defined with respect to ω⋆ +R and ω⋆ +L when expressed in cyclopean coordinates +(x, y, d). More precisely, if +(22) +(ω⋆ +L)⊥ = span +� +� +� +� +� +cos θL +sin θL +0 +� +� , +� +� +−1 +0 +1 +� +� +� +� +� , +(ω⋆ +R)⊥ = span +� +� +� +� +� +cos θR +sin θR +0 +� +� , +� +� +1 +0 +1 +� +� +� +� +� +then +(23) +ωbin = (ω⋆ +L)⊥ ∩ (ω⋆ +R)⊥. +The result of the intersection of these monocular structures identifies a direction, as shown in +Figure 3.3. +We earlier showed that the result of the action of a monocular odd simple cell is to select +directions for the propagation of infomation. We now combine these, for the two eyes, to show +that in the three-dimensional case the binocular neural mechanisms also lead to a direction. We +will see in the next sections that this direction is the direction of the tangent vector to the 3D +stimulus, provided points are corresponding. +3.5. Compatibility with stereo geometry. We consider the direction characterizing the binoc- +ular structure ωbin defined in (21) and we show that it can be associated with the 3D tangent +vector to the 3D curve. The idea is that this tangent vector is orthogonal both to ω⋆ +R and to ω⋆ +L, +and therefore it has the direction of the vector product ω⋆ +L × ω⋆ +R. +Precisely, we consider the normalized tangent vector tL and tR on retinal planes +(24) +tR = (cos θR, sin θR)T +tL = (cos θL, sin θL)T , +to the points (xR, y) and (xL, y) respectively. Taking into account that f is the focal coordinate +of the retinal planes in R3, then we associate to these points the correspondents in R3, namely + +18 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Figure 3.3. Direction detected by ωbin through the intersection of left and right +planes generated by (ω⋆ +R)⊥ and (ω⋆ +L)⊥. Red vector corresponds to the associated +2-form ωbin. +˜mL = (xL − c, y, f)T , ˜mR = (xR + c, y, f)T . Applying equation (7), it is possible to derive the +tangent vector of the three dimensional contour: +(25) +UtL = P −1 +L +˜mL × P −1 +L ˜tL += +� +� +xL +yL +f +� +� × +� +� +cos θL +sin θL +0 +� +� = +� +� +−f sin θL +f cos θL +xL sin θL − yL cos θL +� +� , +UtR = P −1 +R ˜mR × P −1 +R ˜tR += +� +� +xR +yR +f +� +� × +� +� +cos θR +sin θR +0 +� +� = +� +� +−f sin θR +f cos θR +xR sin θR − yR cos θR +� +� , +and the tangent direction is recovered by +(26) +UtL × UtR = f +� +� +xL+xR +2 +sin(θR − θL) − xR−xL +2 +sin(θL + θR) +y sin(θR − θL) − (xR − xL)(cos(θR − θL) − cos(θL + θR)) +f sin(θR − θL) +� +� + +(a) +(wr)GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +19 +Figure 3.4. Three dimensional reconstruction of the space from retinal planes. +The 1- forms ω⋆ +L and ω⋆ +R are identified with the normal to the curves γL and γR. +Their three dimensional counterpart ˜ω⋆ +L and ˜ω⋆ +R identify the tangent vector to +the curve γ : R → R3 by the cross product ˜ω⋆ +L × ˜ω⋆ +R. +If we define +(27) +˜ω⋆ +L := d +fcUtL, +˜ω⋆ +R := d +fcUtR +and the corresponding 2 form ωR3 := ˜ω⋆ +L × ˜ω⋆ +R, using the change of variables (16) we observe that: +(28) +˜ω⋆ +L = ω⋆ +L, +˜ω⋆ +R = ω⋆ +R, +ωR3 = ωbin, +up to a scalar factor. See Appendix C for explicit computation. +In this way, the disparity binocular cells couple in a natural way positions, identified with +points in R3, and orientations in S2, identified with three-dimensional unitary tangent vectors. +As already observed in Remark 3.2, the geometry of the stereo vision is not solved only with +these punctual and directional arguments, but there is the need to take into accounts suitable +type of connections. In [3, 57, 58], Zucker et al. proposed a model that considered the curvature of +monocular structures as an additional variable. Instead, we propose to consider simple monocular +cells selective for orientation, and to insert the notion of curvature directly into the definition of +connection. It is therefore natural to introduce the perceptual space via the manifold R3 ⋊ S2, +and look for appropriate curves. +3.6. A perceptual model in the space of 3D position-orientation. We now derive the +objects in Fig. 1.3. We have clarified (end of section 3.5) that binocular cells are parametrized +by points in R3, and orientations in S2. +An element ξ of the space R3 ⋊ S2 it is defined by +a point p = (p1, p2, p3) in R3 and an unitary vector n ∈ S2. Since the topological dimension + +8* +X +R +CR +★m +R +r220 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +of this geometric object is 2, we introduce the classical spherical coordinates (θ, ϕ) such that +n = (n1, n2, n3)T ∈ S2 can be parameterized as: +(29) +n1 = cos θ sin ϕ +n2 = sin θ sin ϕ +n3 = cos ϕ +with θ ∈ [0, 2π] and ϕ ∈ (0, π). The ambiguity that arises using local coordinate chart is overcome +by the introduction of a second chart, covering the singular points. +Translations and rotations are expressed using the group law of the three-dimensional special +Euclidean group SE(3), defining the group action +(30) +σ : R3 ⋊ S2 × SE(3) −→ R3 ⋊ S2 s.t. σ((p, n), (q, R)) = (Rp + q, Rn), +with (p, n) ∈ R3 ⋊ S2 , (q, R) ∈ SE(3), namely R ∈ SO(3) tridimensional rotation, and q ∈ R3. +3.6.1. Stereo sub-Riemannian geometry. The emergence of a privileged direction in R3 (asso- +ciated with the tangent vector to the stimulus) is the reason why we endow R3 ⋊ S2 with a +sub-Riemannian structure that favors the direction in 3D identified by ωbin. +Formally, we consider admissible movements in R3 ⋊ S2 described by vector fields: +(31) +YR3,ξ = sin ϕ cos θ∂1 + sin ϕ sin θ∂2 + cos ϕ∂3 +Yθ,ξ = − +1 +sin ϕ∂θ +Yϕ,ξ = ∂ϕ +with ξ ∈ R3 ⋊ S2 for ϕ ̸= 0, ϕ ̸= π. The admissible tangent space5 at a point ξ +(32) +Aξ := span{YR3,ξ, Yθ,ξ, Yϕ,ξ} +encodes the coupling between position and orientations, as remarked by Duits and Franken in [27]. +In particular, the vector field YR3 identifies the privileged direction in R3, while Yθ and Yϕ allow +changing this direction, involving just orientation variables of S2. The vector fields {YR3, Yθ, Yϕ} +and their commutators generate the tangent space of R3 ⋊ S2 in a point, allowing to connect +every point of the manifold using privileged directions (Hörmander condition). Furthermore, it +is possible to define a sub-Riemannian structure by choosing a scalar product on the admissible +tangent bundle A: the simplest choice is to declare the vector fields {YR3, Yθ, Yϕ} orthonormal, +considering on S2 the distance inherited from the immersion in R3 with the Euclidean metric. +3.6.2. Change of variables. We have already expressed the change of variable in the variables +(x, y, d) to (r1, r2, r3) in equations (5). However, the cortical coordinates also contain the angular +variables θR and θL which involve the introduction of the spherical coordinates θ, ϕ. +To identify a change of variable among these variables, we first introduce the function +(r1, r2, r3, θ, ϕ) +F−→ (x, y, d, θL, θR) : +(33) +F : +R3 ⋊ S2 +−→ +R3 ⋊ S2 +� +� +� +� +� +� +r1 +r2 +r3 +θ +ϕ +� +� +� +� +� +� +�→ +� +� +� +� +� +� +� +fr1 +r3 +fr2 +r3 +cf +r3 +tan−1( +r3 sin θ cos ϕ−r2 cos ϕ +r3 cos θ sin ϕ−(c+r1) cos ϕ) +tan−1( +r3 sin θ cos ϕ−r2 cos ϕ +r3 cos θ sin ϕ−(c−r1) cos ϕ +� +� +� +� +� +� +� +, +where the retinal right angle θR = tan−1( +r3 sin θ cos ϕ−r2 cos ϕ +r3 cos θ sin ϕ−(c+r1) cos ϕ) and the left retinal angle θL = +tan−1( +r3 sin θ cos ϕ−r2 cos ϕ +r3 cos θ sin ϕ−(c−r1) cos ϕ) are obtained considering equation (6). +5see Appendix A for the definition of admissible tangent space. + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +21 +Analogously, it is possible to define the change of variable (x, y, d, θL, θR) +G +−→ (r1, r2, r3, θ, ϕ): +(34) +G : +R3 ⋊ S2 +−→ +R3 ⋊ S2 +� +� +� +� +� +� +x +y +d +θR +θL +� +� +� +� +� +� +�→ +� +� +� +� +� +� +� +cx +d +cy +d +cf +d +tan−1( 2 sin θR sin θL +sin(θR+θL) ) +tan−1( +√ +sin2(θR+θL)+4 sin2 θR sin2 θL +sin(θR−θL) +) +� +� +� +� +� +� +� +, +where the angles θ = tan−1( 2 sin θR sin θL +sin(θR+θL) ) and ϕ = tan−1( +√ +sin2(θR+θL)+4 sin2 θR sin2 θL +sin(θR−θL) +) are ob- +tained considering that tan θ = (⃗YR3)2 +(⃗YR3)1 and tan ϕ = +√ +(⃗YR3)2 +1+(⃗YR3)2 +2 +(⃗YR3)3 +. +3.6.3. Integral curves. The connectivity of the space is described by admissible curves of the +vector fields spanning A. In particular, a curve Γ : [0, T] −→ R3 ⋊ S2 is said to be admissible6 if: +(35) +˙Γ(t) ∈ AΓ(t), ↔ ˙Γ(t) = a(t)⃗YR3,Γ(t) + b(t)⃗Yθ,Γ(t) + c(t)⃗Yϕ,Γ(t), +where a, b, c are sufficiently smooth function on [0, T]. We will consider a particular case of these +admissible curves, namely constant coefficient integral curves with a(t) = 1, since the vector field +YR3 represents the tangent direction of the 3D stimulus (and so it never vanishes): +(36) +˙Γ(t) = ⃗YR3,Γ(t) + c1⃗Yθ,Γ(t) + c2⃗Yϕ,Γ(t), +with c1 and c2 varying in R. +These curves can be thought of in terms of trajectories in R3 describing a movement in the +⃗YR3 direction, which can eventually change according to ⃗Yθ and ⃗Yϕ. An example of the fan of +integral curves was shown in the Introduction in Figure 1.4. +It is worth noting that in the case described by coefficients c1 and c2 equal to zero, the 3D +trajectories would be straight lines in R3; by varying the coefficients c1 and c2 in R, we allow the +integral curves to follow curved trajectories, twisting and bending in all space directions. +Formally, the amount of "twisting and bending" in space is measured by introducing the +notions of curvature and torsion. We then investigate how these measurements are encoded in +the parameters of the family of integral curves, and what constraints have to be imposed to +obtain different typologies of curves. +Remark 3.4. The 3D projection of the integral curves (36) will be denoted γ and satisfy ˙γ(t) = +(cos θ(t) sin ϕ(t), sin θ(t) sin ϕ(t), cos ϕ(t))T . Classical instruments of differential geometry let us +compute the curvature and the torsion of the curve γ(t): +(37) +k = +� +( ˙ϕ)2 + sin2 θ( ˙θ)2, +τ = 1 +k2 (− cos ϕ sin2 ϕ( ˙θ)3 − sin ϕ ˙ϕ¨θ + ˙θ(−2 cos ϕ( ˙ϕ)2 + sin ϕ ¨ϕ)). +Using the explicit expression of the vector fields Yθ and Yϕ in equation (36), we get +(38) +˙θ = − c1 +sin ϕ, +˙ϕ = c2, +from which it follows that: +(39) +k = +� +c2 +1 + c2 +2 +τ =c2 +1 − c2 +2 +k2 +c1 cotan ϕ. +6sometimes the term horizontal is preferred. + +22 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Proposition 3.2. By varying the parameters c1 and c2 in (39) where we explicit find solutions +of (36), we have: +1. If ϕ = ψ +2 then k = +� +c2 +1, τ = 0, and so the family of curves (36) are circles of radius 1/c2 +1 +on the fronto-parallel plane r3 = cost. +2. If ϕ = ϕ0, with ϕ0 ̸= π/2, then k = +� +c2 +1 and τ = c1 cotan ϕ0, and so the family of curves +(36) are r3-helices. +3. If θ = θ0 then k = +� +c2 +2, τ = 0, and so the family of curves (36) are circles of radius 1/c2 +2 +in the osculating planes. +4. If c1 = ±c2 then τ = 0, and so the family of curves (36) are circles of radius 1/c2 +2 in the +osculating planes. +Proof. The computation follows immediately from the computed curvature and torsion of (39) +and classical results of differential geometry. +□ +(a) +(b) +(c) +Figure 3.5. Examples of integral curves obtained varying parameters c1 and +c2. (A) Arc of circles for ϕ = π/2. (B) r3-helices for ϕ = π/3. (C) Family of +curves with constant curvature k and varying torsion parameter. +Remark 3.5. If we know the value of the curvature k, and we have one free parameter, c2, in +the definition of the integral curves (36), then we are in the setting of Proposition 2.1. In fact, +the coefficient c1 is obtained by imposing c1 = ± +� +k2 − c2 +2, and in particular the component that +remains to be determined is the torsion. +Examples of particular cases of the integral curves (36) according to Proposition 3.2 and +Remark 3.5 are visualized in Figure 3.5. +4. Comparison with experimental data +In this section we present results of compatibility between the proposed sub-Riemannian model +and biological and psychophysical phenomena present in literature. +4.1. Biological Connections. The foundation for building our sub-Riemannian model of stereo +was a sub-Riemannian model of curve continuation. This was motivated by the orientation column +at each position, and the connections between cells in nearby columns. These connections were, +in turn, a direct model of the long-range horizontal connections in visual cortex, for which there +is beautiful biological data (e.g. [12]). We further illustrated, in the Introduction, aspects of the +cortical architecture that support binocular processing. Although the inputs from each eye are +organized into ocular dominance bands, there is no direct evidence for "stereo columns" analogous +to the monocular orientation columns. But, as we shall now show, there is evidence of long-range +connections between binocular cells, and our model informs, concretely, what information should +be carried by these long range connections. Thus, an organization for stereo is suggested, but it +is implicit in the architecture. Nevertheless, there is evidence in support of it. + +r23 +53 +1?GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +23 +(a) +(b) +Figure 4.1. (A) A biocytin injection superimposed on a map of ocular domi- +nance columns, image result from the work in [59]. Binocular zones are in the +middle of monocular zones (coded in black and white). Starting from the injec- +tion site (yellow circle in the center of a binocular zone) the patches’ propagation +(red corresponds to dense while green to sparsely labeled) tends to avoid highly +monocular sites, bypassing the centers of ocular dominance columns, and are +located in binocular zones.(B) 3D interpretation of the physiological image (A). +Just as information propagates to enforce monocular curve continuation, the binocular signal +propagates to form a coherent binocular representations. The Grinwald group established this +for stereo [59] (see also Figure 4.1(A)), using biocytin injections, that propagate directly along +neuronal processes and are deposited at excitatory synapses. Thus, this technique demonstrates +the presence of long-range connections between binocular cells. These results were refined, more +recently, by the Fitzpatrick group [83], using in vivo calcium imaging. As shown in Fig. 1.1(right) +the authors demonstrated both the monocular and the binocular inputs for stereo, and (not +shown) the dependence on orientations. +More precisely, [59] showed selective anisotripic connectivity among binocular regions: the +biocytin tracer does not spread uniformly, but rather is highly directional with distance from +the injection point. (This was the case with monocular biocytin injections as well.) Putting this +together with [83], we interpret the anisotropy as being related to (binocular) orientation ([83]), +which is exactly the behavior of the integral curves of our vector fields. Our 3D association +fields are strongly directional, and information propagates preferentially in the direction of (the +starting point of) the curve. An example can be seen in Figure 4.1, image (B), where the fan of +integral curves (36) is represented, superimposed with colored patches, following the experiment +proposed in [59]. +4.2. Psychophysics and association fields. In this section, we show that the connections +described by the integral curves in our model can be related to the geometric relationships from +psychophysical experiments on perceptual organization of oriented elements in R3. The goal is +to establish that our connections serve as a generalization of the concept of an association field +in 3D. +4.2.1. Towards a notion of association field for 3D contours. The perception of continuity be- +tween two elements of position-orientation in R3 has been studied experimentally. +To start, +Kellman, Garrigan, and Shipley ([48, 49]) introduce 3D relatability, as a way to extend to 3D the +experiments of Field, Heyes and Hess ([30]) in 2D. + +A +C24 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +Figure 4.2. Example of the fan of the 3D relatable edges with initial point E0 +Particularly, in a system of 3D Cartesian coordinates, it is possible to introduce oriented +edges E at the application point (r1, r2, r3)T and with an orientation identified with the an- +gles θ and ϕ. +This orientation can be read, in our case, through the direction expressed by +(cos θ sin ϕ, sin θ sin ϕ, cos ϕ)T . For an initial edge E0, with application point on the origin of the +coordinate system (0, 0, 0)T and orientation lying on the r1-axis, described by θ = 0, ϕ = π/2, +the range of possible orientations (θ, ϕ) 7 for 3D-relatable edges with E0 is given by: +(40) +tan−1 +�r2 +r1 +� +≤ θ ≤ π +2 +and +π +2 ≤ 3π +2 − ϕ ≤ tan−1 +�r3 +r1 +� +. +The bound on these equations identified with the quantity π +2 incorporates the 90 degree constraint +in three dimensions, while the bounds defined by the inverse of the tangent express the absolute +orientation difference between the reference edge E0 and an edge positioned at the arbitrary +oriented point E(r1,r2,r3) so that its linear extension intersects E0; see [48, 49] for further details. +Numerical simulations allow us to visually represent an example of the 3D positions and +orientations that meet the 3D relatability criteria. Starting from an initial edge E0 with endpoints +in (p01, p02, p03)T and orientation on the e1- axis, we represent for an arbitrary point (p1, p2, p3)T +the limit of the relatable orientation (θ, ϕ). Results are shown in Figure 4.2. +Remark 4.1. By projecting on the retinal planes of the 3D fan of relatable points, it is possible +to notice that these projections are in accordance with the notion of 3D compatibility field of in +[3]. See Figure 4.3. +Psychophysical studies, see [25, 32, 34], have investigated the properties of the curves that are +suitable for connecting these relatable points. These curves are well described by being smooth +and monotonic. +In particular, using non-oriented contour elements for contours, Hess et al. +in [34] indicate that contour elements can be effectively grouped based primarily on the good +7The angle ϕ here has been modified to be compatible with our set of coordinates. The relationship between +the angle ˜ϕ in works [48, 49] can be expressed as : ˜ϕ = acos(sin ϕ) + π. + +GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +25 +(a) +(b) +Figure 4.3. (A) Example of 3D association field in the two left and right retinal +planes, generated with the geometry of 3D relatability. +(B) Example of 3D +compatibility field of [3]. +continuation of contour elements in depth. This statement is confirmed by the more recent work +of Deas and Wilcox ([25]), who in addition observe that detection of contours defined by regular +depth continuity is faster than detection of discontinuous contours. All these results support +the existence of depth grouping operations, arguing for the extension of Gestalt principles of +continuity and smoothness in three dimensional space. Finally, on the relationship of the three- +dimensional curves to 2-dimensional association fields, see [49, 51]. These authors have assumed +that the strength of the relatable edges in the co-planar planes of E0 must meet the relations of +the bi-dimensional association fields of [30]. +4.2.2. Compatibility with the sub-Riemannian model. To model associations underlying the 3D +perceptual organization of the previous paragraph, we consider again the constant coefficient +family of integral curves studied in (36): +(41) +˙Γ(t) = ⃗YR3,Γ(t) + c1⃗Yθ,Γ(t) + c2⃗Yϕ,Γ(t), with c1, c2 ∈ R. +Importantly, these curves locally connect the association fan generated by the geometry of 3D +relatability. In particular, Figure 4.4, image(B) shows the family of the horizontal curves con- +necting the initial point E0 with 3D relatable edges. These curves are computed using Matlab +solver function ode45. In analogy with the experiment of Field , Hayes and Hess in [30], we +(a) +(b) +Figure 4.4. (A)3D relatable edges displayed on the right of the initial edge E0. +Unrelatable 3D edges displayed on the left. (B) Horizontal integral curves with +filled lines connect 3D relatable edges with initial point E0. Horizontal integral +curves with dotted lines do not connect 3D unrelatable edges. +choose to represent non-relatable edges to the left of the starting point E0, while on the right + +Projection in the left image +Projection in the right image +E +ORProjection in left image +Projection in right image +10 +3 +w +2 +-2 +E- +-3 +-2.5 +-2 +1.5 +0.5 +0 +0.5 +1.5 +2.5 +-2.5 +-1 +2 +-2 +1.5 +1 +0.5 +0 +0.5 +1 +1.5 +1 +2.5 +X +x 10* +× 10~4rE. +3 +1?26 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +are 3D relatable edges. So, filled lines of the integral curves indicate the correlation between the +central horizontal element E0 and the ones on its right, while dotted lines connect the starting +point E0 with elements not correlated with it, as represented on the left part of the image. +Restricting the curves on the neighborhood of co-planar planes with an arbitrary edge E, we +have different cases. First, on the r1-r2 plane (fronto-parallel) and the r1-r3 plane we have arcs +of circle, as proved with Proposition 3.2. Furthermore, for an arbitrary plane in R3 containing +an edge E, we observe that the curves generating with fixed angle ϕ are helices, and locally +they satisfy the bidimensional constraint in the plane. Examples can be found in Figure 4.5. In +particular, the curves displayed in images (A) and (B) of Figure 4.5 are well in accordance with +the curves of the Citti-Sarti model, depicted in Figure 2.3. +(a) +(b) +(c) +Figure 4.5. (A) Restriction of the fan of the integral curves on the e1-e2 plane. +(B) Restriction of the fan of the integral curves on the e1-e3 plane. (C) Restric- +tion of the fan at ϕ = ϕ0. These curves (black lines) are not planar curves but +helices. However, their projection (white lines) on the coplanar plane with initial +edge satisfies the bidimensional constraints. +4.3. Integration of contours and stereo correspondence problem. Although the goal of +this paper is not to solve the stereo correspondence problem, we can show how our geometry +is helpful in understanding how to match left and right points and features. These ideas are +developed more fully in [11]. +Inspired by the experiment of Hess and Field in [32], we consider a path stimulus γ interpreted +as a contour, embedded in a background of randomly oriented elements: left and right retinal +visual stimuli are depicted in Figure 4.6. We perform a first simplified lift of the retinal images +to a set Ω subset of R3 ⋊ S2. This set contains all the possible corresponding points, obtained by +coupling left and right points which share the same y retinal coordinate, see Figure 4.7, image(A). +The set Ω contains false matches, namely points that do not belong to the original stimulus. It +is the task of correspondence to eliminate these false matches. +We compute for every lifted point the binocular output OB of equation (17). This output can +be seen as a probability measure that gives information on the correspondence of the couple of +left and right points. We can simply evaluate which are the points with the highest probability +of being in correspondence, applying a process of suppression of the non-maximal pairs over the +fiber of disparity. In this way, noise points are removed (Figure 4.7, image (B)). +We now directly exploit good continuation in depth. The remaining noise elements are orthog- +onal to the directions of the elements of the curve that we would like to reconstruct. Calculating +numerically the coefficients c1 and c2 of integral curves (36) that connect all the remaining pairs +of points, we can obtain for every pair the value of curvature and torsion using (39). Figure 4.8 +read it in terms of matrices M representing the values of curvature or torsion for every couple of +points ξi, ξj in the element Mij. In particular, we observe that random points are characterized +by a very high curvature and in general also the torsion deviates from minimum magnitudes. + +1313GOOD CONTINUATION IN 3D: +THE NEUROGEOMETRY OF STEREO VISION +27 +Figure 4.6. Left and right retinal images of the set Ω. Black points are the +projection of the point of the curve γ, while gray points are background random +noise. +(a) +(b) +(c) +Figure 4.7. (A) Lifting of the two left and right retinal images of Figure 4.6 +in the space of position and orientation R3 × S2. (B) Selection of lifted points +according to the binocular output. (C) Points of the stimulus γ connected by +integral curves (36). +So, by discarding these high values, we select only the three-dimensional points of the curve γ, +which are well connected by the integral curves, as shown in image (C) of Figure 4.7. This is in +accordance with the idea developed in [3, 57, 58], where curvature and torsion provide constraints +for reconstruction in 3D. +Summary and Conclusions +Understanding good continuation in depth, like good continuation for planar contours, can +benefit from basic physiological constraints; from psychophysical performance measures, and +from mathematical modeling. In particular, good continuation in the plane is supported by ori- +entation selectivity and cortical architecture (orientation columns), by association field grouping +performance, and by geometric modeling. We maintain that the same should be true for good +continuation in depth. However, while the psychophysical data may be comparable, the physio- +logical data are weaker and the geometry of continuation is not well understood. In this paper, +we introduced the neuro-geometry of stereo vision to fill this gap. It is strongly motivated by an +analogical extension to 3D of 2D geometry, subject to respecting the psychophysics. In the end, +it allowed us to be precise about the type of geometry that is relevant for understanding stereo + +Left Image +Right Image +13r3 +r2328 +M. V. BOLELLI1, G. CITTI1, A. SARTI2, AND S. W. ZUCKER 3 +(a) +(b) +Figure 4.8. Matrices M which element Mij represents the value of curvature/ +torsion for every couple of points ξi, ξj. +The first eight points correspond to +points of the curve γ while the others are random noise. (A) Curvature matrix. +(B) Torsion matrix. +abstractly, and concretely was highly informative toward the physiology. Although a "stereo +columnar architecture" is not obvious from the anatomy, it is well-formed computationally. +The neuro-geometry of binocular cells are described through binocular RPs which are the +product of left and right monocular RPs. Starting from binocular receptive profiles it is possible +to reconstruct the three dimensional space using just the position and orientation of the visual +stimulus recovered in the retinal planes (assuming one has corresponding points). +Technically, we proposed a sub-Riemannian model on the space of position and orientation +R3 ⋊ S2 for the description of the perceptual space of the neural cells involved. This geometrical +structure favors the tangent direction of a 3D curve stimulus. The integral curves of the sub- +Riemannian structure encode the notions of curvature and torsion within their coefficients, and +are introduced to describe the connections between elements. This model can be seen as an +extension in the three-dimensional scene of the 2-dimensional association field. In particular, +the integral curves of the sub-Riemannian structure of the 3D space of position-orientation are +exactly those that locally correspond to psychophysical association fields. +Although the goal of this paper is not to solve the stereo correspondence problem, we have seen +how the geometry we propose is a good starting point to understand how to match left and right +points and features. A future development of the model will consist in defining the probability of +the co-occurrence between two elements, to individuate percepts in 3D space. Individuation of +percepts through harmonic analysis on the sub-Riemannian structure has been proposed in the +past, both for 2D spatial stimuli [77] and in 2D + time spatio-temporal stimuli [6]. It would be +interesting to develop a similar analysis and extend it to stereo vision. +Acknowledgements +MVB, GC, and AS were supported by EU Project, GHAIA, Geometric and Harmonic Analysis +with Interdisciplinary Applications, H2020-MSCA-RISE-2017 +SWZ was supported in part by US NIH EY031059 and by US NSF CRCNS 1822598. + +Curvature +2 +4 +9 +0.5 +8 +10 +12 +0.1 +2 +4 +6 +8 +10 +12Torsion +2 +0 +4 +-0.2 +9 +-0.4 +8 +-0.6 +10 +-0.8 +12 +-1 +2 +4 +9 +8 +10 +1229 +Appendices +A. A gentle introduction to sub-Riemannian geometry +In this paper we exploit techniques from differential geometry, and in particular sub - Riemann- +ian geometry. In this appendix we provide an invitation to these ideas with a rather informal +discussion. For the reader interested in a formal introduction on basic instruments of differential +geometry (arguments of sections A.1 and A.2) please refer to [85]. For a complete and formal +mathematical (comprehensive) introduction to sub-Riemannian geometry we refer to [2], while +for a more informational point of view please consult [79, Ch. 4.2] and [17]. +A.1. Tangent bundle. To start, imagine that you are standing at a point on a smooth surface +in the world, far from any boundaries. Now, you can "walk away" from this point in any (2D) +compass direction; for example, you could walk north or south or any direction in-between. If your +steps were very very short, then the (flat) compass actually characterizes the 2D space of possible +steps you might take. These same ideas are expressed more formally in differential geometry, as +follows. One can attach to every point p of a differentiable manifold M (a generalized surface) +a tangent space TpM (the compass plus some algebra describing vector operations). That is, +the tangent space is a real vector space that contains the possible directions in which one can +tangentially pass through p ∈ M. If the manifold is connected, then the tangent space has, at +every point, the same dimension as the manifold. So, if the manifold is a 2D surface, the tangent +space at a point is a plane. In general, this tangent plane "approximates" the surface only locally. +The elements ⃗Xp of the tangent space TpM at p are called tangent vectors at p. Attached to +a point on the surface, as above, these tangent vectors define the directions in which one could +walk away from the point. But modern differential geometry provides another interpretation: +it is possible to think of the elements of the tangent space in terms of directional derivatives. +Technically, for every smooth function f, Xf(p) = ⃗Xp·∇f(p) will denote the directional derivative +of f in the direction of the vector ⃗ +Xp, with ∇ denoting the gradient vector (expressed in an +appropriate coordinate system) and · scalar product between these vectors. We will also denote +Xp = ⃗Xp · ∇p, omitting the function f. +We now consider pairs of directional derivatives X and Y . If X and Y are partial derivatives, +for every regular function f one has XY f = Y Xf. If X and Y are directional derivatives, in +general XY f ̸= Y Xf. Explicit computation tell us that at every point p +(42) +[X, Y ]f(p) = (XY − Y X)f(p) = (J⃗Yp ⃗Xp − J ⃗ +Xp ⃗Yp) · ∇f(p), +with J ⃗ +Xp and J⃗Yp Jacobian matrices of ⃗Xp and ⃗Yp. The quantity [X, Y ]f is called commutator +since it expresses the fact that the two derivatives do not commute. The same notion can be +expressed in terms of increments: one might visualize an increment from a point p as the head of +a vector ⃗Xp applied at the point p. Then the expression XY − Y X will be geometrical obtained +as follows: place X down at a point, then the other Y at its head, then the the first one backward +finally the second one backward. The issue is whether the quadrilateral is closed. Formally this +is captured by the commutator of two elements X and Y at the point p. +In order to compute the second derivative XY f, we need to know Y f at every point near p. +This lead to the more general notion of vector fields, which are abstractions of the velocity field +of points moving in the manifold. A vector field X attaches to every point p of the manifold M a +vector ⃗Xp from the tangent space at that point, in a smooth manner. There are no abrupt jumps +between points. +Since we related each tangent vector with a derivation above, we can now go further; see +Fig. A.1. +Each vector field can be associated with an ordinary differential equation, whose +solutions are called integral curves of the vector field: they are parametric curves that represent +specific solutions to the ordinary differential equation depicted by the vector field. Think of it as +follows: imagine you are starting at a point, and take an infinitesmimal step in the direction of a + +30 +tangent vector at that point; you will now be at a neighboring point. So, again, you can take a step +from this neighboring point in (possibly) another tangent direction. Continuing this process for a +while, you geometrically trace out integral curves γ : [t1, t2] ⊆ R −→ M. Importantly, the given +vector field X at the point γ(t) is the tangent vector to the curve at that point. Importantly, this +holds true everywhere along the curve, so that the integral curve satisfies a differential equation: +(43) +˙γ(t) = ⃗Xγ(t). +(a) +(b) +(c) +Figure A.1. (A) Tangent planes TpiM (darker planes) at points pi, i = 1, 2, 3 +in the manifold M. (B) Vector field X defined on M: to every point pi, i = 1, 2, 3 +of the manifold M we have a vector ⃗Xpi of the tangent space at that point. (C) +Integral curve γ associated with the vector field X starting from p1 ∈ M. +All the tangent spaces of a manifold may be "glued together" to form a new differentiable +manifold with twice the dimension of the original manifold, called the tangent bundle of the +manifold. As a set, it is given by the disjoint union of the tangent spaces of M, that is: +(44) +TM = +� +p∈M +TpM = {(p, Xp) | p ∈ M, Xp ∈ TpM} . +In particular, an element of TM can be thought of as a pair (p, Xp), where p is a point in M +and Xp is a tangent vector to M at p. There exists a natural projection π : TM → M defined +by π(p, Xp) = p. which maps each element of the tangent space TpM to the single point p. +A.2. Group action on a manifold. The operation of adding (real) numbers has an important +algebraic structure, called a group. It requires, for example, that the sum of any two numbers +is again a number; that there is an inverse operation "-"; and that there is an identity operation +"0" that is, adding to any number yields the same number. +When a group G acts on a manifold (e.g. the real numbers, above), it means that each of +its elements performs a certain operation on all the elements of the manifold in a way that is +compatible with the manifold itself. More precisely, this action is described by a map σ : G×M → +M, (g, x) �→ g · x which is the (left) group action of a group G on a smooth manifold M, if the +map σ is differentiable. +For example, we can take the bidimensional roto-translation group SE(2) = R2×S1 and define +its action on a smooth manifold M ⊆ R2 following the group law: first we apply a rotation and +then a translation of the manifold itself. This is formalized through the map σ : SE(2) × M −→ +M, σ(g, p) = (Rp + q), with g = (q, R) ∈ SE(2), namely a point q ∈ R2 and R bidimensional +rotation of angle θ ∈ S1. A graphical example is shown in Figure A.2. +We are now ready to generalize these familiar ideas to cortical space, with its special position +× orientation structure, or to stereo space. + +Tp1M +Tp2M +p1 +p2 +p3Xp1 +Xp2 +p2 +Xp3Xp1 +Xp2 +p1 +p2 +Xp3 +p331 +Figure A.2. Group action of the roto-translation group SE(2) on the manifold +M (black ellipse): first, the manifold is rotated through a rotation of angle θ +obtaining RθM, and then a translation is applied, moving the rotated manifold +in space realizing RθM + T. +A.3. Sub-Riemannian geometry. A point constraint to move on a manifold, illustrated above, +dictates that one can move only along directions tangent to the manifold, since moving in the nor- +mal direction would leave the manifold. This means that, for every point p, the set of admissible +directions of displacement coincides with the tangent plane TpM. In the presence of further con- +straints, some tangent directions could be forbidden. This leads to introducing, at every point +p, the admissible tangent space Ap, which is the subspace of TpM of admissible directions of +movement. If the tangent space TpM has dimension n, the admissible tangent space Ap will have +dimension m ≤ n. Repeating the same construction for every point of the manifold, we call the +admissible tangent bundle the union of admissible tangent spaces at every point: A = � +p∈M Ap. +If we introduce a scalar product on Ap, then we are able to define a norm on vectors with the +aim to measure the length of such vectors and the distance between points. The manifold with +these properties is usually called sub-Riemannian manifold, while manifolds where movements +are allowed in any direction are called Riemannian manifolds. +(a) +(b) +Figure A.3. (A) Geometric set-up of the motion of a car moving on a plane. +(B)Sub-Riemannian formalization in SE(2). Tangent vector of the path is con- +strained to be in the gray plane, span of ⃗X1,p and ⃗X2,p, admissible directions of +movement. + +Po +T +M +RaM + T(t) = (coso(t), sin(t) +(t) +(t) = (x(t), y(t) +XD += (0, 0, 1) += (cos, sin0, 0) +y32 +Let us explicitly note that while Riemannian geometry arises in presence of a physical con- +straints, sub-Riemannian geometry arises in presence of differential constraints, as for example +in the description of the motion of vehicles. A car moves on a bidimensional plane, but it can +only move in its current direction or it can change its current orientation by rotating the steering +wheel. These are the admissible directions. Moreover, the car cannot move "sideways" (for- +bidden direction): this prevents one from directly reaching any other direction while remaining +in the initial position, restricting the allowable motions to a simultaneous combination of the +two admissible movements. The trajectory described by the vehicle will therefore be a curve, +whose tangent is constrained to follow the two admissible directions. The formalization of this +sub-Riemannian problem takes place in SE(2), considering for every p ∈ SE(2) as admissible +tangent space ApSE(2) the subspace generated by the current direction ⃗X1,p = (cos θ, sin θ, 0)T +and the direction of rotation ⃗X2,p = (0, 0, 1)T . See Figure A.3. +Similarly, we can move from a retinotopic (x, y) position to another retinotopic position, +(x′, y′), moving "up" or "down" through orientation columns from θ to θ′, but we cannot reach +θ′ from θ maintaining the same initial position (running through the same orientation column): +in order to reach the "forbidden direction" we have to walk simultaneously through positions and +orientations. This restriction of movement is what distinguishes a Euclidean (or Riemannian) +geometry from a sub-Riemannian geometry. +B. Proof of Proposition 3.1 +In this appendix, we show ho to prove Proposition 3.1 using tools of differential geometry, and +in particular the concept of differential k−form. +B.1. Differential forms. A differential k-form on an n-dimensional smooth manifold M is any +multilinear function ω : TM k −→ R which takes as input k smooth vector fields and outputs a +scalar element, satisfying the antisymmetry property: +ω(X1, . . . , Xi, . . . , Xj, . . . , Xk) = −ω(X1, . . . , Xj, . . . , Xi, . . . , Xk), +with k ≤ n and k, n ∈ N. +In the special case where ω is a 1-form, it is worth noting that this is an element of the +dual space to TM (cotangent space): ω ∈ TM ∗ ⇐⇒ ω : TM −→ R. If we have coordinates +(x1, . . . , xn) on M, we can express the 1-forms using the dual basis {d x1, . . . , d xn} of TM ∗: +ωp = f1(¯x1, . . . , ¯xn) d x1 + . . . + fn(¯x1, . . . , ¯xn) d xn, with p = (¯x1, . . . , ¯xn), +with fi scalar smooth functions. +Furthermore, it is possible to multiply via the wedge product ∧ a differential k-form, ω, +with a differential l- form, η, obtaining a differential k + l-form ω ∧ η. More precisely, we are +interested in the wedge product of 1-forms ω and η, where the wedge product can be computed +as: ω ∧ η(X, Y ) = ω(X)η(Y ) − ω(Y )η(X), with X and Y vector fields on M. +B.2. Development of the proof. +Proposition B.1. The binocular interaction term OLOR can be associated with the cross product +of the left and right directions defined through (13), namely ω⋆ +pL and ω⋆ +pR of monocular simple +cells: +(45) +OLOR = ω⋆ +pL × ω⋆ +pR. +Proof. As noted in subsubsection 2.2.1, the output of simple cells (11) in SE(2) can then be locally +approximated as O(x, y, θ) = −X3,p(Iσ)(x, y) where Iσ is a smoothed version of I, obtained by +convolving it with a Gaussian kernel, the vector field +(46) +X3,p = − sin θ∂x + cos θ∂y, + +33 +with p = (x, y, θ) ∈ SE(2). Switching to the dual space, the action of simple cells induces a +choice of a 1-form separately on each cell: +(47) +ωp = − sin θ d x + cos θ d y. +Accordingly, it is possible to re-write the binocular interaction term as: +(48) +OLOR = X3,pR(IσR)(xR, y)X3,pL(IσL)(xL, y). +In the following, we will see that this binocular action can be described by a 2-form defined in +terms of the two 1-forms of monocular simple cells. +We will denote with the subscript R the quantities corresponding to the right monocular +structure, and we will use the subscript L for the left one. So, we define vR := (JIσR ⃗X3,pR)X3,pR +using the Jacobian (differential) of the smoothed version of the image I, in such a way that we +have ωpR(vR) = X3,pR(IσR) = (JIσR ⃗X3,pR) since ωpR(X3,pR) = 1 and JIσR ⃗X3,pR ∈ R; the same +reasoning holds for the left structure. It is then possible to recast (48) in the retinal coordinates +as: +(49) +OLOR =ωpL(vL)ωpR(vR) +=ωpL ∧ ωpR(vL, vR) + ωpR(vL)ωpL(vR) +� +�� +� +=0 +, +=ωpL ∧ ωpR(vL, vR), +exploiting the properties of the wedge product and the left and right retinal coordinates. +The retinal coordinates can be expressed in terms of cyclopean coordinates (4) as xR = x − d +and xL = x + d; then, the extended left and right 1-form can be written as: +(50) +ωpR = − sin θR d x + cos θR d y + sin θR d d +ωpL = − sin θL d x + cos θL d y − sin θL d d. +Taking advantage of the isomorphism provided by the Hodge star between vectors and 2-forms +in R3, we relate the exterior and the cross product, using notations (20) 8, in the following way: +(51) +⋆ (ωpL ∧ ωpR) = ω⋆ +pL × ω⋆ +pR, +from which it follows the thesis. +□ +Throughout the paper, to lighten the notation, we will call ωL = ωpL and ωR = ωpR. +B.2.1. Meaning of the mathematical objects. We conclude this section with a consideration on +the mathematical tools introduced and used in this setting, to understand how the mathematical +models proposed by Citti and Sarti, starting with [18], assign these different mathematical objects +to the physical cell, to its action , and to the result of its action. +Remark B.1. It is well known that an odd simple cell (selective for orientation) is activated as a +result of the presence of a stimulus to select its direction (tangent vector to the perceptual curve). +In this setting, the mathematical intuition behind the model proposed in [18] is to identify each +cell with a 1-differential form, which is an element of the cotangent space. Roughly speaking, this +differential form is able to grasp a vector that corresponds to the direction of the stimulus: this is +the result of the action of the cell. Formally, this vector will be an element of the tangent space, +and more precisely it will lay in the kernel of the 1-form. This vector space is then associated +with the action of the cell. +The same reasoning is applied to different families of cells in a series of papers ([78, 6, 1, 8]) +even if these are characterized by distinct sub-Riemannian structures in various manifolds. The +interested reader could refer to [19] for a review. Similarly, we have found the same geometrical +organization in the family of binocular cells. +8Using the notation ω⋆ we identify the vector whose components are the coefficients of the 1-form ω with +respect to the dual basis + +34 +Remark B.2. In this paper, we have dealt with binocular cells which are a combination of +monocular simple cells. To these coupled simple cells (one for the left and one for the right eye) +we formally associate a 2-differential form, the wedge product of the two monocular left and right +1-forms. This 2-form can grasp again a vector, lying in the kernel of this mathematical object, +identifying the three-dimensional stimulus direction. Thus, the same reasoning of Remark B.1 +also applies here to the binocular family of cells. +Translating the results of Remark B.1 into different spaces, with different dimensions, it is then +possible to use the same mathematical objects to explain the behavior of families of different cells, +identifying geometrically the mathematical objects at the basis of the functionality of the family +of studied cells. +C. Change of variables +Let us recover the expression of the 1-forms ˜ωL := UtL and ˜ωR := UtR. Recall here the change +of variable (5): +(52) +� +� +� +� +� +r1 = xc +d +r2 = yc +d +r3 = fc +d +, +and its differential: +(53) +� +� +� +� +� +d r1 = c +d d x − cx +d2 d d +d r2 = c +d d y − cy +d2 d d +d r3 = − fc +d2 d d +. +Writing the quantity UtL, defined in (25), in term of a 1-form in the variables (r1, r2, r3) we have: +(54) +˜ωL = − f sin θL d r1 + f cos θL d r2 + (xL sin θL − y cos θL) d r3. +Changing coordinates: +(55) +˜ωL = − f sin θL +� c +d d x − cx +d2 d d +� ++ f cos θL +� c +d d y − cy +d2 d d +� ++ (xL sin θL − y cos θL) +� +−fc +d2 d d +� +=fc +d (− sin θL d x + cos θL d y − sin θL d d) +=fc +d ωL. +So, up to a scalar factor, we have that ˜ωL = ωL in the variables (x, y, d). The same reasoning +holds for the right structure. +References +[1] S. Abbasi-Sureshjani, M. Favali, G. Citti, A. Sarti, and B. M. ter Haar Romeny. Curvature integration in +a 5d kernel for extracting vessel connections in retinal images. 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Proceedings +of the IEEE, 102(5):812–829, 2014. + diff --git a/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/load_file.txt b/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7654b099da910eda912e464644af2676d8cd988 --- /dev/null +++ b/dNE3T4oBgHgl3EQfeQqG/content/tmp_files/load_file.txt @@ -0,0 +1,1748 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf,len=1747 +page_content='GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Classical good continuation for image curves is based on 2D position and orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is supported by the columnar organization of cortex, by psychophysical experiments, and by rich models of (differential) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Here we extend good continuation to stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We introduce a neurogeometric model, in which the parametrizations involve both spatial and ori- entation disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our model provides insight into the neurobiology, suggesting an implicit organization for neural interactions and a well-defined 3D association field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our model sheds light on the computations underlying the correspondence problem, and illustrates how good continuation in the world generalizes good continuation in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Keywords: Stereo vision, Sub-Riemannian geometry, 3D space of position-orientation, 3D association field, Neurogeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Introduction Binocular vision is the ability of the visual system to provide information about the three- dimensional environment starting from two-dimensional retinal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Disparities are among the main cues for depth perception and stereo vision but, in order to extract them, the brain needs to determine which features in the right eye correspond to those in the left eye, and which do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This generates a coupling problem, which is usually referred to as the stereo correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our goal in this paper is to develop a perceptual organization approach to stereo, extending good continuation for planar curves to that for 3D spatial curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Orientation good continuation in the plane (retinotopic coordinates) is one of the foundational principles of Gestalt perceptual organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It enjoys an extensive history [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is supported by psychophysical investigations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=', [33, 30, 28, 31, 55]), which reveal connections to contour statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' it is supported by physiology (orientation selectivity), which reveals the role for long- range horizontal connections [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' and it is supported by computational modeling ([9, 78]), which reveals a key role for geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Historically, good continuation in depth is much less well devel- oped than good continuation in the plane, despite having comparable origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Quoting Koffka [53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 161-162] : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='a perspective drawing, even when viewed monocularly, does not give the same vivid impression of depth as the same drawing if viewed through a stereoscope with binocular parallax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' for in the stereoscope the tri-dimensional force of the parallax co-operates with the other tri-dimensional forces of organization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' instead of conflict between forces, stereoscopic vision introduces mutual reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our specific goal in this paper is to develop good continuation in depth analogously to the models of contour organization in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Psychophysical investigations suggest this should be feasible ([50, 23, 24, 87]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' our focus will be more mathematical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Specifically, in Koffka’s words, we seek to develop a computational model of "mutual reinforce- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='" Although only one dimension higher than contours in the plane, contours extending in 1Department of Mathematics, University of Bologna, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2CAMS, CNRS - EHESS, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3Departments of Computer Science and Biomedical Engineering, Yale University, New Haven, CT, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='04542v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='NC] 11 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 depth raise subtle new issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' this is why a computational model can be instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' First among the issues is the choice of coordinates which, of course, requires a mathematical framework for specifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the plane, position and orientation are natural;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' smoothness is captured by curvature or the relationship between nearby orientations along a contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For stereo, there is monocular structure in the left eye and in the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Spatial disparity is a standard variable relating them, and it is well known that primate visual systems represent this variable directly [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Spatial disparity is clearly a potential coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' However, other physiological aspects are less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The columnar architecture so powerful for contour organization in the plane is monocular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' it may well be a different story for contours in depth where, at least in V1, there are ocular dominance bands (see next Section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Nevertheless, orientationally-selective cells provide the input for stereo so, at a minimum, both positional disparity and orientation – one orientation for the right eye and (possibly) another for the left – should be involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' While it is traditional to assume only "like" orientations are matched [40, 60, 13, 16, 66], our sensitivity to orientation disparity questions this, making orientation disparity another putative variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We shall show that orientations do play a deep role in stereo, but that it is not necessarily efficent to represent them as a disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Furthermore, this settles a classical debate in stereo psychophysics about ori- entation: since its physiological realization could be confounded with disparity gradients [63, 15], orientation may be redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is not the case, since it is the orientation of the "gradient" that matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus we can describe the main technical goal of this paper: to provide a represen- tation of the geometry of spatial disparity and orientation in support of using good continuation in a manner that both incorporates the biological "givens" and provides a rigorous foundation for the stereo correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' As has been the case with curve organization, we further believe that our modeling will illuminate the underlying connectomics of stereo, at least at the earlier stages, even if the columnar organization is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This would be important if, as is the case with orientation columns in the mouse [74], the neural architectural support for stereo is laid out only implicitly in the connections (rather than in explicit columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Hubel and Wiesel introduced disparity-tuned neurons [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' They observed that single units could be driven from both eyes and that it was possible to plot separate receptive fields (RF) for each eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We emphasize these monocular RFs are tuned to orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see [20] for more on the physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The architecture and the neural connections of the visual cortex underlying the establishment of stereoscopic correspondence in binocular vision have recently been studied in [68], and a review of neural models can be found in [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The classical model for expressing the left/right-eye receptive field combination is the binoc- ular energy model (BEM), first introduced in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It encodes disparities through the receptive profiles of simple cells, raising the possibility of both position and phase disparities [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' How- ever Read and Cumming [73], building upon [4], showed that phase disparity neurons tend to be strongly activated by false correspondence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Therefore, it is widely concluded, the most relevant disparity in the receptive fields is the position alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This, however, neglects the orien- tation difference between the two eyes [65], neglecting the orientation disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Although there are attempts to extend the energy model to incorporate binocular differences in receptive-field orientation [14], they are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The geometrical model we will present incorporates orientation differences directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Many other mathematical models for stereo vision based on neural models have been devel- oped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Many have observed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=', [60]) that orientations should match between the two eyes, although small differences are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This, of course, assumes the structure is frontal-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Subsequently, Jones and Malik [44] used a set of linear filters tuned to different orientations (and scales) but their algorithm was not built on aneurophysiological basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Subsequently, Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' [3, 57, 89] built a more biologically-inspired model that addressed the connections between neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Their differential-geometry model employed position, orientations and curvatures in 2D retinal planes, modeling binocular neurons with orientations given by tangent vectors of Frenet geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our results here are related, although the geometry is deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (We develop this be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=') A more recent work, based on differential geometry and precisely Riemannian geometry, GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 3 is developed in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Before specifying these results, however, we introduce the specific type of geometry that we shall be using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It follows directly from the columnar organization often seen in predators and primates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Columnar architectures and sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We propose a sub- Rie- mannian model for the cortical-inspired geometry underlying stereo vision based on the encod- ing of positional disparities and orientation differences in the information coming from the two eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We build on neuromathematical models, starting from the work of Hoffmann [37] and Koenderink-van Doorn [52], with particular emphasis on the neurogeometry of monocular simple cells ([18, 69, 70, 76, 77, 78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To motivate our mathematical approach, it is instructive to build on an abstraction of visual cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We start with monocular information, segregated into ocular dominance bands [56] in layer 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' these neurons have processes that extend into the superficial layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We cartoon this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, which shows an array of orientation hypercolumns arranged over retinotopic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is colored by dominant eye inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' the binocularly-driven cells tend to be closer to the ocular dominance boundaries, while the monocular cells are toward the centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A zoom emphasizes the orientation distribution along a few of the columns near each position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' horizontal connections (not shown) effect the interactions between these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This raises the basic question in this paper: what is the nature of the interaction among groups of cells representing different orientations at nearby positions and innervated by inputs from the left and right eyes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The physiology suggests (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1(right)) the answer lies in the interactions among both monocular and binocular cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' our model specifies this interaction, starting from the monocular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) (c) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Cartoon of visual cortex, V1, superficial layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Macroscopic organization: A number of (abstracted) orientation hypercolumns, colored by left-eye (green)/right-eye (purple) dominant inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The color grading empha- sizes that at the center of the ocular dominance bands the cells are strongly monocular, while at the boundaries they become binocularly-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) A zoom in to a few orientation columns showing left and right monocular cells at the border of ocular dominance bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Cells in these nearby columns will provide the anatomical substrate for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) More recent work shows that both monocular and binocular inputs matter to these cells (redrawn from [83], using data from ferret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This more advanced wiring suggests the connection structures in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Informal Setup and Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since much of the paper is technical, we here specify, informally, the main ingredients of the model and the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We first list several of the key points, then illustrate them directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Stereo geometry enjoys a mathematical structure that is a formal extension of plane curve geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the plane, points belonging to a curve are described by an orientation 0HMonocular inputs Binocular inputs4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 at a position, and these are naturally represented as elements (orientation, position) of columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In our model, these become abstract fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The collection of fibres across position is a fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Elements of the (monocular) fibre can be thought of as neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For stereo, we shall need fibres that are a "product" of the left and right-eye monocular columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The natural coordinates on the stereo fiber bundle are position, positional disparity and orientations from the left and right eyes respectively, which describe fiber over each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The columnar organization of the stereo system, beyond what is shown in the Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, is completely unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' While visual area MT is suggestive of columns for direction of motion ([61, 22]) and perhaps V4 for slant ([36]), there is no direct evidence of which we are aware in V1 for spatial or orientation disparity columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is the reason why models can be insightful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Curvature provides a kind of "glue" to enable transitions from points on fibres to nearby points on nearby fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These transitions specify "integral curves" through the stereo fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The integral curve viewpoint provides a direction of information flow (information diffuses through the bundle) thereby suggesting underlying circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The integral curves formalize association field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Their parameters describe the spray of curves that is well in accordance with 3D curves as studied in psychophysical experiments in [32, 34, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our formal theory resolves several conjectures in the literature [48, 49, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our formal theory provides a new framework for specifying the correspondence problem, by illustrating how good continuation in the 3-D world generalizes good continuation in the 2-D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is the point where consistent binocular-binocular interactions are most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our formal theory has direct implications for understanding torsional eye movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It suggests, in particular, that the rotational component is not simply a consequence of development, but that it helps to undo inappropriate orientation disparity changes induce by eye movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This provides a novel role for Listing’s Law, and is treated in a companion paper (in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now illustrate these ideas (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Consider a three-dimensional stimulus as a space curve γ : R −→ R3, with a unitary tangent at the point of fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since the tangent is the derivative of a curve, the binocular cells naturally encode the unitary tangent direction ˙γ to the spatial 3D stimulus γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This space tangent projects to a tangent orientation in the left eye1, and perhaps the same or a different orientation in the right eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A nearby space tangent projects to another pair of monocular tangents, illustrated as activity in neighboring columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Note how connections between the binocular neurons support consistency along the space curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is this consistency relationship that we capture with our model of the stereo association field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since space curves live in 3D, two angles are required to specify its space tangent at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In other words, monocular tangent angles span a circle in the plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' space tangent angles span a 2-sphere in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In terms of the projections into the left-eye and the right-eye, the space tangent can be described by the parameters n = (θ, ϕ) of S2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus, we can suitably describe the space of stereo cells – the full set of space tangents at any position in the 3D world – as the manifold of positions and orientations R3 ⋊ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Moving from one position in space to another, and changing the tangent orientation to the one at the new position, amounts to what is called a group action on the appropriate manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We informally introduce these notions in the next subsection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' a more extensive invitation to these ideas is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Sub-Riemannian Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We live in a 3D world in which distances are familiar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' that is, a space of points with a Euclidean distance function defined between any pair of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Apart from 1We are here being loose with language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' By a tangent orientation in the left eye, we mean the orientation of a left-eye innervated column in V1 GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 5 (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Stereo projection of the highlighted tangent vector to the stim- ulus γ ∈ R3 in the left-eye innervated and right-eye innervated monocular orien- tation columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (Each short line denotes a neuron by its orientation preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=') Joint activity across the eyes, which denotes the space tangent, is illustrated by the binocular neuron (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Note the two similar but distinct monocular ori- entations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Connections from the actively stimulated monocular neurons to the binocular neuron are shown as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Stereo projection of a consecu- tive pair of tangents to the stimulus γ ∈ R3 in the left and right retinal columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Each space tangent projects to a different pair of monocular columns because of the spatial disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Consistency in the responses of these four columns cor- responds to consistency between the space tangents attached nearby positions along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This consistency is realized through the binocular neural connection (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' practical considerations we can move in any direction we would like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Cars, however, have much more restricted movement capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' They can move forward or backward, but not sideways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To move in a different direction, cars must turn their wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Here is the basic analogy: in cortical space information can move to a new retinotopic position in a tangent direction, or it can move up or down a column (orientation fibre) to change direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Moving in this fashion, from an orientation at a position to another orientation at a nearby position, is clearly more limited than arbitrary movements in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Euclidean geometry, as above, is an example of a Riemannian geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' the limitations involved in moving through a cortical columnar space r3 Y r1 r2r3 Y r1 r2R YR X6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The full geometry of stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Note how the stereo correspon- dence problem allows to establish the relationship between the 3D tangent point (P, θ, φ) and the projections pL and pR, the disparity and the orientations θL and θR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' specify a sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Just as cars can move along a roads that are mostly smooth, excitatory neurons mainly connect to similarly "like" (in orientation) excitatory neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This chain of neurons indicates a path through sub-Riemannian space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' the fan of such paths is the cortical connectivity which can be considered the neural correlate of association fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Again, for more information please consult Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Moving now out to the world, we must be able to move between all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Repeating the above metaphor more technically, we equip R3 ⋊ S2 with a group action of the three-dimensional Euclidean group of rigid motions SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Notice, importantly, that this group is now acting on the product space of positions and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A bit more is required, though, since the geometry of the stereo vision is not solved only with these punctual and directional arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3 2 R R y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 7 As we showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2 there is the need to take into account the relationships between nearby tangents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' in geometric language this involves a suitable type of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is therefore natural to look at integral curves of the sub-Riemannian structure, which encode in their coefficients the fundamental concept of 3D curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An example of this is shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Notice how the 3D association field envelopes a space curve, in the same way that a 2D association field envelopes a planar curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This figure illustrates, in a basic way, the fundamental result in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The three-dimensional space curve γ is enveloped by the 3D the association field centered at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally, this association field is a fan of integral curves in the sub-Riemmanian geometry computed entirely within the columnar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (It is specifically described by equation (36) with varying c1 and c2 in R, but that will take some work to develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Overview of Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The paper is organized as follows: in Section 2, we describe the geometrical and neuro-mathematical background underlying the problem of stereo vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, we review the standard stereo triangulation technique to relate the coordinate system of one retina with the other, and put them together in order to reconstruct the three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Then, we briefly review the classical neurogeometry of monocular simple cells selective for orientation and the underlying connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The generalization of co-circularity for stereo is also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In Section 3, starting from binocular receptive profiles, we introduce the neuro- mathematical model for binocular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' First we present the cortical fiber bundle of binocular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It follows the differential interpretation of the binocular profiles in terms of the neurogeom- etry of the simple cells, and we show how this is well in accordance with the results of the stereo triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Then, we give a mathematical definition of the manifold R3 ⋊ S2 with the sub- Riemannian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Finally, we study the integral curves and the suitable change of variables that allow us to switch our analysis from cortical to external space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In Section 4 we proceed to the validation of our geometry with respect to psychophysical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We combine information about the psychophysics of 3D perception and formal conjectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' it is here that we formulate a 3D association field analogous to the 2D association field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' At the end, we show an example of C今 r28 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 a lifting of a stimulus and how our integral curves properly connect corresponding points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This illustrates the use of our model as a basis for solving the correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Stereo vision and neuro-mathematical background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Stereo geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this subsection we briefly recall the geometrical configuration under- lying 3D vision, to define the variables that we use in the rest of the paper, mainly referring to [29, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For a complete historical background see [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Stereo variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We consider the global reference system (O, i, j, k) in R3, with O = (0, 0, 0), and coordinates (r1, r2, r3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We introduce the optical centers CL = (−c, 0, 0) and CR = (c, 0, 0), with c real positive element, and we define two reference systems: (CL, iL, jL), (CR, iR, jR), the reference systems of the retinal planes RL and RR with coordinates respectively (xL, y), (xR, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the global system we suppose the retinal planes to be parallel and to have equation r3 = f, with f denoting the focal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This geometrical set-up is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Reconstruction of the 3D space point Q through points QL the retinal plane RL and QR in , RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If we know the coordinate of a point Q = (r1, r2, r3)T in R3, then it is easy to project it in the two planes via perspective projection, having c the coordinate of the optical centers and f focal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This computation defines two projective maps ΠL and ΠR, respectively, for the left and right retinal planes: (1) ΠL : R3 −→ R2 ΠR : R3 −→ R2 � � r1 r2 r3 � � �→ � f(r1+c) r3 fr2 r3 � , � � r1 r2 r3 � � �→ � f(r1−c) r3 fr2 r3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r3 r2 r1 C R YR RR Q Q R RL XR XLGOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A point on the left retinal plane of local coordinates (xL, y)T has global coordinates QL = (−c + xL, y, f)T , and it corresponds to a point Q = (r1, r2, r3)T in the Euclidean R3 such that CL, QL and Q are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This means that the vectors QL − CL = (xL, y, f)T and Q − CL = (r1 + c, r2, r3)T are parallel, obtaining the following relationships: (2) xL = f r1 + c r3 , y = f r2 r3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Analogously, considering QR and CR, we get: (3) xR = f r1 − c r3 , y = f r2 r3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' □ In a standard way, the horizontal disparity is defined as the differences between retinal coor- dinates (4) d := xL − xR 2 , up to a scalar factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Moreover, it is also possible to define the coordinate x as the average of the two retinal coordinates x := xL+xR 2 , leading to the following change of variables: (5) � � � � � x = fr1 r3 y = fr2 r3 d = fc r3 ←→ � � � � � r1 = xc d r2 = yc d r3 = fc d , where the set of coordinates (x, y, d) is known as cyclopean coordinates [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Tangent estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Corresponding points in the retinal planes allow to project back into R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An analogous reasoning can be done for the tangent structure: if we have tangent vectors of corresponding curves in the retinal planes, it is possible to project back and recover an estimate of the 3D tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Let us recall here this result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' a detailed explanation can be found in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Let γL and γR be corresponding left and right retinal curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=', perspective projections of a curve γ ∈ R3 through optical centers CL and CR with focal length f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Knowing the left and right retinal tangent structures, it is possible to recover the direction of the tangent vector ˙γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Starting from a curve γ ∈ R3, we project it in the two retinal planes obtaining γL = ΠL(γ) and γR = ΠR(γ) from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The retinal tangent vectors are obtained through the Jacobian matrix 2 of the left and right retinal projections ˙γL,R(t) = (JΠL,R)γ(t) ˙γ(t): (6) ˙γR(t) = � f(γ3 ˙γ1+(c−γ1) ˙γ3) γ3(t)2 f(γ3 ˙γ2−γ2 ˙γ3) γ2 3 � , ˙γL(t) = � f(γ3 ˙γ1−(c+γ1) ˙γ3) γ3(t)2 f(γ3 ˙γ2−γ2 ˙γ3) γ2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2The Jacobian matrix (JΠ)p evaluated at point p represents how to project displacement vectors (in the sense of derivatives or velocities or directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In details, if ˙γ(t) is the displacement vector in R3, then the matrix product (JΠ)γ(t) ˙γ(t) is another displacement vector, but in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In other words, the Jacobian matrix is the differential of Π at every point where Π is differentiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' common notation includes JΠ or DΠ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Extending the tangent vectors and the points into R3, we get ˜tL = (˙γL1, ˙γL2, 0)T , and ˜mL = (γL1− c, γL2, f)T , and UtL = (PL)−1 ˜mL × (P −1 L )˜tL, with the projection matrix PL = � � 1 0 −c/f 0 1 0 0 0 1 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The same reasoning holds for the right structure, with projection matrix PR = � � 1 0 c/f 0 1 0 0 0 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Then UtR × UtL is a vector parallel to the tangent vector ˙γ: (7) UtR × UtL = � � � � � f 42c(˙γ2γ3 − ˙γ3γ2) γ4 3 � �� � λ(t) ˙γ1, f 42c(˙γ2γ3 − ˙γ3γ2) γ4 3 ˙γ2, f 42c(˙γ2γ3 − ˙γ3γ2) γ4 3 ˙γ3 � � � � � T = λ(t) (˙γ1(t), ˙γ2(t), ˙γ3(t))T = λ(t)˙γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Elements of neuro-mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now provide background on the geometric mod- eling of the monocular system, and good continuation in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our goal is to illustrate the role of sub-Riemannian geometry in the monocular system, which will serve as the basis for generalization in the stereo system starting from the neuro-mathematical model of Citti and Sarti [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Classical neurogeometry of simple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We model the activation map of a cortical neuron’s receptive field (RF) by its receptive profile (RP) ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A classical example is the receptive profiles of simple cells in V1, centered at position (x, y) and orientation θ, modeled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g in [7, 21, 45]) as a bank of Gabor filters ϕ{x,y,θ}, which act on a visual stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally, it is possible to abstract the primary visual cortex as R2 ×S1, or position-orientation space, thereby naturally encoding the Hubel/Wiesel hypercolumnar structure [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An example of this structure is displayed in image (A) of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4 from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Following the neuro-mathematical model of Citti and Sarti [18], the set of simple cells RPs can be obtained via translations of vector (x, y)T and rotation of angle θ from a unique "mother" profile ϕ0(ξ, η) (8) ϕ0(ξ, η) = exp �2πiξ λ � exp � −ξ2 + η2 2σ2 � , a Gabor function with real (even) and imaginary (odd) parts (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Translations and rotations can be expressed as: (9) T(x,y,θ)(ξ, η) = �cos θ − sin θ sin θ cos θ � �ξ η � + �x y � , where T(x,y,θ) denotes the action of the group of rotations and translations SE(2) on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This group operation associates to every point (ξ, η) a new point (˜x, ˜y), according to the law (˜x, ˜y) = T(x,y,θ)(ξ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Hence a general RP can be expressed as (10) ϕ(x,y,θ)(ξ, η) = ϕ0(T −1 (x,y,θ)(ξ, η)), and this represents the action of the group SE(2) on the set of receptive profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The retinal plane R is identified with the R2 plane, whose coordinates are (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' When a visual stimulus I : R −→ R+ of intensity I(x, y) activates the retinal layer, the neurons centered GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 11 (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Even (A) and odd (B) part of Gabor function: the surface of the two-dimensional filters, their common bi-dimensional representation and a mono-dimensional section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' at every point (x, y) produce an output O(x, y, θ), which can be modeled as the integral of the signal I with the set of Gabor filters: (11) O(x, y, θ) = � R ϕ{x,y,θ}(ξ, η)I(ξ, η)dξdη, where the function I represents the retinal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For (x, y) fixed, we will denote ¯θ the point of maximal response: (12) max θ |O(x, y, θ)| = |O(x, y, ¯θ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We will then say that the point (x, y) is lifted to the point (x, y, ¯θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is extremely important conceptually to understand our geometry: it illustrates how an image point, evaluated against an simple cell RP, is lifted to a "cortical" point by introducing the orientation explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If all the points of the image are lifted in the same way, the level lines of the 2D image I are lifted to new curves in the 3D cortical space (x, y, ¯θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We shall now introduce a set of directions for moving on the cortical space (x, y, ¯θ), in the sense of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is important because it will be necessary to move within this space, across both positions and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Biologically, such movements would be the flow of information from one cell in a column to another cell in a nearby column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To begin, in the right hand side of the equation (11) the integral of the signal with the real and imaginary part of the Gabor filter is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The two families of cells have different shapes, hence they detect (or play a role in detecting) different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since the odd-symmetry cells sug- gest boundary detection, we concentrate on them, but this is mainly a convenience for computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The output of a simple cell can then be locally approximated as O(x, y, θ) = −X3,p(Iσ)(x, y), where p = (x, y, θ) ∈ SE(2), Iσ is a smoothed version of I, obtained by convolving it with a Gaussian kernel, and (13) X3,p = − sin θ∂x + cos θ∂y, is the directional derivative in the direction ⃗X3,p = (− sin θ, cos θ, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' From now on, we will denote (by a slight abuse of notation) ω⋆ := ⃗X3,p to remind the reader familiar with the language of 1-forms the correspondence of these quantities, and the relation with the Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3 3The purpose of introducing this notation is also to motivate an implication of the mathematical model in [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Now, think of vector fields as defining a coordinate system at each point in cortical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Then, in addition to above, the vector fields orthogonal to X3,p are: (14) X1,p = cos θ∂x + sin θ∂y, X2,p = ∂θ and they define a 2-dimensional admissible tangent bundle4 to R2 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' One can define a scalar product on this space by imposing the orthonormality of X1,p and X2,p: this determines a sub- Riemannian structure on R2 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Examples of the compatibilities around the central point of the image, derived from planar co-circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Brightness encodes compatibility values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Starting from the central initial oriented point, the solid line indicates a configuration between the patches where the association exists while the dashed line indicates a configuration where it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Association field of Field, Hayes and Hess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The visual signal propagates, in an anisotropic way, along cortical connectivity and connects more strongly cells with comparable orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This propagation establishes the connection between the geometry just developed and 2-dimensional contour integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is a formal- ization of the Gestalt law of good continuation [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It first arose in a simpler form, namely co-circularity in the plane [67], to describe the consistency and the compatibility of neighboring oriented points, in accordance with specific values of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An example of these compatibil- ities can be found in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3, image (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is complemented by psychophysical experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' [42, 84, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, Field et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' in [30] describe the association rules for 2-dimensional contour integration, introducing the concept of association fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A representation of these con- nections can be found in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3, images (B) and (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Note that this is equivalent to the union (over curvature) in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Neurophysiological studies [10, 12, 35, 59, 82] suggest that the cortical correlate of the association field is the long-range horizontal connectivity among cells of similar (but not necessarily identical) orientation preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Based on these findings, Citti and Sarti in [18] modeled cortical propagation as propagation along integral curves of the vector fields X1 and X2, namely curves γ : [0, T] ⊂ R −→ R2 × S1 described by the following differential equation: (15) ˙γ(t) = ⃗X1,γ(t) + k ⃗X2,γ(t), t ∈ [0, T], obtained by varying the parameter k ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (k acts analogously as curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=') An example of these curves is in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Their 2D projection is a close approximation of the association fields (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A related model has been proposed by Duits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' They study the geodesics of the sub- Riemannian structure to take into account all appropriate end-conditions of association fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4as defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3 GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 13 (a) (b) (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Orientation columns of cells in (x, y, θ) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Long- range horizontal connections between cells relate an orientation signal at posi- tion (x, y, θ) to another orientation at (x′, y′, θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Horizontal integral curves in R2 × S1 generated by the sub-Riemannian model [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Projection of the fan of the integral curves in the (x, y) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Generalizing co-circularity for stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The concept of co-circularity in R2 has been de- veloped by observing that a bidimensional curve γ can be locally approximated at 0 via the osculating circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' in [3, 57, 58] generalize this concept with the Frenet differential geometry of a three dimensional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' While in the two-dimensional case the approximation of the curve using the Frenet 2D basis causes the curvature to appear in the coefficient of the Taylor series development (1st order), in the three-dimensional case the coefficients involve both the curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, in [3] the authors propose heuristically to generalize the osculating circle for space curves with an osculating helix, with a preference for r3-helices to improve stability in terms of camera calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this way the orientation disparity is encoded in the behavior of the helix in the 3D space: there is no difference in orientation in the retinal planes if the helix is confined to be in the fronto-parallel plane (the helix becomes a circle), otherwise moving along the 3D curves the retinal projections have different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In [57, 58] they observe that, by introducing the curvature variable as a feature in the two monocular structures, and assuming correspondence, it is possible to reconstruct the 3D Frenet geometry of the curve, starting from the two-dimensional Frenet geometry, up to the torsion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, they prove: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Given two perspective views of a 3D space curve with full calibration, the normal N and curvature k at a curve space point are uniquely determined from the positions, tangents, and curvatures of its projections in two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus the Frenet frame {T, N, B} and curvature k at the space point can be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Hence, using the knowledge of the Frenet basis together with the fundamental addition of the curvature variable, Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' introduced the concept of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This allowed moving the 3D Frenet frame in a consistent way with the corresponding 2D Frenet structures of the left and right retinal planes, to establish stereo correspondence between pairs of (left and right) pairs of tangents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 image (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The model that we propose in this paper is related to, but differs from, what has just been stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, to remain directly compatible with the previous neuro-geometric model, we will work only with the monocular variables of position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Rather than using curvature directly, we shall assume that these variables are encoded within the connections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Long range connections to compatible cells X Rearrangement of orientation hypercolumns by retinotopic position14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Geometrical setup of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The spiral curve 3D projects in the left and right retinal planes together with the Frenet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Stereo correspondence between pairs of (left-right) pairs of tangents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Both figures are taken from [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' mathematically they appear as parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A theoretical result of our model is that the heuristic assumption regarding the r3-helix can now be established rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Let us also mention the paper [1], where the curvature was considered as independent variable and helices have been obtained in the 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The neuromathematical model for stereo vision 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Binocular profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Binocular neurons receive inputs from both the left and right eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To facilitate calculations, we assume these inputs are first combined in simple cells in the primary visual cortex, a widely studied approach ([5, 20, 47, 62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It provides a first approximation in which binocular RPs are described as the product of monocular RPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, image (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is of course a simplification – see [80], for instance – but it is compatible with existing neural findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This binocular model allows us to define disparity and frontoparallel coordinates as (16) � d = xL−xR 2 x = xR+xL 2 , perfectly in accordance with the introduction of cyclopean coordinates in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this way (x, y, d) correspond to the neural correlate of (r1, r2, r3), via the change of variables (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The cortical fiber bundle of binocular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The hypercolumnar structure of monoc- ular simple cells (orientation selective) has been described as a jet fiber bundle in the works of Petitot and Tondut [70], among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We concentrate on the fiber bundle R2 × S1, with fiber S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' [9] among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In our setting, the binocular structure is based on monocular ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' recall the example illus- trations from the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, for each cell on the left eye there is an entire fiber of cells on the right, and vice versa, for each cell on the right there is an entire fiber of cells on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This implies that the binocular space is equipped with a symmetry that involves the left and right structures, allowing us to use the cyclopean coordinates (x, y, d) defined in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Hence, we define the cyclopean retina R, identified with R2, endowed with coordinates (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The structure of the fiber is F = R×S1 ×S1, with coordinates (d, θL, θR) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The total space is T B X fi Z P1 Ci XI Pr f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W Zr 5 XTransport in F Y x pairi y pairi VT xrGOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 15 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Comparisons between binocular interaction RPs and the product of left and right eye RPs, where left and right RPs are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Binocular interaction RPs (Raw data) of a cell is shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Contour plots for the product of left and right eye RPs (L×R) are shown in the right along with 1-dimensional profiles of the left (L) and right (R) eye RPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure adapted from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' defined in a trivial way, E = R×F = R2×R×S1×S1, and the projection π : E −→ R is the trivial projection π(x, y, d, θL, θR) = (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The preimage of the projection E(x,y) := π−1({(x, y)}), for every (x, y) ∈ R, is isomorphic to the fiber F, and the local trivialization property is naturally satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A schematic representation can be found in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The base has been depicted as 1- dimensional, considering the restriction R|x of the cyclopean retina R on the coordinate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The left image displays only the disparity component of the fiber F, encoding the relationships between left and right retinal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The right image shows the presence of the left and right monodimensional orientational fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Binocular energy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To simplify calculations, as stated in the Introduction, we follow the classical binocular energy model [5] for binocular RPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The basic idea is a binocular neuron receives input from each eye;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' if the sum OL + OR of the inputs from the left and right eye is positive, the firing rate of the binocular neuron is proportional to the square of the sum, and it vanishes, if the sum of the inputs is negative: (17) OB = (Pos(OL + OR))2, with Pos(x) = max{x, 0}, OB the binocular output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Raw data (B) AD B AD A kd058m22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='03r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='58 kd449m20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='02r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='59 B B x D Prediction (LxR) E D D AD LxR LxR 口 6(deg) 5 (deg) 6 (deg) 5 (deg)16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Left: schematic representation of the fiber bundle in two dimen- sion, with relationships between left and right retinal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Right: repre- sentation of the selection of a whole fiber of left and right simple cells, for every x and for every d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If OL + OR > 0, then the output of the binocular simple cell can be explicitly written as OB = O2 L + O2 R + 2OLOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The first two terms represent responses due to monocular stimulation while the third term 2OLOR can be interpreted as the binocular interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The activity of a cell is then measured from the output and will be strongest at points that have a higher probability of matching each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The maximum value over d of this quantity is the extracted disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is worth noting that neurophysiological computations of binocular profiles displayed in Fig- ure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 assume the mono-dimensionality of the monocular receptive profile, ignoring information about orientation of monocular simple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' However, this information will be needed to encode different types of orientation disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 (Orientation matters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In 2001, the authors of [14] conducted investigations on the response of binocular neurons to orientation disparity, by extending the energy model of Anzai, Ohzawa and Freeman to incorporate binocular differences in receptive-field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' More recently, the difference between orientations in the receptive fields of the eyes has been confirmed [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The binocular energy model is a type of minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It serves as a starting point, allowing the combination of monocular inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' But is not sufficient to solve the stereo-matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2 (Connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is argued in [68, 75] that, in addition to the neural mechanisms that couple characteristics (such as signals, stimuli, or particular features) relating the left and right monocular structures, there must be a system of connections between binocular cells, which characterizes the processing mechanism of stereo vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see also Samonds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' in [75] in par- ticular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Differential interpretation of binocular RPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is possible to write the interaction term OLOR coming from (17), in terms of the left and right receptive profiles: (18) OLOR = � ϕθL,xL,y(˜xL, ˜yL)IL(˜xL, ˜yL)d˜xLd˜yL � ϕθR,xR,y(˜xR, ˜yR)IR(˜xR, ˜yR)d˜xRd˜yR = � � ϕθL,xL,y(˜xL, ˜yL)ϕθR,xR,y(˜xR, ˜yR)IL(˜xL, ˜yL)IR(˜xR, ˜yR)d˜xRd˜yRd˜xLd˜yL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' d d xR = X-d xL = x+d xR = x-d xL = x+d x xGOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 17 If we fix (˜xR, ˜yR, ˜xL, ˜yL), we derive the expression of the binocular profiles ϕL,R = ϕθR,xR,yϕθL,xL,y as the product of monocular left and right profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is in accordance with the measured pro- files of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The binocular interaction term can be associated with the cross product of the left and right directions defined through (13), namely ω⋆ L and ω⋆ R of monocular simple cells: (19) OLOR = ω⋆ L × ω⋆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The idea is that the binocular output is the combined result of the left and right actions of monocular cells, thus identifying a direction in the space of cyclopean coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The detailed proof of this proposition can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' □ To better understand the geometrical idea behind Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, we recall that the retinal coordinates can be expressed in terms of cyclopean coordinates (4) as xR = x−d and xL = x+d, and so we can write ω⋆ L and ω⋆ R in the 3D space of coordinates (x, y, d) as: (20) ω⋆ R =(− sin θR, cos θR, sin θR)T ω⋆ L = (− sin θL, cos θL, − sin θL)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We define ωbin := ω⋆ L × ω⋆ R as the natural direction characterizing the binocular structure: (21) ωbin = � � sin(θR + θL) 2 sin θR sin θL sin(θR − θL) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The vector ωbin of equation (21) can be interpreted as the intersection of the orthogonal spaces defined with respect to ω⋆ R and ω⋆ L when expressed in cyclopean coordinates (x, y, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' More precisely, if (22) (ω⋆ L)⊥ = span � � � � � cos θL sin θL 0 � � , � � −1 0 1 � � � � � , (ω⋆ R)⊥ = span � � � � � cos θR sin θR 0 � � , � � 1 0 1 � � � � � then (23) ωbin = (ω⋆ L)⊥ ∩ (ω⋆ R)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The result of the intersection of these monocular structures identifies a direction, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We earlier showed that the result of the action of a monocular odd simple cell is to select directions for the propagation of infomation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now combine these, for the two eyes, to show that in the three-dimensional case the binocular neural mechanisms also lead to a direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We will see in the next sections that this direction is the direction of the tangent vector to the 3D stimulus, provided points are corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Compatibility with stereo geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We consider the direction characterizing the binoc- ular structure ωbin defined in (21) and we show that it can be associated with the 3D tangent vector to the 3D curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The idea is that this tangent vector is orthogonal both to ω⋆ R and to ω⋆ L, and therefore it has the direction of the vector product ω⋆ L × ω⋆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Precisely, we consider the normalized tangent vector tL and tR on retinal planes (24) tR = (cos θR, sin θR)T tL = (cos θL, sin θL)T , to the points (xR, y) and (xL, y) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Taking into account that f is the focal coordinate of the retinal planes in R3, then we associate to these points the correspondents in R3, namely 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Direction detected by ωbin through the intersection of left and right planes generated by (ω⋆ R)⊥ and (ω⋆ L)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Red vector corresponds to the associated 2-form ωbin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ˜mL = (xL − c, y, f)T , ˜mR = (xR + c, y, f)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Applying equation (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' it is possible to derive the tangent vector of the three dimensional contour: (25) UtL = P −1 L ˜mL × P −1 L ˜tL = � � xL yL f � � × � � cos θL sin θL 0 � � = � � −f sin θL f cos θL xL sin θL − yL cos θL � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' UtR = P −1 R ˜mR × P −1 R ˜tR = � � xR yR f � � × � � cos θR sin θR 0 � � = � � −f sin θR f cos θR xR sin θR − yR cos θR � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' and the tangent direction is recovered by (26) UtL × UtR = f � � xL+xR 2 sin(θR − θL) − xR−xL 2 sin(θL + θR) y sin(θR − θL) − (xR − xL)(cos(θR − θL) − cos(θL + θR)) f sin(θR − θL) � � (a) (wr)GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 19 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Three dimensional reconstruction of the space from retinal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The 1- forms ω⋆ L and ω⋆ R are identified with the normal to the curves γL and γR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Their three dimensional counterpart ˜ω⋆ L and ˜ω⋆ R identify the tangent vector to the curve γ : R → R3 by the cross product ˜ω⋆ L × ˜ω⋆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If we define (27) ˜ω⋆ L := d fcUtL, ˜ω⋆ R := d fcUtR and the corresponding 2 form ωR3 := ˜ω⋆ L × ˜ω⋆ R, using the change of variables (16) we observe that: (28) ˜ω⋆ L = ω⋆ L, ˜ω⋆ R = ω⋆ R, ωR3 = ωbin, up to a scalar factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' See Appendix C for explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this way, the disparity binocular cells couple in a natural way positions, identified with points in R3, and orientations in S2, identified with three-dimensional unitary tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' As already observed in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2, the geometry of the stereo vision is not solved only with these punctual and directional arguments, but there is the need to take into accounts suitable type of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In [3, 57, 58], Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' proposed a model that considered the curvature of monocular structures as an additional variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Instead, we propose to consider simple monocular cells selective for orientation, and to insert the notion of curvature directly into the definition of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is therefore natural to introduce the perceptual space via the manifold R3 ⋊ S2, and look for appropriate curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A perceptual model in the space of 3D position-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now derive the objects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We have clarified (end of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5) that binocular cells are parametrized by points in R3, and orientations in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An element ξ of the space R3 ⋊ S2 it is defined by a point p = (p1, p2, p3) in R3 and an unitary vector n ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since the topological dimension 8* X R CR ★m R r220 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 of this geometric object is 2, we introduce the classical spherical coordinates (θ, ϕ) such that n = (n1, n2, n3)T ∈ S2 can be parameterized as: (29) n1 = cos θ sin ϕ n2 = sin θ sin ϕ n3 = cos ϕ with θ ∈ [0, 2π] and ϕ ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The ambiguity that arises using local coordinate chart is overcome by the introduction of a second chart, covering the singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Translations and rotations are expressed using the group law of the three-dimensional special Euclidean group SE(3), defining the group action (30) σ : R3 ⋊ S2 × SE(3) −→ R3 ⋊ S2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' σ((p, n), (q, R)) = (Rp + q, Rn), with (p, n) ∈ R3 ⋊ S2 , (q, R) ∈ SE(3), namely R ∈ SO(3) tridimensional rotation, and q ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Stereo sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The emergence of a privileged direction in R3 (asso- ciated with the tangent vector to the stimulus) is the reason why we endow R3 ⋊ S2 with a sub-Riemannian structure that favors the direction in 3D identified by ωbin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally, we consider admissible movements in R3 ⋊ S2 described by vector fields: (31) YR3,ξ = sin ϕ cos θ∂1 + sin ϕ sin θ∂2 + cos ϕ∂3 Yθ,ξ = − 1 sin ϕ∂θ Yϕ,ξ = ∂ϕ with ξ ∈ R3 ⋊ S2 for ϕ ̸= 0, ϕ ̸= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The admissible tangent space5 at a point ξ (32) Aξ := span{YR3,ξ, Yθ,ξ, Yϕ,ξ} encodes the coupling between position and orientations, as remarked by Duits and Franken in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, the vector field YR3 identifies the privileged direction in R3, while Yθ and Yϕ allow changing this direction, involving just orientation variables of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The vector fields {YR3, Yθ, Yϕ} and their commutators generate the tangent space of R3 ⋊ S2 in a point, allowing to connect every point of the manifold using privileged directions (Hörmander condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Furthermore, it is possible to define a sub-Riemannian structure by choosing a scalar product on the admissible tangent bundle A: the simplest choice is to declare the vector fields {YR3, Yθ, Yϕ} orthonormal, considering on S2 the distance inherited from the immersion in R3 with the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We have already expressed the change of variable in the variables (x, y, d) to (r1, r2, r3) in equations (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' However, the cortical coordinates also contain the angular variables θR and θL which involve the introduction of the spherical coordinates θ, ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To identify a change of variable among these variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' we first introduce the function (r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ϕ) F−→ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θR) : (33) F : R3 ⋊ S2 −→ R3 ⋊ S2 � � � � � � r1 r2 r3 θ ϕ � � � � � � �→ � � � � � � � fr1 r3 fr2 r3 cf r3 tan−1( r3 sin θ cos ϕ−r2 cos ϕ r3 cos θ sin ϕ−(c+r1) cos ϕ) tan−1( r3 sin θ cos ϕ−r2 cos ϕ r3 cos θ sin ϕ−(c−r1) cos ϕ � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' where the retinal right angle θR = tan−1( r3 sin θ cos ϕ−r2 cos ϕ r3 cos θ sin ϕ−(c+r1) cos ϕ) and the left retinal angle θL = tan−1( r3 sin θ cos ϕ−r2 cos ϕ r3 cos θ sin ϕ−(c−r1) cos ϕ) are obtained considering equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 5see Appendix A for the definition of admissible tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 21 Analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' it is possible to define the change of variable (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θR) G −→ (r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ϕ): (34) G : R3 ⋊ S2 −→ R3 ⋊ S2 � � � � � � x y d θR θL � � � � � � �→ � � � � � � � cx d cy d cf d tan−1( 2 sin θR sin θL sin(θR+θL) ) tan−1( √ sin2(θR+θL)+4 sin2 θR sin2 θL sin(θR−θL) ) � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' where the angles θ = tan−1( 2 sin θR sin θL sin(θR+θL) ) and ϕ = tan−1( √ sin2(θR+θL)+4 sin2 θR sin2 θL sin(θR−θL) ) are ob- tained considering that tan θ = (⃗YR3)2 (⃗YR3)1 and tan ϕ = √ (⃗YR3)2 1+(⃗YR3)2 2 (⃗YR3)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Integral curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The connectivity of the space is described by admissible curves of the vector fields spanning A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, a curve Γ : [0, T] −→ R3 ⋊ S2 is said to be admissible6 if: (35) ˙Γ(t) ∈ AΓ(t), ↔ ˙Γ(t) = a(t)⃗YR3,Γ(t) + b(t)⃗Yθ,Γ(t) + c(t)⃗Yϕ,Γ(t), where a, b, c are sufficiently smooth function on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We will consider a particular case of these admissible curves, namely constant coefficient integral curves with a(t) = 1, since the vector field YR3 represents the tangent direction of the 3D stimulus (and so it never vanishes): (36) ˙Γ(t) = ⃗YR3,Γ(t) + c1⃗Yθ,Γ(t) + c2⃗Yϕ,Γ(t), with c1 and c2 varying in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These curves can be thought of in terms of trajectories in R3 describing a movement in the ⃗YR3 direction, which can eventually change according to ⃗Yθ and ⃗Yϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An example of the fan of integral curves was shown in the Introduction in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is worth noting that in the case described by coefficients c1 and c2 equal to zero, the 3D trajectories would be straight lines in R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' by varying the coefficients c1 and c2 in R, we allow the integral curves to follow curved trajectories, twisting and bending in all space directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally, the amount of "twisting and bending" in space is measured by introducing the notions of curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We then investigate how these measurements are encoded in the parameters of the family of integral curves, and what constraints have to be imposed to obtain different typologies of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The 3D projection of the integral curves (36) will be denoted γ and satisfy ˙γ(t) = (cos θ(t) sin ϕ(t), sin θ(t) sin ϕ(t), cos ϕ(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Classical instruments of differential geometry let us compute the curvature and the torsion of the curve γ(t): (37) k = � ( ˙ϕ)2 + sin2 θ( ˙θ)2, τ = 1 k2 (− cos ϕ sin2 ϕ( ˙θ)3 − sin ϕ ˙ϕ¨θ + ˙θ(−2 cos ϕ( ˙ϕ)2 + sin ϕ ¨ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Using the explicit expression of the vector fields Yθ and Yϕ in equation (36), we get (38) ˙θ = − c1 sin ϕ, ˙ϕ = c2, from which it follows that: (39) k = � c2 1 + c2 2 τ =c2 1 − c2 2 k2 c1 cotan ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 6sometimes the term horizontal is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' By varying the parameters c1 and c2 in (39) where we explicit find solutions of (36), we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If ϕ = ψ 2 then k = � c2 1, τ = 0, and so the family of curves (36) are circles of radius 1/c2 1 on the fronto-parallel plane r3 = cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If ϕ = ϕ0, with ϕ0 ̸= π/2, then k = � c2 1 and τ = c1 cotan ϕ0, and so the family of curves (36) are r3-helices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If θ = θ0 then k = � c2 2, τ = 0, and so the family of curves (36) are circles of radius 1/c2 2 in the osculating planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If c1 = ±c2 then τ = 0, and so the family of curves (36) are circles of radius 1/c2 2 in the osculating planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The computation follows immediately from the computed curvature and torsion of (39) and classical results of differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' □ (a) (b) (c) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Examples of integral curves obtained varying parameters c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Arc of circles for ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) r3-helices for ϕ = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Family of curves with constant curvature k and varying torsion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If we know the value of the curvature k, and we have one free parameter, c2, in the definition of the integral curves (36), then we are in the setting of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In fact, the coefficient c1 is obtained by imposing c1 = ± � k2 − c2 2, and in particular the component that remains to be determined is the torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Examples of particular cases of the integral curves (36) according to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 are visualized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Comparison with experimental data In this section we present results of compatibility between the proposed sub-Riemannian model and biological and psychophysical phenomena present in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Biological Connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The foundation for building our sub-Riemannian model of stereo was a sub-Riemannian model of curve continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This was motivated by the orientation column at each position, and the connections between cells in nearby columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These connections were, in turn, a direct model of the long-range horizontal connections in visual cortex, for which there is beautiful biological data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We further illustrated, in the Introduction, aspects of the cortical architecture that support binocular processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Although the inputs from each eye are organized into ocular dominance bands, there is no direct evidence for "stereo columns" analogous to the monocular orientation columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' But, as we shall now show, there is evidence of long-range connections between binocular cells, and our model informs, concretely, what information should be carried by these long range connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus, an organization for stereo is suggested, but it is implicit in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Nevertheless, there is evidence in support of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' r23 53 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 23 (a) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) A biocytin injection superimposed on a map of ocular domi- nance columns, image result from the work in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Binocular zones are in the middle of monocular zones (coded in black and white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Starting from the injec- tion site (yellow circle in the center of a binocular zone) the patches’ propagation (red corresponds to dense while green to sparsely labeled) tends to avoid highly monocular sites, bypassing the centers of ocular dominance columns, and are located in binocular zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) 3D interpretation of the physiological image (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Just as information propagates to enforce monocular curve continuation, the binocular signal propagates to form a coherent binocular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The Grinwald group established this for stereo [59] (see also Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1(A)), using biocytin injections, that propagate directly along neuronal processes and are deposited at excitatory synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus, this technique demonstrates the presence of long-range connections between binocular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These results were refined, more recently, by the Fitzpatrick group [83], using in vivo calcium imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1(right) the authors demonstrated both the monocular and the binocular inputs for stereo, and (not shown) the dependence on orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' More precisely, [59] showed selective anisotripic connectivity among binocular regions: the biocytin tracer does not spread uniformly, but rather is highly directional with distance from the injection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (This was the case with monocular biocytin injections as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=') Putting this together with [83], we interpret the anisotropy as being related to (binocular) orientation ([83]), which is exactly the behavior of the integral curves of our vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Our 3D association fields are strongly directional, and information propagates preferentially in the direction of (the starting point of) the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' An example can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, image (B), where the fan of integral curves (36) is represented, superimposed with colored patches, following the experiment proposed in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Psychophysics and association fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this section, we show that the connections described by the integral curves in our model can be related to the geometric relationships from psychophysical experiments on perceptual organization of oriented elements in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The goal is to establish that our connections serve as a generalization of the concept of an association field in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Towards a notion of association field for 3D contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The perception of continuity be- tween two elements of position-orientation in R3 has been studied experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To start, Kellman, Garrigan, and Shipley ([48, 49]) introduce 3D relatability, as a way to extend to 3D the experiments of Field, Heyes and Hess ([30]) in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A C24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Example of the fan of the 3D relatable edges with initial point E0 Particularly, in a system of 3D Cartesian coordinates, it is possible to introduce oriented edges E at the application point (r1, r2, r3)T and with an orientation identified with the an- gles θ and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This orientation can be read, in our case, through the direction expressed by (cos θ sin ϕ, sin θ sin ϕ, cos ϕ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For an initial edge E0, with application point on the origin of the coordinate system (0, 0, 0)T and orientation lying on the r1-axis, described by θ = 0, ϕ = π/2, the range of possible orientations (θ, ϕ) 7 for 3D-relatable edges with E0 is given by: (40) tan−1 �r2 r1 � ≤ θ ≤ π 2 and π 2 ≤ 3π 2 − ϕ ≤ tan−1 �r3 r1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The bound on these equations identified with the quantity π 2 incorporates the 90 degree constraint in three dimensions, while the bounds defined by the inverse of the tangent express the absolute orientation difference between the reference edge E0 and an edge positioned at the arbitrary oriented point E(r1,r2,r3) so that its linear extension intersects E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see [48, 49] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Numerical simulations allow us to visually represent an example of the 3D positions and orientations that meet the 3D relatability criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Starting from an initial edge E0 with endpoints in (p01, p02, p03)T and orientation on the e1- axis, we represent for an arbitrary point (p1, p2, p3)T the limit of the relatable orientation (θ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' By projecting on the retinal planes of the 3D fan of relatable points, it is possible to notice that these projections are in accordance with the notion of 3D compatibility field of in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Psychophysical studies, see [25, 32, 34], have investigated the properties of the curves that are suitable for connecting these relatable points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These curves are well described by being smooth and monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, using non-oriented contour elements for contours, Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' in [34] indicate that contour elements can be effectively grouped based primarily on the good 7The angle ϕ here has been modified to be compatible with our set of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The relationship between the angle ˜ϕ in works [48, 49] can be expressed as : ˜ϕ = acos(sin ϕ) + π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 25 (a) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Example of 3D association field in the two left and right retinal planes, generated with the geometry of 3D relatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Example of 3D compatibility field of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' continuation of contour elements in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This statement is confirmed by the more recent work of Deas and Wilcox ([25]), who in addition observe that detection of contours defined by regular depth continuity is faster than detection of discontinuous contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' All these results support the existence of depth grouping operations, arguing for the extension of Gestalt principles of continuity and smoothness in three dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Finally, on the relationship of the three- dimensional curves to 2-dimensional association fields, see [49, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These authors have assumed that the strength of the relatable edges in the co-planar planes of E0 must meet the relations of the bi-dimensional association fields of [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Compatibility with the sub-Riemannian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To model associations underlying the 3D perceptual organization of the previous paragraph, we consider again the constant coefficient family of integral curves studied in (36): (41) ˙Γ(t) = ⃗YR3,Γ(t) + c1⃗Yθ,Γ(t) + c2⃗Yϕ,Γ(t), with c1, c2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Importantly, these curves locally connect the association fan generated by the geometry of 3D relatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4, image(B) shows the family of the horizontal curves con- necting the initial point E0 with 3D relatable edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These curves are computed using Matlab solver function ode45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In analogy with the experiment of Field , Hayes and Hess in [30], we (a) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A)3D relatable edges displayed on the right of the initial edge E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Unrelatable 3D edges displayed on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Horizontal integral curves with filled lines connect 3D relatable edges with initial point E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Horizontal integral curves with dotted lines do not connect 3D unrelatable edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' choose to represent non-relatable edges to the left of the starting point E0, while on the right Projection in the left image Projection in the right image E ORProjection in left image Projection in right image 10 3 w 2 2 E- 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 1 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 X x 10* × 10~4rE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 3 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='26 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 are 3D relatable edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, filled lines of the integral curves indicate the correlation between the central horizontal element E0 and the ones on its right, while dotted lines connect the starting point E0 with elements not correlated with it, as represented on the left part of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Restricting the curves on the neighborhood of co-planar planes with an arbitrary edge E, we have different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' First, on the r1-r2 plane (fronto-parallel) and the r1-r3 plane we have arcs of circle, as proved with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Furthermore, for an arbitrary plane in R3 containing an edge E, we observe that the curves generating with fixed angle ϕ are helices, and locally they satisfy the bidimensional constraint in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Examples can be found in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, the curves displayed in images (A) and (B) of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 are well in accordance with the curves of the Citti-Sarti model, depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Restriction of the fan of the integral curves on the e1-e2 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Restriction of the fan of the integral curves on the e1-e3 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Restric- tion of the fan at ϕ = ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These curves (black lines) are not planar curves but helices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' However, their projection (white lines) on the coplanar plane with initial edge satisfies the bidimensional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Integration of contours and stereo correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Although the goal of this paper is not to solve the stereo correspondence problem, we can show how our geometry is helpful in understanding how to match left and right points and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These ideas are developed more fully in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Inspired by the experiment of Hess and Field in [32], we consider a path stimulus γ interpreted as a contour, embedded in a background of randomly oriented elements: left and right retinal visual stimuli are depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We perform a first simplified lift of the retinal images to a set Ω subset of R3 ⋊ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This set contains all the possible corresponding points, obtained by coupling left and right points which share the same y retinal coordinate, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='7, image(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The set Ω contains false matches, namely points that do not belong to the original stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is the task of correspondence to eliminate these false matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We compute for every lifted point the binocular output OB of equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This output can be seen as a probability measure that gives information on the correspondence of the couple of left and right points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We can simply evaluate which are the points with the highest probability of being in correspondence, applying a process of suppression of the non-maximal pairs over the fiber of disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this way, noise points are removed (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='7, image (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now directly exploit good continuation in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The remaining noise elements are orthog- onal to the directions of the elements of the curve that we would like to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Calculating numerically the coefficients c1 and c2 of integral curves (36) that connect all the remaining pairs of points, we can obtain for every pair the value of curvature and torsion using (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='8 read it in terms of matrices M representing the values of curvature or torsion for every couple of points ξi, ξj in the element Mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, we observe that random points are characterized by a very high curvature and in general also the torsion deviates from minimum magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 1313GOOD CONTINUATION IN 3D: THE NEUROGEOMETRY OF STEREO VISION 27 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Left and right retinal images of the set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Black points are the projection of the point of the curve γ, while gray points are background random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Lifting of the two left and right retinal images of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6 in the space of position and orientation R3 × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Selection of lifted points according to the binocular output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Points of the stimulus γ connected by integral curves (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, by discarding these high values, we select only the three-dimensional points of the curve γ, which are well connected by the integral curves, as shown in image (C) of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is in accordance with the idea developed in [3, 57, 58], where curvature and torsion provide constraints for reconstruction in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Summary and Conclusions Understanding good continuation in depth, like good continuation for planar contours, can benefit from basic physiological constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' from psychophysical performance measures, and from mathematical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, good continuation in the plane is supported by ori- entation selectivity and cortical architecture (orientation columns), by association field grouping performance, and by geometric modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We maintain that the same should be true for good continuation in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' However, while the psychophysical data may be comparable, the physio- logical data are weaker and the geometry of continuation is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this paper, we introduced the neuro-geometry of stereo vision to fill this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is strongly motivated by an analogical extension to 3D of 2D geometry, subject to respecting the psychophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the end, it allowed us to be precise about the type of geometry that is relevant for understanding stereo Left Image Right Image 13r3 r2328 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' BOLELLI1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' CITTI1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' SARTI2, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' ZUCKER 3 (a) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Matrices M which element Mij represents the value of curvature/ torsion for every couple of points ξi, ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The first eight points correspond to points of the curve γ while the others are random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Curvature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Torsion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' abstractly, and concretely was highly informative toward the physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Although a "stereo columnar architecture" is not obvious from the anatomy, it is well-formed computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The neuro-geometry of binocular cells are described through binocular RPs which are the product of left and right monocular RPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Starting from binocular receptive profiles it is possible to reconstruct the three dimensional space using just the position and orientation of the visual stimulus recovered in the retinal planes (assuming one has corresponding points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Technically, we proposed a sub-Riemannian model on the space of position and orientation R3 ⋊ S2 for the description of the perceptual space of the neural cells involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This geometrical structure favors the tangent direction of a 3D curve stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The integral curves of the sub- Riemannian structure encode the notions of curvature and torsion within their coefficients, and are introduced to describe the connections between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This model can be seen as an extension in the three-dimensional scene of the 2-dimensional association field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, the integral curves of the sub-Riemannian structure of the 3D space of position-orientation are exactly those that locally correspond to psychophysical association fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Although the goal of this paper is not to solve the stereo correspondence problem, we have seen how the geometry we propose is a good starting point to understand how to match left and right points and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A future development of the model will consist in defining the probability of the co-occurrence between two elements, to individuate percepts in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Individuation of percepts through harmonic analysis on the sub-Riemannian structure has been proposed in the past, both for 2D spatial stimuli [77] and in 2D + time spatio-temporal stimuli [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It would be interesting to develop a similar analysis and extend it to stereo vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Acknowledgements MVB, GC, and AS were supported by EU Project, GHAIA, Geometric and Harmonic Analysis with Interdisciplinary Applications, H2020-MSCA-RISE-2017 SWZ was supported in part by US NIH EY031059 and by US NSF CRCNS 1822598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Curvature 2 4 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='5 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 2 4 6 8 10 12Torsion 2 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='8 12 1 2 4 9 8 10 1229 Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A gentle introduction to sub-Riemannian geometry In this paper we exploit techniques from differential geometry, and in particular sub - Riemann- ian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this appendix we provide an invitation to these ideas with a rather informal discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For the reader interested in a formal introduction on basic instruments of differential geometry (arguments of sections A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2) please refer to [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For a complete and formal mathematical (comprehensive) introduction to sub-Riemannian geometry we refer to [2], while for a more informational point of view please consult [79, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To start, imagine that you are standing at a point on a smooth surface in the world, far from any boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Now, you can "walk away" from this point in any (2D) compass direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' for example, you could walk north or south or any direction in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If your steps were very very short, then the (flat) compass actually characterizes the 2D space of possible steps you might take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These same ideas are expressed more formally in differential geometry, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' One can attach to every point p of a differentiable manifold M (a generalized surface) a tangent space TpM (the compass plus some algebra describing vector operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' That is, the tangent space is a real vector space that contains the possible directions in which one can tangentially pass through p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If the manifold is connected, then the tangent space has, at every point, the same dimension as the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, if the manifold is a 2D surface, the tangent space at a point is a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In general, this tangent plane "approximates" the surface only locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The elements ⃗Xp of the tangent space TpM at p are called tangent vectors at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Attached to a point on the surface, as above, these tangent vectors define the directions in which one could walk away from the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' But modern differential geometry provides another interpretation: it is possible to think of the elements of the tangent space in terms of directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Technically, for every smooth function f, Xf(p) = ⃗Xp·∇f(p) will denote the directional derivative of f in the direction of the vector ⃗ Xp, with ∇ denoting the gradient vector (expressed in an appropriate coordinate system) and · scalar product between these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We will also denote Xp = ⃗Xp · ∇p, omitting the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We now consider pairs of directional derivatives X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If X and Y are partial derivatives, for every regular function f one has XY f = Y Xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If X and Y are directional derivatives, in general XY f ̸= Y Xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Explicit computation tell us that at every point p (42) [X, Y ]f(p) = (XY − Y X)f(p) = (J⃗Yp ⃗Xp − J ⃗ Xp ⃗Yp) · ∇f(p), with J ⃗ Xp and J⃗Yp Jacobian matrices of ⃗Xp and ⃗Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The quantity [X, Y ]f is called commutator since it expresses the fact that the two derivatives do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The same notion can be expressed in terms of increments: one might visualize an increment from a point p as the head of a vector ⃗Xp applied at the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Then the expression XY − Y X will be geometrical obtained as follows: place X down at a point, then the other Y at its head, then the the first one backward finally the second one backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The issue is whether the quadrilateral is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally this is captured by the commutator of two elements X and Y at the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In order to compute the second derivative XY f, we need to know Y f at every point near p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This lead to the more general notion of vector fields, which are abstractions of the velocity field of points moving in the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A vector field X attaches to every point p of the manifold M a vector ⃗Xp from the tangent space at that point, in a smooth manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' There are no abrupt jumps between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Since we related each tangent vector with a derivation above, we can now go further;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Each vector field can be associated with an ordinary differential equation, whose solutions are called integral curves of the vector field: they are parametric curves that represent specific solutions to the ordinary differential equation depicted by the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Think of it as follows: imagine you are starting at a point, and take an infinitesmimal step in the direction of a 30 tangent vector at that point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' you will now be at a neighboring point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, again, you can take a step from this neighboring point in (possibly) another tangent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Continuing this process for a while, you geometrically trace out integral curves γ : [t1, t2] ⊆ R −→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Importantly, the given vector field X at the point γ(t) is the tangent vector to the curve at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Importantly, this holds true everywhere along the curve, so that the integral curve satisfies a differential equation: (43) ˙γ(t) = ⃗Xγ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) (c) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Tangent planes TpiM (darker planes) at points pi, i = 1, 2, 3 in the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B) Vector field X defined on M: to every point pi, i = 1, 2, 3 of the manifold M we have a vector ⃗Xpi of the tangent space at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (C) Integral curve γ associated with the vector field X starting from p1 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' As a set, it is given by the disjoint union of the tangent spaces of M, that is: (44) TM = � p∈M TpM = {(p, Xp) | p ∈ M, Xp ∈ TpM} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In particular, an element of TM can be thought of as a pair (p, Xp), where p is a point in M and Xp is a tangent vector to M at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' There exists a natural projection π : TM → M defined by π(p, Xp) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' which maps each element of the tangent space TpM to the single point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Group action on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The operation of adding (real) numbers has an important algebraic structure, called a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It requires, for example, that the sum of any two numbers is again a number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' that there is an inverse operation "-";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' and that there is an identity operation "0" that is, adding to any number yields the same number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' When a group G acts on a manifold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' the real numbers, above), it means that each of its elements performs a certain operation on all the elements of the manifold in a way that is compatible with the manifold itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' More precisely, this action is described by a map σ : G×M → M, (g, x) �→ g · x which is the (left) group action of a group G on a smooth manifold M, if the map σ is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' For example, we can take the bidimensional roto-translation group SE(2) = R2×S1 and define its action on a smooth manifold M ⊆ R2 following the group law: first we apply a rotation and then a translation of the manifold itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This is formalized through the map σ : SE(2) × M −→ M, σ(g, p) = (Rp + q), with g = (q, R) ∈ SE(2), namely a point q ∈ R2 and R bidimensional rotation of angle θ ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A graphical example is shown in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We are now ready to generalize these familiar ideas to cortical space, with its special position × orientation structure, or to stereo space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Tp1M Tp2M p1 p2 p3Xp1 Xp2 p2 Xp3Xp1 Xp2 p1 p2 Xp3 p331 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Group action of the roto-translation group SE(2) on the manifold M (black ellipse): first, the manifold is rotated through a rotation of angle θ obtaining RθM, and then a translation is applied, moving the rotated manifold in space realizing RθM + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A point constraint to move on a manifold, illustrated above, dictates that one can move only along directions tangent to the manifold, since moving in the nor- mal direction would leave the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This means that, for every point p, the set of admissible directions of displacement coincides with the tangent plane TpM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the presence of further con- straints, some tangent directions could be forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This leads to introducing, at every point p, the admissible tangent space Ap, which is the subspace of TpM of admissible directions of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If the tangent space TpM has dimension n, the admissible tangent space Ap will have dimension m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Repeating the same construction for every point of the manifold, we call the admissible tangent bundle the union of admissible tangent spaces at every point: A = � p∈M Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If we introduce a scalar product on Ap, then we are able to define a norm on vectors with the aim to measure the length of such vectors and the distance between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The manifold with these properties is usually called sub-Riemannian manifold, while manifolds where movements are allowed in any direction are called Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (a) (b) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (A) Geometric set-up of the motion of a car moving on a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' (B)Sub-Riemannian formalization in SE(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Tangent vector of the path is con- strained to be in the gray plane, span of ⃗X1,p and ⃗X2,p, admissible directions of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Po T M RaM + T(t) = (coso(t), sin(t) (t) (t) = (x(t), y(t) XD = (0, 0, 1) = (cos, sin0, 0) y32 Let us explicitly note that while Riemannian geometry arises in presence of a physical con- straints, sub-Riemannian geometry arises in presence of differential constraints, as for example in the description of the motion of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A car moves on a bidimensional plane, but it can only move in its current direction or it can change its current orientation by rotating the steering wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' These are the admissible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Moreover, the car cannot move "sideways" (for- bidden direction): this prevents one from directly reaching any other direction while remaining in the initial position, restricting the allowable motions to a simultaneous combination of the two admissible movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The trajectory described by the vehicle will therefore be a curve, whose tangent is constrained to follow the two admissible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The formalization of this sub-Riemannian problem takes place in SE(2), considering for every p ∈ SE(2) as admissible tangent space ApSE(2) the subspace generated by the current direction ⃗X1,p = (cos θ, sin θ, 0)T and the direction of rotation ⃗X2,p = (0, 0, 1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' See Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Similarly, we can move from a retinotopic (x, y) position to another retinotopic position, (x′, y′), moving "up" or "down" through orientation columns from θ to θ′, but we cannot reach θ′ from θ maintaining the same initial position (running through the same orientation column): in order to reach the "forbidden direction" we have to walk simultaneously through positions and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This restriction of movement is what distinguishes a Euclidean (or Riemannian) geometry from a sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 In this appendix, we show ho to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 using tools of differential geometry, and in particular the concept of differential k−form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' A differential k-form on an n-dimensional smooth manifold M is any multilinear function ω : TM k −→ R which takes as input k smooth vector fields and outputs a scalar element, satisfying the antisymmetry property: ω(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xk) = −ω(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , Xk), with k ≤ n and k, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the special case where ω is a 1-form, it is worth noting that this is an element of the dual space to TM (cotangent space): ω ∈ TM ∗ ⇐⇒ ω : TM −→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' If we have coordinates (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , xn) on M, we can express the 1-forms using the dual basis {d x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , d xn} of TM ∗: ωp = f1(¯x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , ¯xn) d x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' + fn(¯x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , ¯xn) d xn, with p = (¯x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' , ¯xn), with fi scalar smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Furthermore, it is possible to multiply via the wedge product ∧ a differential k-form, ω, with a differential l- form, η, obtaining a differential k + l-form ω ∧ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' More precisely, we are interested in the wedge product of 1-forms ω and η, where the wedge product can be computed as: ω ∧ η(X, Y ) = ω(X)η(Y ) − ω(Y )η(X), with X and Y vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Development of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The binocular interaction term OLOR can be associated with the cross product of the left and right directions defined through (13), namely ω⋆ pL and ω⋆ pR of monocular simple cells: (45) OLOR = ω⋆ pL × ω⋆ pR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' As noted in subsubsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1, the output of simple cells (11) in SE(2) can then be locally approximated as O(x, y, θ) = −X3,p(Iσ)(x, y) where Iσ is a smoothed version of I, obtained by convolving it with a Gaussian kernel, the vector field (46) X3,p = − sin θ∂x + cos θ∂y, 33 with p = (x, y, θ) ∈ SE(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Switching to the dual space, the action of simple cells induces a choice of a 1-form separately on each cell: (47) ωp = − sin θ d x + cos θ d y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Accordingly, it is possible to re-write the binocular interaction term as: (48) OLOR = X3,pR(IσR)(xR, y)X3,pL(IσL)(xL, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In the following, we will see that this binocular action can be described by a 2-form defined in terms of the two 1-forms of monocular simple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We will denote with the subscript R the quantities corresponding to the right monocular structure, and we will use the subscript L for the left one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, we define vR := (JIσR ⃗X3,pR)X3,pR using the Jacobian (differential) of the smoothed version of the image I, in such a way that we have ωpR(vR) = X3,pR(IσR) = (JIσR ⃗X3,pR) since ωpR(X3,pR) = 1 and JIσR ⃗X3,pR ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' the same reasoning holds for the left structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is then possible to recast (48) in the retinal coordinates as: (49) OLOR =ωpL(vL)ωpR(vR) =ωpL ∧ ωpR(vL, vR) + ωpR(vL)ωpL(vR) � �� � =0 , =ωpL ∧ ωpR(vL, vR), exploiting the properties of the wedge product and the left and right retinal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The retinal coordinates can be expressed in terms of cyclopean coordinates (4) as xR = x − d and xL = x + d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' then, the extended left and right 1-form can be written as: (50) ωpR = − sin θR d x + cos θR d y + sin θR d d ωpL = − sin θL d x + cos θL d y − sin θL d d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Taking advantage of the isomorphism provided by the Hodge star between vectors and 2-forms in R3, we relate the exterior and the cross product, using notations (20) 8, in the following way: (51) ⋆ (ωpL ∧ ωpR) = ω⋆ pL × ω⋆ pR, from which it follows the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' □ Throughout the paper, to lighten the notation, we will call ωL = ωpL and ωR = ωpR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Meaning of the mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' We conclude this section with a consideration on the mathematical tools introduced and used in this setting, to understand how the mathematical models proposed by Citti and Sarti, starting with [18], assign these different mathematical objects to the physical cell, to its action , and to the result of its action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' It is well known that an odd simple cell (selective for orientation) is activated as a result of the presence of a stimulus to select its direction (tangent vector to the perceptual curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this setting, the mathematical intuition behind the model proposed in [18] is to identify each cell with a 1-differential form, which is an element of the cotangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Roughly speaking, this differential form is able to grasp a vector that corresponds to the direction of the stimulus: this is the result of the action of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Formally, this vector will be an element of the tangent space, and more precisely it will lay in the kernel of the 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This vector space is then associated with the action of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The same reasoning is applied to different families of cells in a series of papers ([78, 6, 1, 8]) even if these are characterized by distinct sub-Riemannian structures in various manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The interested reader could refer to [19] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Similarly, we have found the same geometrical organization in the family of binocular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' 8Using the notation ω⋆ we identify the vector whose components are the coefficients of the 1-form ω with respect to the dual basis 34 Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' In this paper, we have dealt with binocular cells which are a combination of monocular simple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' To these coupled simple cells (one for the left and one for the right eye) we formally associate a 2-differential form, the wedge product of the two monocular left and right 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' This 2-form can grasp again a vector, lying in the kernel of this mathematical object, identifying the three-dimensional stimulus direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Thus, the same reasoning of Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 also applies here to the binocular family of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Translating the results of Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content='1 into different spaces, with different dimensions, it is then possible to use the same mathematical objects to explain the behavior of families of different cells, identifying geometrically the mathematical objects at the basis of the functionality of the family of studied cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Change of variables Let us recover the expression of the 1-forms ˜ωL := UtL and ˜ωR := UtR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Recall here the change of variable (5): (52) � � � � � r1 = xc d r2 = yc d r3 = fc d , and its differential: (53) � � � � � d r1 = c d d x − cx d2 d d d r2 = c d d y − cy d2 d d d r3 = − fc d2 d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Writing the quantity UtL, defined in (25), in term of a 1-form in the variables (r1, r2, r3) we have: (54) ˜ωL = − f sin θL d r1 + f cos θL d r2 + (xL sin θL − y cos θL) d r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' Changing coordinates: (55) ˜ωL = − f sin θL � c d d x − cx d2 d d � + f cos θL � c d d y − cy d2 d d � + (xL sin θL − y cos θL) � −fc d2 d d � =fc d (− sin θL d x + cos θL d y − sin θL d d) =fc d ωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' So, up to a scalar factor, we have that ˜ωL = ωL in the variables (x, y, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' The same reasoning holds for the right structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfeQqG/content/2301.04542v1.pdf'} 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sha256:38d35fbcba8ad3033b58c1700bb978bb3f5fd5d96af5d8bcca91afe82bfd8857 +size 215369 diff --git a/ftE0T4oBgHgl3EQfXQCS/vector_store/index.pkl b/ftE0T4oBgHgl3EQfXQCS/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..97f65f3f7ce631d45285288071dbc6a15ce3b0dc --- /dev/null +++ b/ftE0T4oBgHgl3EQfXQCS/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fbf8f5eea6b7a5d8b4a4a73a158a308e80013a018727f1a7092f738d0bdb32c8 +size 113947 diff --git a/ftE1T4oBgHgl3EQfewRy/content/tmp_files/2301.03210v1.pdf.txt b/ftE1T4oBgHgl3EQfewRy/content/tmp_files/2301.03210v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee55f16e14d0d6e8c396a25de970d082b2c260d4 --- /dev/null +++ b/ftE1T4oBgHgl3EQfewRy/content/tmp_files/2301.03210v1.pdf.txt @@ -0,0 +1,1762 @@ +Probing the structural evolution along the fission path in the superheavy +nucleus 256Sg +Ting-Ting Li,1 Hua-Lei Wang,1, ∗ Zhen-Zhen Zhang,1 and Min-Liang Liu2, 3 +1School of Physics and Microelectronics, +Zhengzhou University, Zhengzhou 450001, China. +2Key Laboratory of High Precision Nuclear Spectroscopy, +Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China. +3School of Nuclear Science and Technology, +University of Chinese Academy of Sciences, Beijing 100049, China. +(Dated: January 10, 2023) +Abstract +The evolution of structure property along the fission path in the superheavy nucleus 256Sg is pre- +dicted through the multi-dimensional potential-energy(or Routhian)-surface calculations, in which the phe- +nomenological deformed Woods-Saxon potential is adopted. Calculated nuclear deformations and fission +barriers for 256 +106Sg150 and its neighbors, e.g., 258,260Sg, 254Rf and 252No are presented and compared with +other theoretical results. A series of energy maps and curves are provided and used to evaluate the corre- +sponding shape-instability properties, especially in the directions of triaxial γ and different hexadecapole +deformations (e.g., α40, α42 and α44). It is found that the triaxial deformation may help the nucleus bypass +the first fission-barrier of the axial case. After the first minimum in the nuclear energy surface, the fission +pathway of the nucleus can be affected by γ and hexadecapole deformation degrees of freedom. In addition, +microscopic single-particle structure, pairing and Coriolis effects are briefly investigated and discussed. +Keywords: structure evolution, fission path; fission barrier; superheavy nuclei; macroscopic- +microscopic model. +PACS numbers: +∗wanghualei@zzu.edu.cn(Corresponding author) +1 +arXiv:2301.03210v1 [nucl-th] 9 Jan 2023 + +1. Introduction +The evolution of nuclear structure properties with some degree of freedom (e.g., nucleon num- +ber, spin, temperature, etc) is one of the most significant issues in nuclear physics [1], especially +towards the superheavy mass region. Great progress has been made in the synthesis of superheavy +nuclei with the development of the radioactive beam facility, heavy-ion accelerator and highly- +effective detector systems [2–4]. Spontaneous fission is usually one of important decay modes +in a superheavy nucleus and the barrier along the fission path is critical to understand the fission +process [5, 6]. For instance, the survival probability of a synthesized superheavy nucleus in the +heavy-ion fusion reaction is directly related to such a barrier, during the cooling process of a com- +pound nucleus, which plays a decisive role in the competition between nucleon evaporation and +fission (a small change of the fission barrier may result in several orders of magnitude difference +in survival probability) [7]. Nevertheless, it is still rather difficult to give an accurate description +for the fission barrier so far. To a large extent, the barrier size and shape can be determined by the +fission path in the nuclear energy surface. +Up to now, there are several types of models which are widely used for investigating nuclear +fission phenomena, including e.g., the macroscopic-microscopic (MM) models [8–12], the non- +relativistic energy density functionals based on zero-range Skyrme and finite-range Gogny inter- +actions [13–18], the extended Thomas-Fermi plus Strutinsky integral methods [19, 20] , and the +covariant density functional theory [5, 21, 22]. The MM methods usually have the high descriptive +power as well as simplicity of calculation and thus are still used by many researchers so far. In +such an approach, the empirical one-body nuclear mean-filed (e.g., the Nilsson and Woods-Saxon +potentials) Hamiltonian is used to solve the microscopic single-particle levels and wave functions +and a macroscopic liquid-drop model (e.g., the standard liquid-drop model [23], the finite-range +droplet model [24], and the Lublin-Strasboug drop model [25], etc) is combined to describe the +nuclear bulk property. In recent years, the model parameters, including their uncertainties and +propagations, in both phenomenological Woods-Saxon potential and the macroscopic liquid-drop +model are still studied and optimized, e.g., cf Refs. [11, 26–30]. Indeed, the parameters of MM +models are mainly from the fitting of available single-particle levels of several spherical nuclei +and several thousand nuclear-mass data. They are generally successful near the β-stability line, +especially in the medium and heavy nuclear regions. Without the preconceived knowledge, e.g., +about the measured densities and single-particle energies, it may be needed to test whether the +modeling and model parameters of a phenomenological one-body potential are still valid enough. +Part of our aim of this work is to test the theoretical method in such aspects. +Prior to this work, 16 Sg isotopes from A = 258 to 273 were synthesized by the fusion- +evaporation reactions, e.g., 238U(30Si,xn)268−xSg [31]. It was reported that the lightest even-even +Sg isotope, 258Sg, has a revised half-life of 2.8+0.8 +−0.5 ms [32]. Naturally, one expects that based on +the fusion-evaporation mechanism, the superheavy nuclide 256Sg will be synthesized as the next +2 + +candidate which is the nearest even-even nucleus to the known ones in this isotopic chain. Keeping +this in mind, we predict the properties of structure evolution along the possible fission path for the +superheavy nuclide 256Sg in this project. In our previous studies, we systematically investigated +the octupole correlation properties for 42 even-even nuclei with 102 ≤ Z ≤ 112 [33] and the tri- +axial effects on the inner fission barriers in 95 tranuranium even-even nuclei 94 ≤ Z ≤ 118 [34]. +The triaxiality and Coriolis effects on the fission barrier in isovolumic nuclei with A = 256 were +investigated, where the 256Sg was calculated but just focused on the first (inner) fission barrier [35]. +In Ref. [36], we investigated the effects of various deformations (e.g., β2, γ and β4) on the first +barrier in even-even nuclei with N = 152 and 94 ≤ Z ≤ 108. In addition, we studied the +collective rotational effects including the α-decay-chain nuclei (from 216Po and 272Cn) [37] and +254−258Rf [38] by the similar calculation. The primary purpose of this study is to investigate the +effects of different deformation parameters, especially the axial and non-axial hexadepole defor- +mations, on the fission path of 256Sg by analyzing the topography of the energy surfaces calculated +in a reasonable subspace of collective coordinates (it is impossible to calculate in the full defor- +mation space). The probe of the shape evolution along the fission path on the energy landscape +will be useful for understanding the formation mechanism of the fission barrier. We provide the +analysis of the single-particle structures, shell and pairing evolutions, especially at the minima +and saddles. Sobiczewski et al [39] systematically investigated the static inner barrier of heaviest +nuclei with proton number 98 ≤ Z ≤ 126 and neutron number 134 ≤ N ≤ 192 in a multidimen- +sional deformation space and pointed out that the inclusion of the non-axial hexadecapole shapes +lowers the barrier by up to about 1.5 MeV. In the synthesis of the superheavy nuclei, nuclear hex- +adecapole deformations were revealed to have an important influence on production cross sections +of superheavy nuclei by e.g., affecting the driving potentials and the fusion probabilities [40, 41]. +This paper is organized as follows: In Sect.2, we briefly describe the outline of the theoretical +framework and the details of the numerical calculations. The results of the calculations and their +relevant discussion are given in Sect.3. Finally, the concluding remarks will be given in Sect.4. +2. Theoretical framework +In what follows, we recall the unified procedure and give the necessary references related to the +present theoretical calculation, which may be somewhat helpful for some readers to clarify some +details (e.g., the various variants of the pairing-energy contribution within the framework of the +macroscopic-microscopic method). We employ potential-energy(or Routhian)-surface calculation +to study the present project. This method is based on the macroscopic-microscopic model [42, 43] +and the cranking approximation [44–46], which is one of widely used and powerful tools in nuclear +structure research, especially for rotating nuclei. The usual expression for the total energy in the +rotating coordinate frame (namely, the so-called total Routhian) reads [47] +Eω(Z, N, ˆβ) = Eω +macr(Z, N, ˆβ) + δEω +micro(Z, N, ˆβ), +(1) +3 + +where Eω(Z, N, ˆβ) represents the total Routhian of a nucleus (Z, N) at frequency ω and defor- +mation ˆβ. The first term on the right-hand side in Eq. (1) denotes the macroscopic (liquid drop, or +LD) energy with the rigid-body moment of inertia calculated classically at a given deformation, as- +suming a uniform density distribution; δEω +micro represents the contribution due to the microscopic +effects under rotation. After rearrangement employing elementary transformations [48–52], the +total Routhian can be rewritten as, +Eω(Z, N, ˆβ) = Eω=0(Z, N, ˆβ) ++ [⟨ ˆHω(Z, N, ˆβ)⟩ − ⟨ ˆHω=0(Z, N, ˆβ)⟩] +− 1 +2ω2[Jmacr(A, ˆβ) − JStru(Z, N, ˆβ)]. +(2) +The notations for the quantities in Eq. (2) are standard [47, 53]. The term Eω=0(Z, N, ˆβ) is the +static total energy (corresponding ω = 0) which consists of a macroscopic LD part ELD(Z, N, ˆβ) +and a shell correction δEshell(Z, N, ˆβ) and a pairing-energy contribution δEpair(Z, N, ˆβ) (neglect- +ing the superscript ω = 0) . The second term in the square brackets represents the energy change +of the cranked Hamiltonian ˆHω(Z, N, ˆβ) due to rotation [47, 53]. In Eq. (2), it is usually and +reasonably assumed that the average pairing energy of the liquid-drop term and the Strutinsky- +smeared pairing energy cancel each other [47]. Therefore, one can further write Eq. (2) as [cf. +Ref. [54] and references therein], +Eω(Z, N, ˆβ) = ELD(Z, N, ˆβ) ++ δEshell(Z, N, ˆβ) + δEpair(Z, N, ˆβ) ++ [⟨ ˆHω(Z, N, ˆβ)⟩ − ⟨ ˆHω=0(Z, N, ˆβ)⟩]. +(3) +As known, several phenomenological LD models (such as standard liquid drop model [23], finite- +range droplet model [42], Lublin-Strasbourg drop model [25]) with slight difference have been +developed for calculating the smoothly varying part. In these LD models, the dominating terms +are mainly associated with the volume energy, the surface energy and the Coulomb energy. In the +present work, the macroscopic energy is given by the standard LD model with the parameters used +by Myers and Swiatecki [23]. +The single-particle levels used below are calculated by solving numerically the Schr¨odinger +equation with the Woods-Saxon (WS) Hamiltonian [55] +HWS = T + Vcent(⃗r; ˆβ) + Vso(⃗r, ⃗p,⃗s; ˆβ) ++VCoul(⃗r, ˆβ), +(4) +where the Coulomb potential VCoul(⃗r, ˆβ) defined as a classical electrostatic potential of a uniformly +charged drop is added for protons. The central part of the WS potential is calculated as +Vcent(⃗r, ˆβ) = V0[1 ± κ(N − Z)/(N + Z)] +1 + exp[distΣ(⃗r, ˆβ)/a] +, +(5) +4 + +where the plus and minus signs hold for protons and neutrons, respectively and the parameter a +denotes the diffuseness of the nuclear surface. The term distΣ(⃗r, ˆβ) represents the distance of a +point ⃗r from the nuclear surface Σ parameterized in term of the multipole expansion of spherical +harmonics Yλµ(θ, φ) (which are convenient to describe the geometrical properties), that is, +Σ : R(θ, φ) = r0A1/3c(ˆβ) +� +1 + +� +λ ++λ +� +µ=−λ +αλµY ∗ +λµ(θ, φ) +� +, +(6) +where the function c(ˆβ) ensures the conservation of the nuclear volume with a change in the nu- +clear shape and ˆβ denotes the set of all the deformation parameters {αλµ}. For a given nucleus +with mass number A, a limiting value of λ < A1/3 is often estimated. In the present shape +parametrization, we consider quadrupole and hexadecapole degrees of freedom, including non- +axial deformations, namely, ˆβ ≡ {α20, α2±2, α40, α4±2, α4±4}. The quantity R(θ, φ) denotes the +distance of any point on the nuclear surface from the origin of the coordinate system. Because only +the even λ and even µ components are taken into account, the present parametrisation will preserve +three symmetry planes. After requesting the hexadecpole degrees of freedom to be functions of +the scalars in the quadrupole tensor α2µ, one can reduce the number of independent coefficients to +three, namely, β2, γ and β4, which obey the relationships [56] +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +α20 = β2 cos γ +α22 = α2−2 = − 1 +√ +2β2 sin γ +α40 = 1 +6β4(5 cos2 γ + 1) +α42 = α4−2 = − 1 +12 +√ +30β4 sin 2γ +α44 = α4−4 = +1 +12 +√ +70β4 sin2 γ. +(7) +The (β2, γ, β4) parametrization has all the symmetry properties of Bohr’s (β2, γ) parametriza- +tion [57]. The spin-orbit potential, which can strongly affects the level order, is defined by +Vso(⃗r, ⃗p,⃗s; ˆβ) = −λ +� ℏ +2mc +�2 +× +� +∇V0[1 ± κ(N − Z)/(N + Z)] +1 + exp[distΣso(⃗r, ˆβ)/aso] +� +× ⃗p · ⃗s, +(8) +where λ denotes the strength parameter of the effective spin-orbit force acting on the individual +nucleons. The new surface Σso is different from the one in Eq. (6) due to the different radius +parameter. In the present work, the WS parameters are taken from Refs. [56, 58], as listed in +Table I. +In computing the Woods-Saxon Hamiltonian matrix, the eigenfunctions of the axially deformed +harmonic oscillator potential in the cylindrical coordinate system are adopted as the basis func- +5 + +Table I: The adopted WS parameters for both protons and neutrons (for more details, cf e.g, Ref. [56]). Note +that nuclear shape does not sensitively depend on the parameter sets in well-deformed nuclei, especially +those with large stiffness. +V0 (MeV) +κ +r0 (fm) a(fm) +λ +(r0)so (fm) aso (fm) +53.754 +0.791 +1.190 +0.637 29.494 +1.190 +0.637 +tions [59], +|nρnzΛΣ⟩ = ψΛ +nρ(ρ)ψnz(z)ψΛ(ϕ)χ(Σ), +(9) +where +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ψΛ +nρ(ρ) = +√ +nρ! +√ +(nρ+|Λ|)!(2mωρ/ℏ)1/2 +×e− η2 +2 ηΛL|Λ| +nρ (η), +ψnz(z) = +1 +√√π2nz nz!(2mωz/ℏ)1/4 +×e− ξ2 +2 Hnz(ξ), +ψΛ(ϕ) = +1 +√ +2πeiΛϕ, +(10) +and χ(Σ) represents the spin wave functions, cf. e.g., Sec. 3.1 in Ref. [59] for more details. In +our calculation, the eigenfunctions with the principal quantum number N ≤ 12 and 14 have been +chosen as a basis for protons and neutrons, respectively. It is found that, by such a basis cutoff, the +results are sufficiently stable with respect to a possible enlargement of the basis space. In addition, +the time reversal (resulting in the Kramers degeneracy) and spatial symmetries (e.g., the existence +of three symmetry x − y, y − z and z − x planes) are used for simplifying the Hamiltonian matrix +calculation. +The shell correction δEshell(Z, N, ˆβ), as seen in Eq. (3), is usually the most important correc- +tion to the LD energy. Strutinsky first proposed a phenomenological expression, +δEshell(Z, N, ˆβ) = +� +ei − +� +e˜g(e)de, +(11) +where ei denotes the calculated single-particle levels and ˜g(e) is the so-called smooth level density. +Obviously, the smooth level distribution function is the most important quantity, which was early +defined as, +˜g(e, γ) ≡ +1 +γ√π +� +i +exp[−(e − ei)2 +γ2 +], +(12) +where γ indicates the smoothing parameter without much physical significance. To eliminate any +possibly strong γ-parameter dependence for the final result, the mathematical form of the smooth +level density ˜g(e) has been optimized by introducing a phenomenological curvature-correction +6 + +polynomial Pp(x)[49, 60–62]. Then, the ˜g(e) expression will take the form +˜g(e, γ, p) = +1 +γ√π +� +i=1 +Pp(e − ei +γ +) × exp[−(e − ei)2 +γ2 +], +(13) +where the corrective polynomial Pp(x) can be expanded in terms of the Hermite or Laguerre +polynomials. The corresponding coefficients of the expansion can be obtained by using the or- +thogonality properties of these polynomials and Strutinsky condition (i.e., see the APPENDIX in +Ref.[63]). In fact, this method can be considered standard so far. For instance, the integration in +Eq. (12) can be calculated as follows (see Ref.[64] for more details), +� +e˜g(e, γ, p)de = +� +˜e(n)dn += +� +i=1 +{1 +2ei[1 + erf( +˜λ − ei +γ +)] +− +1 +2√πγexp[−(˜λ − ei)2 +γ2 +] +− 1 +√πexp[−(˜λ − ei)2 +γ2 +] +× +p +� +m=1 +cm[1 +2γHm( +˜λ − ei +γ +) ++eiHm−1( +˜λ − ei +γ +) ++mγHm−2( +˜λ − ei +γ +)]}. +(14) +Of course, there are some other methods developed for the shell correction calculations, e.g., the +semiclassical Wigner-Kirkwood expansion method [56, 65] and the Green’s function method [66]. +In this work, the widely used Strutinsky method is adopted though its known problems which ap- +pear for mean-field potentials of finite depth as well as for nuclei close to the proton or neutron +drip lines. The smooth density is calculated with a sixth-order Hermite polynomial and a smooth- +ing range γ = 1.20ℏω0, where ℏω0 = 41/A1/3 MeV, indicating a satisfactory independence of the +shell correction on the parameters γ and p [64]. +Besides the shell correction, the pairing-energy contribution is also one of important single- +particle corrections. Due to the short-range interaction of nucleon pairs in time-reversed orbitals, +the total potential energy in nuclei relative to the energy without pairing always decreases. There +exist various variants of the pairing-energy contribution in the microscopic-energy calculations, +as is recently pointed out in Ref. [11]. Typically, several kinds of the phenomenological pairing +energy expressions (namely, pairing correlation and pairing correction energies employing or not +employing the particle number projection technique) are widely adopted in the applications of the +macroscopic-microscopic approach [11]. To avoid the confusions, it may be somewhat necessary +7 + +to simply review the ‘standard’ definitions for pairing correlation and pairing correction, e.g., cf +Refs. [11, 64]. For instance, the former is given by the difference between e.g., BCS energy +of the system at pairing ∆ ̸= 0 and its partner expression at ∆ = 0; similar to the Strutinsky +shell correction, the later represents the difference between the above pairing correlation and its +Strutinsky-type smoothed out partner. +In the present work, the contribution δEpair(Z, N, ˆβ) in Eq. (3) is the pairing correlation energy +as mentioned above. The pairing is treated by the Lipkin-Nogami (LN) method [67], which helps +avoiding not only the spurious pairing phase transition but also the particle number fluctuation +encountered in the simpler BCS calculation. In the LN technique [53, 67], it aims at minimizing +the expectation value of the following model Hamiltonian +ˆH = ˆHWS + ˆHpair − λ1 ˆN − λ2 ˆN 2. +(15) +Here, ˆHpair indicates the pairing interaction Hamiltonian including monopole and doubly stretched +quadrupole pairing forces [68–70]: +¯v(λµ) +αβγδ = −Gλµg(λµ) +α¯β g∗(λµ) +γ¯δ +, +(16) +where +g(λµ) +α¯β += +� +δα¯β +λ = 0, µ = 0, +⟨α| �Qµ|¯β⟩ +λ = 2, µ = 0, 1, 2. +(17) +The monopole pairing strength G00 is determined by the average gap method [68] and the +quadrupole pairing strengths G2µ are obtained by restoring the Galilean invariance broken by the +seniority pairing force [70]. To some extent, the quadrupole pairing can affect rotational band- +head energies, moments of inertia, band-crossing frequencies and signature inversion in odd-odd +nuclei [69, 71–73]. The pairing window, including dozens of single-particle levels, the respec- +tive states (e.g. half of the particle number Z or N) just below and above the Fermi energy, is +adopted empirically for both protons and neutrons. The pairing gap ∆, Fermi energy λ (namely, +λ1 + 2λ2(Ntotal + 1)), particle number fluctuation constant λ2, occupation probabilities v2 +k, and +shifted single-particle energies εk can be determined from the following 2(N2 − N1) + 5 coupled +nonlinear equations [67, 68], +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ntotal = 2 �N2 +k=N1 v2 +k + 2(N1 − 1), +∆ = G �N2 +k=N1 ukvk, +v2 +k = 1 +2 +� +1 − +εk−λ +√ +(εk−λ)2+∆2 +� +, +εk = ek + (4λ − G)v2 +k, +λ2 = G +4 +� +(�N2 +k=N1 u3 +kvk)(�N2 +k=N1 ukv3 +k)−�N2 +k=N1 u4 +kv4 +k +(�N2 +k=N1 u2 +kv2 +k)2−�N2 +k=N1 u4 +kv4 +k +� +, +(18) +8 + +where u2 +k = 1 − v2 +k and k = N1, N1 + 1, · · · , N2. The LN pairing energy for the system of +even-even nuclei at “paired solution” (pairing gap ∆ ̸= 0) can be given by [42, 67] +ELN = +� +k +2vk +2ek − ∆2 +G − G +� +k +vk +4 +−4λ2 +� +k +uk +2vk +2, +(19) +where vk2, ek, ∆ and λ2 represent the occupation probabilities, single-particle energies, pairing +gap and number-fluctuation constant, respectively. Correspondingly, the partner expression at +“no-pairing solution” (∆ = 0) reads +ELN(∆ = 0) = +� +k +2ek − GN +2 . +(20) +The pairing correlation is defined as the difference between paired solution ELN and no-pairing +solution ELN(∆ = 0). +In the cranking calculation, we only consider the one-dimensional approximation, supposing +that the nuclear system is constrained to rotate around a fixed axis (e.g. the x−axis with the largest +moment of inertia) at a given frequency ω. The cranking Hamiltonian follows the form +Hω = HWS + Hpair − ωjx − λ1 ˆN − λ2 ˆN 2. +(21) +The resulting cranking LN equation takes the form of the well known Hartree–Fock–Bogolyubov– +like (HFB) equation which can be solved by using the HFB cranking (HFBC) method [74] (also +see, e.g., Ref [1], for a detailed description). The HFB-like equations have the following form +(see, e.g., Ref. [53]): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +β>0 +�� +(eα − λ)δαβ − ω(jx)αβ − Gρ∗ +¯α¯β + 4λ2ραβ +� +×Uβk − ∆δαβV¯βk +� += EkUαk, +� +β>0 +�� +(eα − λ)δαβ − ω(jx)αβ − Gραβ + 4λ2ρ∗ +¯α¯β +� +×V¯βk + ∆∗δαβUβk +� += EkV¯αk, +(22) +where ∆ = G � +α>0 κα¯α, λ = λ1 +2λ2(N +1) and Ek = εk −λ2. Further, εk is the quasi-particle +energy and α (¯α) denotes the states of signature r = −i (r = +i). The quantities ρ and κ respec- +tively correspond to the density matrix and pairing tensor. While solving the HFBC equations, +pairing is treated self-consistently at each frequency ω and each grid point in the selected defor- +mation space (namely, pairing self-consistency). Symmetries of the rotating potential are used +to simplify the cranking equations. For instance, in the present reflection-symmetric case, both +9 + +signature, r, and intrinsic parity, π are good quantum numbers. Finally, the energy in the rotating +framework can be given by +Eω = Tr(e − ωjx)ρ − ∆2 +G − G +� +α,β>0 +ρα,βρ˜α,˜β +−2λ2Trρ(1 − ρ). +(23) +Accordingly, one can obtain the energy relative to the non-rotating (ω = 0) state, as seen in the +last term of Eq. (3). It should certainly be mentioned that the above derivations are used for the +quasi-particle vacuum configuration of even-even nuclear system. However, it is convenient to +extend the formalism to one or many quasi-particle excited configuration(s) by only modifying +the density matrix and pairing tensor and keeping the form of all the equations untouched. After +the numerically calculated Routhians at any fixed ω are interpolated using, e.g., a cubic spline +function between the lattice points, the equilibrium deformation can be determined by minimizing +the multi-dimensional potential-energy map. +3. Results And Discussion +The calculations of nuclear potential energy and/or Routhian surfaces are very helpful for un- +derstanding the structure properties (including the fission path) in nuclei. It is well known that the- +oretical description of fission is usually based on the analysis of the topography of the energy maps. +The evolution of the potential energy surface as a function of the collective coordinates is of impor- +tance. We performed the nuclear potential-energy calculations using the deformed Woos-Saxon +mean-field Hamiltonian in the deformation spaces (β2, γ, α4µ=0,2,4) and (β2, γ, β4). More elab- +orated investigation will include the parameters related to reflection asymmetric shapes because +they are required for the description of the asymmetry in fission-fragment mass-distribution [75]. +In Fig. 1, the results of potential energy surfaces projected on (β2, γ) plane and respectively mini- +mized over the hexadecapole deformation α40, α42, α44 and β4 are illustrated for 256 +106Sg150. In these +maps, the β2 and γ deformation variables are directly presented as the horizontal and vertical co- +ordinates in a Cartesian coordinate system, instead of the usual Cartesian quadrupole coordinates +[X = β2sin(γ + 30◦), Y = β2cos(γ + 30◦)] and the (β2, γ) plane in the polar coordinate sys- +tem. For the static energy surfaces, for guiding eyes, the γ domain [−60◦, 60◦] is adopted though, +in principle, half is enough. One can see that two minima (at β2 ≈ 0.24 and 0.7) appear and +the double-humped barrier is reproduced but the second peak is lower than those in the actinide +region [76]. Calculated energy map shows that the hexadecapole deformation has no influence +on the first minimum but can decrease the second minimum. It is found that the γ destroy will +strongly change the fission path, especially, between two minima. +In order to understand how dependent calculated total energies are on these hexadecapole defor- +mations α4µ=0,2,4 (we focus here on the even-µ components), Figure 2 illustrates the corresponding +10 + +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(a) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(b) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(c) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(d) +Figure 1: +The projections of calculated total energy on the (β2, γ) plane of quadrupole axial and triaxial +(γ) deformations for 256 +106Sg150. At each deformation grid, a minimization has been performed over the +hexadecapole deformation degrees of freedom α40, α42, α44 and β4 in the subplots (a), (b), (c) and (d), +respectively. The energy interval between neighbouring contour lines is 1 MeV. The red dash line denotes +the possible fission pathway. See the text for more details. +2D maps projected on (β2, α4µ=0,2,4) and (β2, β4) planes for 256 +106Sg150. To separately investigate the +effects of different hexadecapole deformation parameters on the energy surfaces, in the left four +subfigures of Fig. 2, we performed the calculations in 2D deformation spaces displayed by the +horizontal and vertical coordinates, ignoring other degrees of freedom. It needs to be stressed that +the hexadecapole deformation β4 involves the fixed relationships of {α4µ=0,2,4} and γ, cf. Eq. 7. +For instance, three deformation parameters {α4µ=0,2,4} can be determined in terms of a pair of +given β4 and γ values. It can be seen from the left panel of Fig. 2 that only α40 (equivalently +β4 at γ = 0◦) deformation changes the fission pathway. It seems that the non-axial deformation +parameters α42 and α44 have no influence on the fission trajectory at this moment. In the right +11 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α40 +-10 +-5 + 0 + 5 + 10 +(a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α40 +-10 +-5 + 0 + 5 + 10 +(e) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α42 +-10 +-5 + 0 + 5 + 10 +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α42 +-10 +-5 + 0 + 5 + 10 +(f) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α44 +-10 +-5 + 0 + 5 + 10 +(c) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +-0.15 +0.00 +0.15 +0.30 +α44 +-10 +-5 + 0 + 5 + 10 +(g) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-10 +-5 + 0 + 5 + 10 +(d) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-10 +-5 + 0 + 5 + 10 +(h) +Figure 2: Similar to Fig. 1 but projections on (β2,α40), (β2,α42),(β2,α44) and (β2,β4) planes for 256 +106Sg150. +Note that in the right four subfigures (e),(f),(g) and (h), the minimization was performed over the triaxial +deformation γ at each mesh grid. In (a),(b),(c) and (d) subplots, the triaxial destroy was not considered. See +text for more information. +part, at each deformation point of the corresponding map, the minimization was performed over +triaxial deformation γ. Indeed, one can find that non-zero {α4µ=0,2,4} values appear along the +fission pathway, indicating the three {α4µ=0,2,4} deformations play a role during the calculations; +see, e.g., Fig. 2(e)-(g). For simplicity of calculation and simultaneously including the effects of +such three hexadecapole deformation parameters, total energy projection on the (β2, β4) plane is +illustrated in Fig. 2(h), minimized over γ. It was often suggested that the 3-dimensional space +(β2, γ, β4) is the most important, e.g., cf. Ref. [39]. Similar to the γ deformation, the β4 defor- +mation has an obvious influence on the fission pathway after the first minimum for this nucleus. +Moreover, the β4 deformation always keeps a non-zero value after the first minimum. +From the 2D energy β2 vs γ and β2 vs β4 maps, we can obtain the further energy projection +e.g., on the β2 direction. By such an operation, the total energy curve will be given, which is +usually useful for extracting the information of fission barrier. Figure 3 illustrates four types of +total energy curves in functions of β2 for five selected nuclei 256,258,260Sg, 254Rf and 252No. Note +that the blue, grey, red and green lines respectively correspond to those curves whose energies are +minimized over γ and β4; γ; β4; and none. By them, one can see the evolution of the energy curves +12 + +-10 + 0 + 10 +106 +256Sg150 +γ=0 , β4=0 +γ=0 , β4min +γmin , β4=0 +γmin , β4min +0.4 +0.8 +1.2 +106 +258Sg152 +β2 +0.4 +0.8 +1.2 +106 +260Sg154 +β2 +-10 + 0 + 10 +104 +254Rf150 +Etot (MeV) +-10 + 0 + 10 +0.0 +0.4 +0.8 +1.2 +102 +252No150 +β2 +Figure 3: Four types of deformation energy curves as the function of quadrupole axial deformation β2 for +256 +106Sg150 and its two isotopic and isotonic neighbours, namely, 258 +106Sg152, 260 +106Sg154, 254 +104Rf150 and 252 +102No150. +At each β2 point, the minimization was performed over γ and/or β4. The legends denote that whether or +not total energy at each β2 was minimized and, if so, with respect to what deformation parameter(s). See +text for further explanations. +from both isotopic and isotonic directions. It seems that from the isotonic direction, 256 +106Sg150 is +the critical nucleus in which the hexadecapole deformation β4 always play a role after the first +minimum. From this figure, we can obtain the equilibrium deformations of different minima +and maxima, further the height of fission barriers. The impact of the triaxial and hexadecapole +deformations on the energy curves can clearly evaluated. The inclusion of different deformation +parameters can affect not only the height but also the shape of the fission barrier. As noted in +Ref. [75], the tunneling probability through the fission barrier will depend exponentially on the +square root of its height times its width, when approximated by a square potential barrier. One can +find that the triaxial deformation can decrease the barrier hight, especially for the inner barrier e.g. +in 256Sg. Nevertheless, the hexadecapole deformation (responsible for necking [77]) decreases +both the height and the width of the fission barrier. Even, as seen in 256,258Sg, the least-energy +fission path is strongly modified by the hexadecapole deformation after their first minima. After +the second saddles, the effect of the hexadecapole deformation becomes significant in all selected +nuclei. However, it was found that the octupole deformation will play an important role at the +second saddle and after that, leading to a change of the obtained mass asymmetry at the scission +point [7, 33, 75]. +In Table II, the present results (calculated quadrupole deformation β2 and fission barrier Bf) for +five selected nuclei are confronted with other accepted theories (the experimental data are scarce +so far), including the results of the heavy-nuclei (HN) model [9, 78], the fold-Yukawa (FY) single- +particle potential and the finite-range droplet model (FRDM) [79], the Hartree-Fock-BCS (HF- +13 + +Table II: The results of potential-energy-surface (PES) calculations for ground-state equilibrium deforma- +tion parameter β2 and inner fission barriers Bf for the 5 selected even-even nuclei, together with some other +theoretical calculations for comparison; see the text for more descriptions. +Nuclei +β2 +Bf/MeV +PES +HN [78] +FF [79] +HFBCS [80] +ETFSI [81] +PES +HN [9] +FFL [8] +ETFSI [20] +260 +106Sg154 +0.243 +0.247 +0.242 +0.31 +0.25 +6.49 +6.28 +5.84 +4.6 +258 +106Sg152 +0.242 +0.247 +0.252 +0.27 +0.25 +6.16 +6.22 +5.93 +4.7 +256 +106Sg150 +0.243 +0.246 +0.252 +0.25 +0.27 +5.88 +5.46 +5.30 +— +254 +104Rf150 +0.243 +0.247 +0.252 +0.27 +0.27 +6.44 +5.74 +5.87 +5.3 +252 +102No150 +0.243 +0.249 +0.250 +0.30 +0.26 +7.01 +6.52 +6.50 +5.8 +BCS) [80], the fold-Yukawa (FY) single-particle potential and the finite-range liquid-drop model +(FRLDM) [8], and the extended Thomas-Fermi plus Strutinsky integral (ETFSI) [20, 81] methods. +Comparison shows that these results are somewhat model-dependent but in good agreement with +each other to a large extent. It can be found that the HFBCS calculation gave the larger equilibrium +deformations and our calculation has the higher inner fission-barriers. Our calculated deformations +may be underestimated to some extent, cf. Ref. [82]. As discussed by Dudek et al. [83], the un- +derestimated quadrupole deformation β2 should be slightly modified by the empirical relationship +1.10β2-0.03(β2)3. Within the framework of the same model, it can be seen that the selected five +nuclei almost have the same β2 in the PES, HN and FF (FY+FRDM) [79] calculations. In the +HFBCS and ETFSI calculations, the nucleus 256Sg has the largest and the smallest β2 values in +the five nuclei, respectively, but the differences are still rather small. Concerning the inner fission +barriers, it seems that the present calculation may relatively overestimate the barriers. However, +the present calculation has the same trends to the results given by HN and FFL (FY+FRLDM) [8] +calculations. For instance, the nucleus 256Sg has the smallest inner barrier in these five nuclei, in +good agreement with those in HN and FFL calculations. In our previous publication [34], a lower +Bf about 4.8 MeV was obtained by using the universal parameter set. This value is lower about +1 MeV than the present calculation (5.88 MeV, as seen in the table) and lower than the values by +HN and FFL calculations. The further experimental information is desirable. Interestingly, though +the inner barrier of 256Sg is the lowest, its outer barrier (∼ 2.72 MeV) is higher than those in its +isotopic neighbors 258,260Sg (∼ 2.52 and 2.29 MeV). It is certainly expected that the outer barrier +of 256Sg can relatively increase the survival probability of this superheavy nucleus, benifiting for +the observation in experiment to some extent. +In macroscopic-microscopic model, as is well known, the total energy is mainly determined +by the liquid-drop energy and shell correction. In Fig. 4, to understand their evolution properties +from light to heavy nuclei, we show the macroscopic energy and microscopic shell correction for +arbitrarily selected nine nuclei along the β-stability line (cf. Ref. [29]). As excepted, one can see +14 + +-10 + 0 + 10 + 20 +(a) +Energy (MeV) +-10 + 0 + 10 + 20 +(a) +Energy (MeV) +48 +22Ti26 +130 + 58Ce80 +240 + 94Pu146 +-10 + 0 + 10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(b) +β2 +76 +34Se42 +172 + 70Yb102 +276 +106Sg170 +-10 + 0 + 10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +108 + 46Pd62 +206 + 82Pb124 +312 +118Og194 +Figure 4: Macroscopic energies (a) and Shell correction energies (b) as the function of quadrupole axial +deformation β2 for several selected nuclei (see the legends, or cf. Ref. [29]) along the β-stability line. Note +that during the calculation other deformation parameters are set to be zero. +that with increasing mass number A the macroscopic energy (the important contribution of fission +barrier) is decreasing at a given β2 (e.g., ∼ 0.4, about the position of the first barrier;cf. Fig. 3) +deformation, indeed, almost approaching zero in the superheavy region [e.g., with Z >∼ 104, see +276 +106Sg170 in Fig. 4(a), indicating the disappearance of the macroscopic fission barrier]. In particular, +the calculated liquid-drop energy rapidly descends with increasing β2 in the “heavier” superheavy +nucleus 312 +118Sg194 which denotes that it is more difficult to bound such a heavy nucleon-system. +Figure 3(b) illustrates the corresponding shell corrections for the selected nuclei mentioned above. +Indeed, the energy staggering is rather large and combining the smoothed macroscopic energy, the +potential pocket(s) can appear, which is the formation mechanism of superheavy nuclei. +In Fig. 5, we provide the further evolution information on the total energy and its different +components in functions of the quadrupole deformation β2 for 256 +106Sg150. Figure 5(a) illustrates that +total energy, together with the macroscopic liquid-drop energy Eld, shell correction δEshell and +pairing correlation δEpair. For simplicity, other deformation degrees of freedom are ignored. In +this nucleus, as seen, the macroscopic energy fully makes no contribution to the fission barrier. +The barrier is mainly formed by the quantum shell effect. The inclusion of short-range pairing +interaction always decreases the total energy, showing an irregular but relatively smoothed change +(decreasing the barrier here). With increasing β2, the shell effect tends to disapear. In the subfigure +Fig. 5(b), we show the total Routhian and the rotational contribution at ground-state and two se- +15 + +-20 +-10 + 0 + 10 + 20 +256 +106Sg150 +(a) +Etot +Eld +-20 +-10 + 0 + 10 + 20 +256 +106Sg150 +(a) +δEshell +δEpair +-20 +-10 + 0 + 10 + 20 +(b) +ω=0.00 : +ω=0.15 : +ω=0.30 : +Energy (MeV) +Erou +Erou +Erou +-20 +-10 + 0 + 10 + 20 +(b) +ω=0.00 : +ω=0.15 : +ω=0.30 : +Energy (MeV) +δH +δH +δH +-20 +-10 + 0 + 10 + 20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(c) +F=0.95 : +F=1.00 : +F=1.05 : +β2 +Etot +Etot +Etot +-20 +-10 + 0 + 10 + 20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(c) +F=0.95 : +F=1.00 : +F=1.05 : +β2 +δEpair +δEpair +δEpair +Figure 5: (a) Total energy Etot curve (together with its macroscopic liquid-drop energy Eld and microscopic +shell correction and pairing correlation energies, namely, δEshell and δEpair) vs β2 deformation for the +nucleus 256 +106Sg150. For simplicity, other deformation degrees of freedom were closed during the calculation. +(b) Similar to (a) but for the total Routhian (Erou) curves and the corresponding rotational contribution δH +at three selected frequencies ℏω = 0.00, 0.15 and 0.30 MeV. (c)Similar to (a) but for the total energy and +the corresponding pairing correlation δE at three selected pairing-strength factor F = 0.95, 1.00 and 1.05 +(the adjusted pairing strength G = FG0). +lected frquencies ℏω = 0.15 and 0.30 MeV, aiming to see the effect of the Coriolis force. One can +see that, similar to the trend of the pairing correlation, the energy due to rotation will decrease the +barrier because the energy difference e.g., at the positions of the first barrier and the first minimum +is a negative value. It should be noted that the selected rotational frequencies respectively corre- +spond to the values before and after the first band-crossing frequency in such a normal-deformed +superheavy nucleus, e.g., cf. Ref. [38]. Along the curve, the ground-state or yrast configuration +for the nucleus may be rather different (see, e.g., Fig. 6, the occupied single-particle levels below +the Fermi surface will generally be rather different). In Fig. 5(c), the total energy and its pairing +correlation energy are illustrated with different pairing strengths by adjusting the factor F (e.g., in +G = FG0, where G0 is the orginal pairing strength). It can be noticed that the pairing correlation +energy will decrease with increasing pairing strength G. Both at the barrier and the minimum, the +effects seem to be very similar. At the large deformation region, the pairing correlation tends to a +constant. +16 + +-3 +-2 +-1 + 0 + 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +92 +114 +104 +106 +104 +104 +(a) +2f7/2 +1i13/2 +Single proton levels (MeV) +β2 +-10 +-9 +-8 +-7 +-6 +-5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +126 +164 +148 +150 +150 +150 +140 +(b) +2d5/2 +2g7/2 +1j15/2 +1i11/2 +2g9/2 +Single neutron levels (MeV) +β2 +Figure 6: Calculated proton (a) and neutron (b) single-particle energies as functions of the quadrupole +deformation β2 for 256 +106Sg150, focusing on the domain near the Fermi surface. The levels with positive +and negative parities are respectively denoted by red solid and blue dotted lines. Spherical single-particle +orbitals (i.e., at β2 = 0.0) in the window of interest are labeled by the quantum numbers nlj. +The microscopic structure of nuclei is primarily determined by the single-particle levels, es- +pecially near the Fermi level [84]. Experimentally, one can detect and investigate single-particle +states by e.g., the inelastic electron scattering [like (e, e′p)], the direct stripping and pick-up re- +actions [typically (p, d) and (d, p) reactions], β-decay rates, and so on [85, 86]. Because the +measured single-particle states may be not pure, a rigorous definition of these states is given by +the Green’s function formalism (cf. Ref. [84]), showing that it is necessary to extract the spectro- +scopic factor. Such a quantity will provide an illustration of how much a single-particle level can +be considered as a pure state and whether or not the correlations (e.g., the short- and long-range +ones) beyond the mean field appear. Theoretically, the single-particle levels correspond to the +eigenstates of the mean-field Hamiltonian (e.g., the Woods-Saxon-type one in this work). They +are also the building blocks of the many-body wave functions, e.g., in self-consistent Hartree- +Fock calculation. In Fig. 6, the single-particle levels near the proton and neutron Fermi surfaces +are respectively illustrated in (a) and (b) parts. A set of conserved quantum numbers (associated +with a complete set of commuting observables) are usually used for labeling the corresponding +single-particle levels and wave functions. For instance, the spherical single-particle levels are de- +noted by the spherical quantum numbers n, l and j (corresponding the principal quantum number, +the orbital angular momentum, and the total angular momentum, respectively). Similar to atomic +spectroscopy, the notations s, p, d, f, g, h · · · (corresponding to l = 0, 1, 2, 3, 4, 5 · · · , re- +spectively) are used. Due to the strong spin-orbit coupling, the single particle state with l will +split into two states with j = l ± 1/2 (The degeneracy of each spherical single-particle level can +be calculated by 2j + 1). In the present work, one can see that the expected shell structure and +shell closure can be well reproduced. When deformed shape occurs, the 2j+1 degeneracy will +be broken and the spherical single-particle level will split into j + 1/2 components (each one is +17 + +typically double degenerate due to Kramers degeneracy). These deformed single-particle levels +are generally described by asymptotic Nilsson quantum numbers Ωπ[NnzΛ], where N is the total +oscillator shell quantum number; nz stands the number of oscillator quanta in the z direction (the +direction of the symmetry axis); Λ is the projection of angular momentum along the symmetry +axis; Σ is the projection of intrinsic spin along the symmetry axis; Ω is the projection of total +angular momentum j (including orbital l and spin s) on the symmetry axis and Ω = Λ + Σ. Note +that the Nilsson labels are not given owing to space limitations. Similar to magnetic field, in the +rotational coordinate system, the Coriolis force resulted from the non-inertial reference frame can +also break the time reversal symmetry and mix the Nilsson states. Then, the single-particle Routhi- +ans can only be labeled by the conserved parity and signature (π, α) or (π, r) (cf. Ref. [1] for the +rigorous definition). It should be pointed out here that we did not perform the virtual crossing +removal [87] of single-particle levels with same symmetries in these plots but this will not affect +the identification of the single-particle levels. From Fig. 6, one can see that the shell gaps appear +at the energy-minimum positions with lower level-densities and the higher level-densities occur +at the saddle positions (cf. e.g., Fig. 5). The deformed neutron shells at N = 152 and 162 are +reproduced [4]. +For a clear display about the level density near the minimum and saddle points, Figure 7 +presents the proton and neutron single-particle levels at these corresponding deformation points. +Note that the Fermi levels (the green levels) at the four typical points A, B, C and D are shifted to +zero for comparison. The levels in Fig. 7(a) and (c) correspond to deformation conditions same to +those in Figs. 5 and 6 where only the β2 deformation is considered. In the right two subfigures of +Fig. 7, at each β2 point, the “realistic” β4 value is taken into account (the equilibrium deformation +is adopted after potential-energy minimization over β4). Relative to the left two ones, the levels +are rearranged to an extent by the hexadecapole deformation degree of freedom. As excepted, the +level density is lower (higher) near the minimum (saddle) point, indicating the occurrence of a +largely negative (positive) shell correction. +Figure 8 illustrates the total Routhian surfaces projected on the β vs γ plane for 256 +106Sg150 at +several typical rotational frequencies. At each grid in the maps, the minimization of the total +Routhian was performed over β4. It needs to be stressed that the energy domains denoted by the +color palettes are different in Figs. 8(c) and (d) for a better display. Under rotation, the triaxial +deformation parameter γ covers the range from −120◦ to 60◦ because the three sectors (−120◦, +−60◦), (−60◦, 0◦) and (0◦, 60◦) will represent rotation about the long, medium and short axes, +respectively (the nucleus with triaxial shape). The nucleus with four γ values −120◦, −60◦, 0◦ +and 60◦ has the axially symmetric shape but different rotational orientation (cf. e.g., Ref. [88]). +For instance, the triaxial deformation parameter γ = −120◦ during rotation denotes that a prolate +nucleus with a non-collective rotation (namely, rotating around its symmetry axis; see, e.g., the +low-frequency part on the fission path in Fig. 8(d)). The 1D cranking is limited in the present +study. From this figure, one can see the evolution properties of the triaxiality and rotation axis for +18 + +-2 +-1 + 0 + 1 + 2 +A +B +C +D +Single proton levels (MeV) +(a) +-2 +-1 + 0 + 1 + 2 +A +B +C +D +Single proton levels (MeV) +(b) +-2 +-1 + 0 + 1 + 2 +A +B +C +D +Single neutron levels (MeV) +(c) +-2 +-1 + 0 + 1 + 2 +A +B +C +D +Single neutron levels (MeV) +(d) +Figure 7: (a) Calculated proton single-particle levels for 256 +106Sg150 at the four typical β2 deformation points +(A, the 1st minimum; B, the 1st maximum;, C, the 2nd minimum; and D, the 2nd maximum) along the +energy curve , see e.g., Fig. 3. In this plot, only β2 deformation is considered for simplicity, corresponding +to the blue energy curve in Fig. 3. (b) Similar to (a) but, in this plot, the energy is minimized over β4 for each +β2 points, corresponding to the red energy curve in Fig. 3. (c) Similar to (a) but for neutron single-particle +levels. (d) Similar to (b) but for neutron single-particle levels. +both the equilibrium shape and states along the fission path. +To investigate the hexadecapole-deformation effect under rotation, the total Routhian surfaces +projected on the (β2, β4) plane are shown in Fig. 9 for 256 +106Sg150 at four selected rotational frequen- +cies ℏω = 0.0, 0.1, 0.2 and 0.3 MeV, respectively. Note that the color palletes are slightly adjusted, +similar to those in Fig. 8. It can be seen that the hexadecapole deformation β4 can strongly de- +crease the total Routhian along the fission path, especially at high rotational frequency and large +quadrupole deformation. In other words, the fission pathway will be modified by the the hexade- +capole deformation β4. It should be pointed out that from this figure one can find that part of the +fission pathway evolutes along the border (with β4=0.30) of the calculation domain, indicating the +nucleus may possess a larger β4 at this moment. Figure 10 illustrates the total Routhian curves +in functions of β2 for 256 +106Sg150 at the selected rotational frequencies mentioned above. The size +and shape of the inner and outer barriers and their evolution with rotation can be evaluated con- +19 + +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-120 +-90 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(a) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-120 +-90 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-10 +-5 + 0 + 5 + 10 +(b) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-120 +-90 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-15 +-10 +-5 + 0 + 5 +(c) +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +β2 +-120 +-90 +-60 +-30 + 0 + 30 + 60 +γ (deg) +-20 +-15 +-10 +-5 + 0 + 5 +(d) +Figure 8: Similar to Fig. 1(d) but for total Routhian projections of 256 +106Sg150 at rotational frequencies ℏω = +0.0 (a), 0.1 (b), 0.2 (c) and 0.3 (d) MeV, respectively. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-10 +-5 + 0 + 5 + 10 +(a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-10 +-5 + 0 + 5 + 10 +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-15 +-10 +-5 + 0 + 5 +(c) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +β2 +-0.15 +0.00 +0.15 +0.30 +β4 +-20 +-15 +-10 +-5 + 0 + 5 +(d) +Figure 9: Similar to Fig. 2(h) but for total Routhian projections of 256 +106Sg150 at rotational frequencies ℏω = +0.0 (a), 0.1 (b), 0.2 (c) and 0.3 (d) MeV, respectively. +20 + +-30 +-20 +-10 + 0 + 10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +106 +256Sg150 +Etot (MeV) +β2 +ω=0.0 +ω=0.1 +ω=0.2 +ω=0.3 +Figure 10: The calculated total Routhian curves against β2 for 256 +106Sg150 at four selected frequencies ℏω = +0.0, 0.1, 0.2 and 0.3 MeV. At each β2 point, the minimization was performed over γ and β4. +veniently. In the previous studies, e.g., in Refs. [6, 7, 33] , it was pointed out that the octupole +correlation may further decrease the outer barrier in this mass region based on the PES calculation +and fission fragment analysis. The outer barrier for this nucleus may finally be very low. It will be +an open problem whether it will be able to play a certain role in blocking the fission process. +4. Conclusions +We evaluate the structure evolution along the fission pathway for 256Sg by using the multi- +dimensional potential-energy(or Routhian)-surface calculations, focusing on the effects of triaxial +and hexadecapole deformation and Coriolis force. Nuclear shape and microscopic single-particle +structure are investigated and analyzed. The present results are compared with other theories. The +properties of nuclear shape and fission barrier are analyzed by comparing with its neighboring +even-even nuclei, showing a reasonable agreement. Based on the deformation energy or Routhian +curves, the fission barriers are analyzed, focusing on their shapes, heights, and evolution with +rotation. It is found that the triaxial deformation γ decreases the potential energy on the land- +scape near the saddles but the hexadecapole deformation β4 (especially the axial α40 component) +modifies the least-energy fission path after the first minimum, especially in 256Sg. In addition, in +contrast to the inner barrier, the outer barriers seem to have an increasing trend from 260Sg to 256Sg +which may be benefit for blocking the fission of 256Sg to some extent. Next, it will be necessary +to simultaneously consider the reflection asymmetry in a more reasonable deformation subspace. +Acknowledgement +This work was supported by the National Natural Science Foundation of China (Nos. +11975209, U2032211 and 12075287), the Physics Research and Development Program of +21 + +Zhengzhou University (No. 32410017), and the Project of Youth Backbone Teachers of Col- +leges and Universities of Henan Province (No. 2017GGJS008). Some of the calculations were +conducted at the National Supercomputing Center in Zhengzhou. +Conflict of Interest +The authors declare that they have no known competing financial interests or personal relation- +ships that could have appeared to influence the work reported in this paper. +22 + +References +[1] M J A de Voigt, J Dudek and Z Szyma´nski Rev. Mod. 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C 92 024303 (2015) +26 + diff --git a/ftE1T4oBgHgl3EQfewRy/content/tmp_files/load_file.txt b/ftE1T4oBgHgl3EQfewRy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75ab55e2c34dec62381a436b9a8986fea8d3d6c3 --- /dev/null +++ b/ftE1T4oBgHgl3EQfewRy/content/tmp_files/load_file.txt @@ -0,0 +1,1144 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf,len=1143 +page_content='Probing the structural evolution along the fission path in the superheavy nucleus 256Sg Ting-Ting Li,1 Hua-Lei Wang,1, ∗ Zhen-Zhen Zhang,1 and Min-Liang Liu2, 3 1School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (Dated: January 10, 2023) Abstract The evolution of structure property along the fission path in the superheavy nucleus 256Sg is pre- dicted through the multi-dimensional potential-energy(or Routhian)-surface calculations, in which the phe- nomenological deformed Woods-Saxon potential is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Calculated nuclear deformations and fission barriers for 256 106Sg150 and its neighbors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', 258,260Sg, 254Rf and 252No are presented and compared with other theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' A series of energy maps and curves are provided and used to evaluate the corre- sponding shape-instability properties, especially in the directions of triaxial γ and different hexadecapole deformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', α40, α42 and α44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is found that the triaxial deformation may help the nucleus bypass the first fission-barrier of the axial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' After the first minimum in the nuclear energy surface, the fission pathway of the nucleus can be affected by γ and hexadecapole deformation degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In addition, microscopic single-particle structure, pairing and Coriolis effects are briefly investigated and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Keywords: structure evolution, fission path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' fission barrier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' superheavy nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' macroscopic- microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' PACS numbers: ∗wanghualei@zzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='cn(Corresponding author) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='03210v1 [nucl-th] 9 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Introduction The evolution of nuclear structure properties with some degree of freedom (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', nucleon num- ber, spin, temperature, etc) is one of the most significant issues in nuclear physics [1], especially towards the superheavy mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Great progress has been made in the synthesis of superheavy nuclei with the development of the radioactive beam facility, heavy-ion accelerator and highly- effective detector systems [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Spontaneous fission is usually one of important decay modes in a superheavy nucleus and the barrier along the fission path is critical to understand the fission process [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the survival probability of a synthesized superheavy nucleus in the heavy-ion fusion reaction is directly related to such a barrier, during the cooling process of a com- pound nucleus, which plays a decisive role in the competition between nucleon evaporation and fission (a small change of the fission barrier may result in several orders of magnitude difference in survival probability) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Nevertheless, it is still rather difficult to give an accurate description for the fission barrier so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To a large extent, the barrier size and shape can be determined by the fission path in the nuclear energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Up to now, there are several types of models which are widely used for investigating nuclear fission phenomena, including e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the macroscopic-microscopic (MM) models [8–12], the non- relativistic energy density functionals based on zero-range Skyrme and finite-range Gogny inter- actions [13–18], the extended Thomas-Fermi plus Strutinsky integral methods [19, 20] , and the covariant density functional theory [5, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The MM methods usually have the high descriptive power as well as simplicity of calculation and thus are still used by many researchers so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In such an approach, the empirical one-body nuclear mean-filed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the Nilsson and Woods-Saxon potentials) Hamiltonian is used to solve the microscopic single-particle levels and wave functions and a macroscopic liquid-drop model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the standard liquid-drop model [23], the finite-range droplet model [24], and the Lublin-Strasboug drop model [25], etc) is combined to describe the nuclear bulk property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In recent years, the model parameters, including their uncertainties and propagations, in both phenomenological Woods-Saxon potential and the macroscopic liquid-drop model are still studied and optimized, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', cf Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [11, 26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Indeed, the parameters of MM models are mainly from the fitting of available single-particle levels of several spherical nuclei and several thousand nuclear-mass data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' They are generally successful near the β-stability line, especially in the medium and heavy nuclear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Without the preconceived knowledge, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', about the measured densities and single-particle energies, it may be needed to test whether the modeling and model parameters of a phenomenological one-body potential are still valid enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Part of our aim of this work is to test the theoretical method in such aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Prior to this work, 16 Sg isotopes from A = 258 to 273 were synthesized by the fusion- evaporation reactions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', 238U(30Si,xn)268−xSg [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It was reported that the lightest even-even Sg isotope, 258Sg, has a revised half-life of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='5 ms [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Naturally, one expects that based on the fusion-evaporation mechanism, the superheavy nuclide 256Sg will be synthesized as the next 2 candidate which is the nearest even-even nucleus to the known ones in this isotopic chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Keeping this in mind, we predict the properties of structure evolution along the possible fission path for the superheavy nuclide 256Sg in this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In our previous studies, we systematically investigated the octupole correlation properties for 42 even-even nuclei with 102 ≤ Z ≤ 112 [33] and the tri- axial effects on the inner fission barriers in 95 tranuranium even-even nuclei 94 ≤ Z ≤ 118 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The triaxiality and Coriolis effects on the fission barrier in isovolumic nuclei with A = 256 were investigated, where the 256Sg was calculated but just focused on the first (inner) fission barrier [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [36], we investigated the effects of various deformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', β2, γ and β4) on the first barrier in even-even nuclei with N = 152 and 94 ≤ Z ≤ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In addition, we studied the collective rotational effects including the α-decay-chain nuclei (from 216Po and 272Cn) [37] and 254−258Rf [38] by the similar calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The primary purpose of this study is to investigate the effects of different deformation parameters, especially the axial and non-axial hexadepole defor- mations, on the fission path of 256Sg by analyzing the topography of the energy surfaces calculated in a reasonable subspace of collective coordinates (it is impossible to calculate in the full defor- mation space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The probe of the shape evolution along the fission path on the energy landscape will be useful for understanding the formation mechanism of the fission barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' We provide the analysis of the single-particle structures, shell and pairing evolutions, especially at the minima and saddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Sobiczewski et al [39] systematically investigated the static inner barrier of heaviest nuclei with proton number 98 ≤ Z ≤ 126 and neutron number 134 ≤ N ≤ 192 in a multidimen- sional deformation space and pointed out that the inclusion of the non-axial hexadecapole shapes lowers the barrier by up to about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the synthesis of the superheavy nuclei, nuclear hex- adecapole deformations were revealed to have an important influence on production cross sections of superheavy nuclei by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', affecting the driving potentials and the fusion probabilities [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' This paper is organized as follows: In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2, we briefly describe the outline of the theoretical framework and the details of the numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The results of the calculations and their relevant discussion are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Finally, the concluding remarks will be given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Theoretical framework In what follows, we recall the unified procedure and give the necessary references related to the present theoretical calculation, which may be somewhat helpful for some readers to clarify some details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the various variants of the pairing-energy contribution within the framework of the macroscopic-microscopic method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' We employ potential-energy(or Routhian)-surface calculation to study the present project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' This method is based on the macroscopic-microscopic model [42, 43] and the cranking approximation [44–46], which is one of widely used and powerful tools in nuclear structure research, especially for rotating nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The usual expression for the total energy in the rotating coordinate frame (namely, the so-called total Routhian) reads [47] Eω(Z, N, ˆβ) = Eω macr(Z, N, ˆβ) + δEω micro(Z, N, ˆβ), (1) 3 where Eω(Z, N, ˆβ) represents the total Routhian of a nucleus (Z, N) at frequency ω and defor- mation ˆβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The first term on the right-hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (1) denotes the macroscopic (liquid drop, or LD) energy with the rigid-body moment of inertia calculated classically at a given deformation, as- suming a uniform density distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' δEω micro represents the contribution due to the microscopic effects under rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' After rearrangement employing elementary transformations [48–52], the total Routhian can be rewritten as, Eω(Z, N, ˆβ) = Eω=0(Z, N, ˆβ) + [⟨ ˆHω(Z, N, ˆβ)⟩ − ⟨ ˆHω=0(Z, N, ˆβ)⟩] − 1 2ω2[Jmacr(A, ˆβ) − JStru(Z, N, ˆβ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2) The notations for the quantities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2) are standard [47, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The term Eω=0(Z, N, ˆβ) is the static total energy (corresponding ω = 0) which consists of a macroscopic LD part ELD(Z, N, ˆβ) and a shell correction δEshell(Z, N, ˆβ) and a pairing-energy contribution δEpair(Z, N, ˆβ) (neglect- ing the superscript ω = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The second term in the square brackets represents the energy change of the cranked Hamiltonian ˆHω(Z, N, ˆβ) due to rotation [47, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2), it is usually and reasonably assumed that the average pairing energy of the liquid-drop term and the Strutinsky- smeared pairing energy cancel each other [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Therefore, one can further write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2) as [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [54] and references therein], Eω(Z, N, ˆβ) = ELD(Z, N, ˆβ) + δEshell(Z, N, ˆβ) + δEpair(Z, N, ˆβ) + [⟨ ˆHω(Z, N, ˆβ)⟩ − ⟨ ˆHω=0(Z, N, ˆβ)⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (3) As known, several phenomenological LD models (such as standard liquid drop model [23], finite- range droplet model [42], Lublin-Strasbourg drop model [25]) with slight difference have been developed for calculating the smoothly varying part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In these LD models, the dominating terms are mainly associated with the volume energy, the surface energy and the Coulomb energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the present work, the macroscopic energy is given by the standard LD model with the parameters used by Myers and Swiatecki [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The single-particle levels used below are calculated by solving numerically the Schr¨odinger equation with the Woods-Saxon (WS) Hamiltonian [55] HWS = T + Vcent(⃗r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' ˆβ) + Vso(⃗r, ⃗p,⃗s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' ˆβ) +VCoul(⃗r, ˆβ), (4) where the Coulomb potential VCoul(⃗r, ˆβ) defined as a classical electrostatic potential of a uniformly charged drop is added for protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The central part of the WS potential is calculated as Vcent(⃗r, ˆβ) = V0[1 ± κ(N − Z)/(N + Z)] 1 + exp[distΣ(⃗r, ˆβ)/a] , (5) 4 where the plus and minus signs hold for protons and neutrons, respectively and the parameter a denotes the diffuseness of the nuclear surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The term distΣ(⃗r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' ˆβ) represents the distance of a point ⃗r from the nuclear surface Σ parameterized in term of the multipole expansion of spherical harmonics Yλµ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' φ) (which are convenient to describe the geometrical properties),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Σ : R(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' φ) = r0A1/3c(ˆβ) � 1 + � λ +λ � µ=−λ αλµY ∗ λµ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' φ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (6) where the function c(ˆβ) ensures the conservation of the nuclear volume with a change in the nu- clear shape and ˆβ denotes the set of all the deformation parameters {αλµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For a given nucleus with mass number A, a limiting value of λ < A1/3 is often estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the present shape parametrization, we consider quadrupole and hexadecapole degrees of freedom, including non- axial deformations, namely, ˆβ ≡ {α20, α2±2, α40, α4±2, α4±4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The quantity R(θ, φ) denotes the distance of any point on the nuclear surface from the origin of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Because only the even λ and even µ components are taken into account, the present parametrisation will preserve three symmetry planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' After requesting the hexadecpole degrees of freedom to be functions of the scalars in the quadrupole tensor α2µ, one can reduce the number of independent coefficients to three, namely, β2, γ and β4, which obey the relationships [56] � � � � � � � � � � � � � � � � � � � � � � � α20 = β2 cos γ α22 = α2−2 = − 1 √ 2β2 sin γ α40 = 1 6β4(5 cos2 γ + 1) α42 = α4−2 = − 1 12 √ 30β4 sin 2γ α44 = α4−4 = 1 12 √ 70β4 sin2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (7) The (β2, γ, β4) parametrization has all the symmetry properties of Bohr’s (β2, γ) parametriza- tion [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The spin-orbit potential, which can strongly affects the level order, is defined by Vso(⃗r, ⃗p,⃗s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' ˆβ) = −λ � ℏ 2mc �2 × � ∇V0[1 ± κ(N − Z)/(N + Z)] 1 + exp[distΣso(⃗r, ˆβ)/aso] � × ⃗p · ⃗s, (8) where λ denotes the strength parameter of the effective spin-orbit force acting on the individual nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The new surface Σso is different from the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (6) due to the different radius parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the present work, the WS parameters are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [56, 58], as listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In computing the Woods-Saxon Hamiltonian matrix, the eigenfunctions of the axially deformed harmonic oscillator potential in the cylindrical coordinate system are adopted as the basis func- 5 Table I: The adopted WS parameters for both protons and neutrons (for more details, cf e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that nuclear shape does not sensitively depend on the parameter sets in well-deformed nuclei, especially those with large stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' V0 (MeV) κ r0 (fm) a(fm) λ (r0)so (fm) aso (fm) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='791 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='637 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='494 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='637 tions [59], |nρnzΛΣ⟩ = ψΛ nρ(ρ)ψnz(z)ψΛ(ϕ)χ(Σ), (9) where � � � � � � � � � � � � � � � � � � � � � � � � � ψΛ nρ(ρ) = √ nρ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' √ (nρ+|Λ|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2mωρ/ℏ)1/2 ×e− η2 2 ηΛL|Λ| nρ (η), ψnz(z) = 1 √√π2nz nz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (2mωz/ℏ)1/4 ×e− ξ2 2 Hnz(ξ), ψΛ(ϕ) = 1 √ 2πeiΛϕ, (10) and χ(Σ) represents the spin wave functions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [59] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In our calculation, the eigenfunctions with the principal quantum number N ≤ 12 and 14 have been chosen as a basis for protons and neutrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is found that, by such a basis cutoff, the results are sufficiently stable with respect to a possible enlargement of the basis space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In addition, the time reversal (resulting in the Kramers degeneracy) and spatial symmetries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the existence of three symmetry x − y, y − z and z − x planes) are used for simplifying the Hamiltonian matrix calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The shell correction δEshell(Z, N, ˆβ), as seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (3), is usually the most important correc- tion to the LD energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Strutinsky first proposed a phenomenological expression, δEshell(Z, N, ˆβ) = � ei − � e˜g(e)de, (11) where ei denotes the calculated single-particle levels and ˜g(e) is the so-called smooth level density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Obviously, the smooth level distribution function is the most important quantity, which was early defined as, ˜g(e, γ) ≡ 1 γ√π � i exp[−(e − ei)2 γ2 ], (12) where γ indicates the smoothing parameter without much physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To eliminate any possibly strong γ-parameter dependence for the final result, the mathematical form of the smooth level density ˜g(e) has been optimized by introducing a phenomenological curvature-correction 6 polynomial Pp(x)[49, 60–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Then, the ˜g(e) expression will take the form ˜g(e, γ, p) = 1 γ√π � i=1 Pp(e − ei γ ) × exp[−(e − ei)2 γ2 ], (13) where the corrective polynomial Pp(x) can be expanded in terms of the Hermite or Laguerre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The corresponding coefficients of the expansion can be obtained by using the or- thogonality properties of these polynomials and Strutinsky condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', see the APPENDIX in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='[63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In fact, this method can be considered standard so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (12) can be calculated as follows (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [64] for more details), � e˜g(e, γ, p)de = � ˜e(n)dn = � i=1 {1 2ei[1 + erf( ˜λ − ei γ )] − 1 2√πγexp[−(˜λ − ei)2 γ2 ] − 1 √πexp[−(˜λ − ei)2 γ2 ] × p � m=1 cm[1 2γHm( ˜λ − ei γ ) +eiHm−1( ˜λ − ei γ ) +mγHm−2( ˜λ − ei γ )]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (14) Of course, there are some other methods developed for the shell correction calculations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the semiclassical Wigner-Kirkwood expansion method [56, 65] and the Green’s function method [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In this work, the widely used Strutinsky method is adopted though its known problems which ap- pear for mean-field potentials of finite depth as well as for nuclei close to the proton or neutron drip lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The smooth density is calculated with a sixth-order Hermite polynomial and a smooth- ing range γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='20ℏω0, where ℏω0 = 41/A1/3 MeV, indicating a satisfactory independence of the shell correction on the parameters γ and p [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Besides the shell correction, the pairing-energy contribution is also one of important single- particle corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Due to the short-range interaction of nucleon pairs in time-reversed orbitals, the total potential energy in nuclei relative to the energy without pairing always decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' There exist various variants of the pairing-energy contribution in the microscopic-energy calculations, as is recently pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Typically, several kinds of the phenomenological pairing energy expressions (namely, pairing correlation and pairing correction energies employing or not employing the particle number projection technique) are widely adopted in the applications of the macroscopic-microscopic approach [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To avoid the confusions, it may be somewhat necessary 7 to simply review the ‘standard’ definitions for pairing correlation and pairing correction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', cf Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [11, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the former is given by the difference between e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', BCS energy of the system at pairing ∆ ̸= 0 and its partner expression at ∆ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' similar to the Strutinsky shell correction, the later represents the difference between the above pairing correlation and its Strutinsky-type smoothed out partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the present work, the contribution δEpair(Z, N, ˆβ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (3) is the pairing correlation energy as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The pairing is treated by the Lipkin-Nogami (LN) method [67], which helps avoiding not only the spurious pairing phase transition but also the particle number fluctuation encountered in the simpler BCS calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the LN technique [53, 67], it aims at minimizing the expectation value of the following model Hamiltonian ˆH = ˆHWS + ˆHpair − λ1 ˆN − λ2 ˆN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (15) Here, ˆHpair indicates the pairing interaction Hamiltonian including monopole and doubly stretched quadrupole pairing forces [68–70]: ¯v(λµ) αβγδ = −Gλµg(λµ) α¯β g∗(λµ) γ¯δ , (16) where g(λµ) α¯β = � δα¯β λ = 0, µ = 0, ⟨α| �Qµ|¯β⟩ λ = 2, µ = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (17) The monopole pairing strength G00 is determined by the average gap method [68] and the quadrupole pairing strengths G2µ are obtained by restoring the Galilean invariance broken by the seniority pairing force [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To some extent, the quadrupole pairing can affect rotational band- head energies, moments of inertia, band-crossing frequencies and signature inversion in odd-odd nuclei [69, 71–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The pairing window, including dozens of single-particle levels, the respec- tive states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' half of the particle number Z or N) just below and above the Fermi energy, is adopted empirically for both protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The pairing gap ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Fermi energy λ (namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' λ1 + 2λ2(Ntotal + 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' particle number fluctuation constant λ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' occupation probabilities v2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' and shifted single-particle energies εk can be determined from the following 2(N2 − N1) + 5 coupled nonlinear equations [67,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' � � � � � � � � � � � � � � � � � � � � � � � � � Ntotal = 2 �N2 k=N1 v2 k + 2(N1 − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' ∆ = G �N2 k=N1 ukvk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' v2 k = 1 2 � 1 − εk−λ √ (εk−λ)2+∆2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' εk = ek + (4λ − G)v2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' λ2 = G 4 � (�N2 k=N1 u3 kvk)(�N2 k=N1 ukv3 k)−�N2 k=N1 u4 kv4 k (�N2 k=N1 u2 kv2 k)2−�N2 k=N1 u4 kv4 k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (18) 8 where u2 k = 1 − v2 k and k = N1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' N1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The LN pairing energy for the system of even-even nuclei at “paired solution” (pairing gap ∆ ̸= 0) can be given by [42, 67] ELN = � k 2vk 2ek − ∆2 G − G � k vk 4 −4λ2 � k uk 2vk 2, (19) where vk2, ek, ∆ and λ2 represent the occupation probabilities, single-particle energies, pairing gap and number-fluctuation constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Correspondingly, the partner expression at “no-pairing solution” (∆ = 0) reads ELN(∆ = 0) = � k 2ek − GN 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (20) The pairing correlation is defined as the difference between paired solution ELN and no-pairing solution ELN(∆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the cranking calculation, we only consider the one-dimensional approximation, supposing that the nuclear system is constrained to rotate around a fixed axis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' the x−axis with the largest moment of inertia) at a given frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The cranking Hamiltonian follows the form Hω = HWS + Hpair − ωjx − λ1 ˆN − λ2 ˆN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (21) The resulting cranking LN equation takes the form of the well known Hartree–Fock–Bogolyubov– like (HFB) equation which can be solved by using the HFB cranking (HFBC) method [74] (also see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Ref [1], for a detailed description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The HFB-like equations have the following form (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [53]): � � � � � � � � � � � � � � � � � � � � � � � � � � β>0 �� (eα − λ)δαβ − ω(jx)αβ − Gρ∗ ¯α¯β + 4λ2ραβ � ×Uβk − ∆δαβV¯βk � = EkUαk, � β>0 �� (eα − λ)δαβ − ω(jx)αβ − Gραβ + 4λ2ρ∗ ¯α¯β � ×V¯βk + ∆∗δαβUβk � = EkV¯αk, (22) where ∆ = G � α>0 κα¯α, λ = λ1 +2λ2(N +1) and Ek = εk −λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Further, εk is the quasi-particle energy and α (¯α) denotes the states of signature r = −i (r = +i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The quantities ρ and κ respec- tively correspond to the density matrix and pairing tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' While solving the HFBC equations, pairing is treated self-consistently at each frequency ω and each grid point in the selected defor- mation space (namely, pairing self-consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Symmetries of the rotating potential are used to simplify the cranking equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, in the present reflection-symmetric case, both 9 signature, r, and intrinsic parity, π are good quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Finally, the energy in the rotating framework can be given by Eω = Tr(e − ωjx)ρ − ∆2 G − G � α,β>0 ρα,βρ˜α,˜β −2λ2Trρ(1 − ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (23) Accordingly, one can obtain the energy relative to the non-rotating (ω = 0) state, as seen in the last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It should certainly be mentioned that the above derivations are used for the quasi-particle vacuum configuration of even-even nuclear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' However, it is convenient to extend the formalism to one or many quasi-particle excited configuration(s) by only modifying the density matrix and pairing tensor and keeping the form of all the equations untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' After the numerically calculated Routhians at any fixed ω are interpolated using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', a cubic spline function between the lattice points, the equilibrium deformation can be determined by minimizing the multi-dimensional potential-energy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Results And Discussion The calculations of nuclear potential energy and/or Routhian surfaces are very helpful for un- derstanding the structure properties (including the fission path) in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is well known that the- oretical description of fission is usually based on the analysis of the topography of the energy maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The evolution of the potential energy surface as a function of the collective coordinates is of impor- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' We performed the nuclear potential-energy calculations using the deformed Woos-Saxon mean-field Hamiltonian in the deformation spaces (β2, γ, α4µ=0,2,4) and (β2, γ, β4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' More elab- orated investigation will include the parameters related to reflection asymmetric shapes because they are required for the description of the asymmetry in fission-fragment mass-distribution [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 1, the results of potential energy surfaces projected on (β2, γ) plane and respectively mini- mized over the hexadecapole deformation α40, α42, α44 and β4 are illustrated for 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In these maps, the β2 and γ deformation variables are directly presented as the horizontal and vertical co- ordinates in a Cartesian coordinate system, instead of the usual Cartesian quadrupole coordinates [X = β2sin(γ + 30◦), Y = β2cos(γ + 30◦)] and the (β2, γ) plane in the polar coordinate sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For the static energy surfaces, for guiding eyes, the γ domain [−60◦, 60◦] is adopted though, in principle, half is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' One can see that two minima (at β2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='24 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='7) appear and the double-humped barrier is reproduced but the second peak is lower than those in the actinide region [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Calculated energy map shows that the hexadecapole deformation has no influence on the first minimum but can decrease the second minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is found that the γ destroy will strongly change the fission path, especially, between two minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In order to understand how dependent calculated total energies are on these hexadecapole defor- mations α4µ=0,2,4 (we focus here on the even-µ components), Figure 2 illustrates the corresponding 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 60 30 0 30 60 γ (deg) 10 5 0 5 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 60 30 0 30 60 γ (deg) 10 5 0 5 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 60 30 0 30 60 γ (deg) 10 5 0 5 10 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 60 30 0 30 60 γ (deg) 10 5 0 5 10 (d) Figure 1: The projections of calculated total energy on the (β2, γ) plane of quadrupole axial and triaxial (γ) deformations for 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' At each deformation grid, a minimization has been performed over the hexadecapole deformation degrees of freedom α40, α42, α44 and β4 in the subplots (a), (b), (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The energy interval between neighbouring contour lines is 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The red dash line denotes the possible fission pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' See the text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2D maps projected on (β2, α4µ=0,2,4) and (β2, β4) planes for 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To separately investigate the effects of different hexadecapole deformation parameters on the energy surfaces, in the left four subfigures of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2, we performed the calculations in 2D deformation spaces displayed by the horizontal and vertical coordinates, ignoring other degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It needs to be stressed that the hexadecapole deformation β4 involves the fixed relationships of {α4µ=0,2,4} and γ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, three deformation parameters {α4µ=0,2,4} can be determined in terms of a pair of given β4 and γ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It can be seen from the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2 that only α40 (equivalently β4 at γ = 0◦) deformation changes the fission pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It seems that the non-axial deformation parameters α42 and α44 have no influence on the fission trajectory at this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the right 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 α40 10 5 0 5 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 α40 10 5 0 5 10 (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 α44 10 5 0 5 10 (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 10 5 0 5 10 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 10 5 0 5 10 (h) Figure 2: Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 1 but projections on (β2,α40), (β2,α42),(β2,α44) and (β2,β4) planes for 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that in the right four subfigures (e),(f),(g) and (h), the minimization was performed over the triaxial deformation γ at each mesh grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In (a),(b),(c) and (d) subplots, the triaxial destroy was not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' See text for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' part, at each deformation point of the corresponding map, the minimization was performed over triaxial deformation γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Indeed, one can find that non-zero {α4µ=0,2,4} values appear along the fission pathway, indicating the three {α4µ=0,2,4} deformations play a role during the calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2(e)-(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For simplicity of calculation and simultaneously including the effects of such three hexadecapole deformation parameters, total energy projection on the (β2, β4) plane is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2(h), minimized over γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It was often suggested that the 3-dimensional space (β2, γ, β4) is the most important, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Similar to the γ deformation, the β4 defor- mation has an obvious influence on the fission pathway after the first minimum for this nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Moreover, the β4 deformation always keeps a non-zero value after the first minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' From the 2D energy β2 vs γ and β2 vs β4 maps, we can obtain the further energy projection e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', on the β2 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' By such an operation, the total energy curve will be given, which is usually useful for extracting the information of fission barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Figure 3 illustrates four types of total energy curves in functions of β2 for five selected nuclei 256,258,260Sg, 254Rf and 252No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that the blue, grey, red and green lines respectively correspond to those curves whose energies are minimized over γ and β4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' β4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' and none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' By them, one can see the evolution of the energy curves 12 10 0 10 106 256Sg150 γ=0 , β4=0 γ=0 , β4min γmin , β4=0 γmin , β4min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 106 258Sg152 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 106 260Sg154 β2 10 0 10 104 254Rf150 Etot (MeV) 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 102 252No150 β2 Figure 3: Four types of deformation energy curves as the function of quadrupole axial deformation β2 for 256 106Sg150 and its two isotopic and isotonic neighbours, namely, 258 106Sg152, 260 106Sg154, 254 104Rf150 and 252 102No150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' At each β2 point, the minimization was performed over γ and/or β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The legends denote that whether or not total energy at each β2 was minimized and, if so, with respect to what deformation parameter(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' See text for further explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' from both isotopic and isotonic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It seems that from the isotonic direction, 256 106Sg150 is the critical nucleus in which the hexadecapole deformation β4 always play a role after the first minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' From this figure, we can obtain the equilibrium deformations of different minima and maxima, further the height of fission barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The impact of the triaxial and hexadecapole deformations on the energy curves can clearly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The inclusion of different deformation parameters can affect not only the height but also the shape of the fission barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' As noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [75], the tunneling probability through the fission barrier will depend exponentially on the square root of its height times its width, when approximated by a square potential barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' One can find that the triaxial deformation can decrease the barrier hight, especially for the inner barrier e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' in 256Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Nevertheless, the hexadecapole deformation (responsible for necking [77]) decreases both the height and the width of the fission barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Even, as seen in 256,258Sg, the least-energy fission path is strongly modified by the hexadecapole deformation after their first minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' After the second saddles, the effect of the hexadecapole deformation becomes significant in all selected nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' However, it was found that the octupole deformation will play an important role at the second saddle and after that, leading to a change of the obtained mass asymmetry at the scission point [7, 33, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Table II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' the present results (calculated quadrupole deformation β2 and fission barrier Bf) for five selected nuclei are confronted with other accepted theories (the experimental data are scarce so far),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' including the results of the heavy-nuclei (HN) model [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 78],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' the fold-Yukawa (FY) single- particle potential and the finite-range droplet model (FRDM) [79],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' the Hartree-Fock-BCS (HF- 13 Table II: The results of potential-energy-surface (PES) calculations for ground-state equilibrium deforma- tion parameter β2 and inner fission barriers Bf for the 5 selected even-even nuclei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' together with some other theoretical calculations for comparison;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' see the text for more descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Nuclei β2 Bf/MeV PES HN [78] FF [79] HFBCS [80] ETFSI [81] PES HN [9] FFL [8] ETFSI [20] 260 106Sg154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='49 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='84 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 258 106Sg152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='93 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='7 256 106Sg150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 — 254 104Rf150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 252 102No150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 BCS) [80], the fold-Yukawa (FY) single-particle potential and the finite-range liquid-drop model (FRLDM) [8], and the extended Thomas-Fermi plus Strutinsky integral (ETFSI) [20, 81] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Comparison shows that these results are somewhat model-dependent but in good agreement with each other to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It can be found that the HFBCS calculation gave the larger equilibrium deformations and our calculation has the higher inner fission-barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Our calculated deformations may be underestimated to some extent, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' As discussed by Dudek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [83], the un- derestimated quadrupole deformation β2 should be slightly modified by the empirical relationship 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='10β2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='03(β2)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Within the framework of the same model, it can be seen that the selected five nuclei almost have the same β2 in the PES, HN and FF (FY+FRDM) [79] calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the HFBCS and ETFSI calculations, the nucleus 256Sg has the largest and the smallest β2 values in the five nuclei, respectively, but the differences are still rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Concerning the inner fission barriers, it seems that the present calculation may relatively overestimate the barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' However, the present calculation has the same trends to the results given by HN and FFL (FY+FRLDM) [8] calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the nucleus 256Sg has the smallest inner barrier in these five nuclei, in good agreement with those in HN and FFL calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In our previous publication [34], a lower Bf about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 MeV was obtained by using the universal parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' This value is lower about 1 MeV than the present calculation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='88 MeV, as seen in the table) and lower than the values by HN and FFL calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The further experimental information is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Interestingly, though the inner barrier of 256Sg is the lowest, its outer barrier (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='72 MeV) is higher than those in its isotopic neighbors 258,260Sg (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='52 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='29 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is certainly expected that the outer barrier of 256Sg can relatively increase the survival probability of this superheavy nucleus, benifiting for the observation in experiment to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In macroscopic-microscopic model, as is well known, the total energy is mainly determined by the liquid-drop energy and shell correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 4, to understand their evolution properties from light to heavy nuclei, we show the macroscopic energy and microscopic shell correction for arbitrarily selected nine nuclei along the β-stability line (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' As excepted, one can see 14 10 0 10 20 (a) Energy (MeV) 10 0 10 20 (a) Energy (MeV) 48 22Ti26 130 58Ce80 240 94Pu146 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 (b) β2 76 34Se42 172 70Yb102 276 106Sg170 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 108 46Pd62 206 82Pb124 312 118Og194 Figure 4: Macroscopic energies (a) and Shell correction energies (b) as the function of quadrupole axial deformation β2 for several selected nuclei (see the legends, or cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [29]) along the β-stability line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that during the calculation other deformation parameters are set to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' that with increasing mass number A the macroscopic energy (the important contribution of fission barrier) is decreasing at a given β2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4, about the position of the first barrier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3) deformation, indeed, almost approaching zero in the superheavy region [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', with Z >∼ 104, see 276 106Sg170 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 4(a), indicating the disappearance of the macroscopic fission barrier].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In particular, the calculated liquid-drop energy rapidly descends with increasing β2 in the “heavier” superheavy nucleus 312 118Sg194 which denotes that it is more difficult to bound such a heavy nucleon-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Figure 3(b) illustrates the corresponding shell corrections for the selected nuclei mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Indeed, the energy staggering is rather large and combining the smoothed macroscopic energy, the potential pocket(s) can appear, which is the formation mechanism of superheavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 5, we provide the further evolution information on the total energy and its different components in functions of the quadrupole deformation β2 for 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Figure 5(a) illustrates that total energy, together with the macroscopic liquid-drop energy Eld, shell correction δEshell and pairing correlation δEpair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For simplicity, other deformation degrees of freedom are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In this nucleus, as seen, the macroscopic energy fully makes no contribution to the fission barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The barrier is mainly formed by the quantum shell effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The inclusion of short-range pairing interaction always decreases the total energy, showing an irregular but relatively smoothed change (decreasing the barrier here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' With increasing β2, the shell effect tends to disapear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the subfigure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 5(b), we show the total Routhian and the rotational contribution at ground-state and two se- 15 20 10 0 10 20 256 106Sg150 (a) Etot Eld 20 10 0 10 20 256 106Sg150 (a) δEshell δEpair 20 10 0 10 20 (b) ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 : ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 : ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 : Energy (MeV) Erou Erou Erou 20 10 0 10 20 (b) ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 : ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 : ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 : Energy (MeV) δH δH δH 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 (c) F=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='95 : F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 : F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='05 : β2 Etot Etot Etot 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 (c) F=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='95 : F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 : F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='05 : β2 δEpair δEpair δEpair Figure 5: (a) Total energy Etot curve (together with its macroscopic liquid-drop energy Eld and microscopic shell correction and pairing correlation energies, namely, δEshell and δEpair) vs β2 deformation for the nucleus 256 106Sg150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For simplicity, other deformation degrees of freedom were closed during the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (b) Similar to (a) but for the total Routhian (Erou) curves and the corresponding rotational contribution δH at three selected frequencies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (c)Similar to (a) but for the total energy and the corresponding pairing correlation δE at three selected pairing-strength factor F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='95, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='05 (the adjusted pairing strength G = FG0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' lected frquencies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 MeV, aiming to see the effect of the Coriolis force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' One can see that, similar to the trend of the pairing correlation, the energy due to rotation will decrease the barrier because the energy difference e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', at the positions of the first barrier and the first minimum is a negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It should be noted that the selected rotational frequencies respectively corre- spond to the values before and after the first band-crossing frequency in such a normal-deformed superheavy nucleus, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Along the curve, the ground-state or yrast configuration for the nucleus may be rather different (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 6, the occupied single-particle levels below the Fermi surface will generally be rather different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 5(c), the total energy and its pairing correlation energy are illustrated with different pairing strengths by adjusting the factor F (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', in G = FG0, where G0 is the orginal pairing strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It can be noticed that the pairing correlation energy will decrease with increasing pairing strength G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Both at the barrier and the minimum, the effects seem to be very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' At the large deformation region, the pairing correlation tends to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 16 3 2 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 92 114 104 106 104 104 (a) 2f7/2 1i13/2 Single proton levels (MeV) β2 10 9 8 7 6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 126 164 148 150 150 150 140 (b) 2d5/2 2g7/2 1j15/2 1i11/2 2g9/2 Single neutron levels (MeV) β2 Figure 6: Calculated proton (a) and neutron (b) single-particle energies as functions of the quadrupole deformation β2 for 256 106Sg150, focusing on the domain near the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The levels with positive and negative parities are respectively denoted by red solid and blue dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Spherical single-particle orbitals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', at β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0) in the window of interest are labeled by the quantum numbers nlj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The microscopic structure of nuclei is primarily determined by the single-particle levels, es- pecially near the Fermi level [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Experimentally, one can detect and investigate single-particle states by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the inelastic electron scattering [like (e, e′p)], the direct stripping and pick-up re- actions [typically (p, d) and (d, p) reactions], β-decay rates, and so on [85, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Because the measured single-particle states may be not pure, a rigorous definition of these states is given by the Green’s function formalism (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [84]), showing that it is necessary to extract the spectro- scopic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Such a quantity will provide an illustration of how much a single-particle level can be considered as a pure state and whether or not the correlations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the short- and long-range ones) beyond the mean field appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Theoretically, the single-particle levels correspond to the eigenstates of the mean-field Hamiltonian (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the Woods-Saxon-type one in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' They are also the building blocks of the many-body wave functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', in self-consistent Hartree- Fock calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 6, the single-particle levels near the proton and neutron Fermi surfaces are respectively illustrated in (a) and (b) parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' A set of conserved quantum numbers (associated with a complete set of commuting observables) are usually used for labeling the corresponding single-particle levels and wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the spherical single-particle levels are de- noted by the spherical quantum numbers n, l and j (corresponding the principal quantum number, the orbital angular momentum, and the total angular momentum, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Similar to atomic spectroscopy, the notations s, p, d, f, g, h · · · (corresponding to l = 0, 1, 2, 3, 4, 5 · · · , re- spectively) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Due to the strong spin-orbit coupling, the single particle state with l will split into two states with j = l ± 1/2 (The degeneracy of each spherical single-particle level can be calculated by 2j + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the present work, one can see that the expected shell structure and shell closure can be well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' When deformed shape occurs, the 2j+1 degeneracy will be broken and the spherical single-particle level will split into j + 1/2 components (each one is 17 typically double degenerate due to Kramers degeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' These deformed single-particle levels are generally described by asymptotic Nilsson quantum numbers Ωπ[NnzΛ], where N is the total oscillator shell quantum number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' nz stands the number of oscillator quanta in the z direction (the direction of the symmetry axis);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Λ is the projection of angular momentum along the symmetry axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Σ is the projection of intrinsic spin along the symmetry axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ω is the projection of total angular momentum j (including orbital l and spin s) on the symmetry axis and Ω = Λ + Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that the Nilsson labels are not given owing to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Similar to magnetic field, in the rotational coordinate system, the Coriolis force resulted from the non-inertial reference frame can also break the time reversal symmetry and mix the Nilsson states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Then, the single-particle Routhi- ans can only be labeled by the conserved parity and signature (π, α) or (π, r) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [1] for the rigorous definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It should be pointed out here that we did not perform the virtual crossing removal [87] of single-particle levels with same symmetries in these plots but this will not affect the identification of the single-particle levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 6, one can see that the shell gaps appear at the energy-minimum positions with lower level-densities and the higher level-densities occur at the saddle positions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The deformed neutron shells at N = 152 and 162 are reproduced [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For a clear display about the level density near the minimum and saddle points, Figure 7 presents the proton and neutron single-particle levels at these corresponding deformation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that the Fermi levels (the green levels) at the four typical points A, B, C and D are shifted to zero for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The levels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 7(a) and (c) correspond to deformation conditions same to those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 5 and 6 where only the β2 deformation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the right two subfigures of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 7, at each β2 point, the “realistic” β4 value is taken into account (the equilibrium deformation is adopted after potential-energy minimization over β4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Relative to the left two ones, the levels are rearranged to an extent by the hexadecapole deformation degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' As excepted, the level density is lower (higher) near the minimum (saddle) point, indicating the occurrence of a largely negative (positive) shell correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Figure 8 illustrates the total Routhian surfaces projected on the β vs γ plane for 256 106Sg150 at several typical rotational frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' At each grid in the maps, the minimization of the total Routhian was performed over β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It needs to be stressed that the energy domains denoted by the color palettes are different in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 8(c) and (d) for a better display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Under rotation, the triaxial deformation parameter γ covers the range from −120◦ to 60◦ because the three sectors (−120◦, −60◦), (−60◦, 0◦) and (0◦, 60◦) will represent rotation about the long, medium and short axes, respectively (the nucleus with triaxial shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The nucleus with four γ values −120◦, −60◦, 0◦ and 60◦ has the axially symmetric shape but different rotational orientation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' For instance, the triaxial deformation parameter γ = −120◦ during rotation denotes that a prolate nucleus with a non-collective rotation (namely, rotating around its symmetry axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', the low-frequency part on the fission path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 8(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The 1D cranking is limited in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' From this figure, one can see the evolution properties of the triaxiality and rotation axis for 18 2 1 0 1 2 A B C D Single proton levels (MeV) (a) 2 1 0 1 2 A B C D Single proton levels (MeV) (b) 2 1 0 1 2 A B C D Single neutron levels (MeV) (c) 2 1 0 1 2 A B C D Single neutron levels (MeV) (d) Figure 7: (a) Calculated proton single-particle levels for 256 106Sg150 at the four typical β2 deformation points (A, the 1st minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' B, the 1st maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', C, the 2nd minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' and D, the 2nd maximum) along the energy curve , see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In this plot, only β2 deformation is considered for simplicity, corresponding to the blue energy curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (b) Similar to (a) but, in this plot, the energy is minimized over β4 for each β2 points, corresponding to the red energy curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (c) Similar to (a) but for neutron single-particle levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' (d) Similar to (b) but for neutron single-particle levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' both the equilibrium shape and states along the fission path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' To investigate the hexadecapole-deformation effect under rotation, the total Routhian surfaces projected on the (β2, β4) plane are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 9 for 256 106Sg150 at four selected rotational frequen- cies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Note that the color palletes are slightly adjusted, similar to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It can be seen that the hexadecapole deformation β4 can strongly de- crease the total Routhian along the fission path, especially at high rotational frequency and large quadrupole deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In other words, the fission pathway will be modified by the the hexade- capole deformation β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It should be pointed out that from this figure one can find that part of the fission pathway evolutes along the border (with β4=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30) of the calculation domain, indicating the nucleus may possess a larger β4 at this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Figure 10 illustrates the total Routhian curves in functions of β2 for 256 106Sg150 at the selected rotational frequencies mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The size and shape of the inner and outer barriers and their evolution with rotation can be evaluated con- 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 120 90 60 30 0 30 60 γ (deg) 10 5 0 5 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 120 90 60 30 0 30 60 γ (deg) 10 5 0 5 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 120 90 60 30 0 30 60 γ (deg) 15 10 5 0 5 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 120 90 60 30 0 30 60 γ (deg) 20 15 10 5 0 5 (d) Figure 8: Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 1(d) but for total Routhian projections of 256 106Sg150 at rotational frequencies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 (a), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1 (b), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 (c) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 (d) MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 10 5 0 5 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 10 5 0 5 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 15 10 5 0 5 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='30 β4 20 15 10 5 0 5 (d) Figure 9: Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2(h) but for total Routhian projections of 256 106Sg150 at rotational frequencies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 (a), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1 (b), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 (c) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 (d) MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 20 30 20 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 106 256Sg150 Etot (MeV) β2 ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0 ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1 ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 Figure 10: The calculated total Routhian curves against β2 for 256 106Sg150 at four selected frequencies ℏω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='3 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' At each β2 point, the minimization was performed over γ and β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' veniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In the previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=', in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' [6, 7, 33] , it was pointed out that the octupole correlation may further decrease the outer barrier in this mass region based on the PES calculation and fission fragment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The outer barrier for this nucleus may finally be very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It will be an open problem whether it will be able to play a certain role in blocking the fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Conclusions We evaluate the structure evolution along the fission pathway for 256Sg by using the multi- dimensional potential-energy(or Routhian)-surface calculations, focusing on the effects of triaxial and hexadecapole deformation and Coriolis force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Nuclear shape and microscopic single-particle structure are investigated and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The present results are compared with other theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' The properties of nuclear shape and fission barrier are analyzed by comparing with its neighboring even-even nuclei, showing a reasonable agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Based on the deformation energy or Routhian curves, the fission barriers are analyzed, focusing on their shapes, heights, and evolution with rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' It is found that the triaxial deformation γ decreases the potential energy on the land- scape near the saddles but the hexadecapole deformation β4 (especially the axial α40 component) modifies the least-energy fission path after the first minimum, especially in 256Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' In addition, in contrast to the inner barrier, the outer barriers seem to have an increasing trend from 260Sg to 256Sg which may be benefit for blocking the fission of 256Sg to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Next, it will be necessary to simultaneously consider the reflection asymmetry in a more reasonable deformation subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Natural Science Foundation of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 11975209, U2032211 and 12075287), the Physics Research and Development Program of 21 Zhengzhou University (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 32410017), and the Project of Youth Backbone Teachers of Col- leges and Universities of Henan Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 2017GGJS008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Some of the calculations were conducted at the National Supercomputing Center in Zhengzhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' Conflict of Interest The authors declare that they have no known competing financial interests or personal relation- ships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} +page_content=' 22 References [1] M J A de Voigt, J Dudek and Z Szyma´nski Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfewRy/content/2301.03210v1.pdf'} 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P. Huang1, +T. Hussain71, N H¨usken27,34, W. Imoehl27, M. Irshad69,56, J. Jackson27, S. Jaeger4, S. Janchiv31, E. Jang53, +J. H. Jeong53, J. H. Jeong10A, Q. Ji1, Q. P. Ji20, X. B. Ji1,61, X. L. Ji1,56, Y. Y. Ji48, Z. K. Jia69,56, P. C. Jiang45,g, +S. S. Jiang38, T. J. Jiang17, X. S. Jiang1,56,61, Y. Jiang61, J. B. Jiao48, Z. Jiao23, S. Jin41, Y. Jin64, M. Q. Jing1,61, +T. Johansson73, X. K.1, S. Kabana32, N. Kalantar-Nayestanaki62, X. L. Kang9, X. S. Kang39, R. Kappert62, +M. Kavatsyuk62, B. C. Ke79, A. Khoukaz66, R. Kiuchi1, R. Kliemt14, L. Koch36, O. B. Kolcu60A, B. Kopf4, +M. Kuessner4, A. Kupsc43,73, W. K¨uhn36, J. J. Lane65, J. S. Lange36, P. Larin19, A. Lavania26, L. Lavezzi72A,72C, +T. T. Lei69,k, Z. H. Lei69,56, H. Leithoff34, M. Lellmann34, T. Lenz34, C. Li42, C. Li46, C. H. Li38, Cheng Li69,56, +D. M. Li79, F. Li1,56, G. Li1, H. Li69,56, H. B. Li1,61, H. J. Li20, H. N. Li54,i, Hui Li42, J. R. Li59, J. S. Li57, +J. W. Li48, Ke Li1, L. J Li1,61, L. K. Li1, Lei Li3, M. H. 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Zu69,56 +(BESIII Collaboration) +1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China +2 Beihang University, Beijing 100191, People’s Republic of China +3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China +4 Bochum Ruhr-University, D-44780 Bochum, Germany +5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA +6 Central China Normal University, Wuhan 430079, People’s Republic of China +7 Central South University, Changsha 410083, People’s Republic of China +8 China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China +9 China University of Geosciences, Wuhan 430074, People’s Republic of China +10 Chung-Ang University, Seoul, 06974, Republic of Korea +11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan +12 Fudan University, Shanghai 200433, People’s Republic of China +13 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia +14 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany +15 Guangxi Normal University, Guilin 541004, People’s Republic of China +16 Guangxi University, Nanning 530004, People’s Republic of China +17 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China +18 Hebei University, Baoding 071002, People’s Republic of China +19 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany +20 Henan Normal University, Xinxiang 453007, People’s Republic of China +21 Henan University of Science and Technology, Luoyang 471003, People’s Republic of China +22 Henan University of Technology, Zhengzhou 450001, People’s Republic of China +23 Huangshan College, Huangshan 245000, People’s Republic of China +24 Hunan Normal University, Changsha 410081, People’s Republic of China +25 Hunan University, Changsha 410082, People’s Republic of China +26 Indian Institute of Technology Madras, Chennai 600036, India +27 Indiana University, Bloomington, Indiana 47405, USA +28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; + +3 +(B)INFN Sezione di Perugia, I-06100, Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy +29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, +Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy +30 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China +31 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330, Mongolia +32 Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile +33 Jilin University, Changchun 130012, People’s Republic of China +34 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany +35 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia +36 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany +37 Lanzhou University, Lanzhou 730000, People’s Republic of China +38 Liaoning Normal University, Dalian 116029, People’s Republic of China +39 Liaoning University, Shenyang 110036, People’s Republic of China +40 Nanjing Normal University, Nanjing 210023, People’s Republic of China +41 Nanjing University, Nanjing 210093, People’s Republic of China +42 Nankai University, Tianjin 300071, People’s Republic of China +43 National Centre for Nuclear Research, Warsaw 02-093, Poland +44 North China Electric Power University, Beijing 102206, People’s Republic of China +45 Peking University, Beijing 100871, People’s Republic of China +46 Qufu Normal University, Qufu 273165, People’s Republic of China +47 Shandong Normal University, Jinan 250014, People’s Republic of China +48 Shandong University, Jinan 250100, People’s Republic of China +49 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China +50 Shanxi Normal University, Linfen 041004, People’s Republic of China +51 Shanxi University, Taiyuan 030006, People’s Republic of China +52 Sichuan University, Chengdu 610064, People’s Republic of China +53 Soochow University, Suzhou 215006, People’s Republic of China +54 South China Normal University, Guangzhou 510006, People’s Republic of China +55 Southeast University, Nanjing 211100, People’s Republic of China +56 State Key Laboratory of Particle Detection and Electronics, +Beijing 100049, Hefei 230026, People’s Republic of China +57 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China +58 Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand +59 Tsinghua University, Beijing 100084, People’s Republic of China +60 Turkish Accelerator Center Particle Factory Group, (A)Istinye University, 34010, Istanbul, +Turkey; (B)Near East University, Nicosia, North Cyprus, 99138, Mersin 10, Turkey +61 University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China +62 University of Groningen, NL-9747 AA Groningen, The Netherlands +63 University of Hawaii, Honolulu, Hawaii 96822, USA +64 University of Jinan, Jinan 250022, People’s Republic of China +65 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom +66 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany +67 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom +68 University of Science and Technology Liaoning, Anshan 114051, People’s Republic of China +69 University of Science and Technology of China, Hefei 230026, People’s Republic of China +70 University of South China, Hengyang 421001, People’s Republic of China +71 University of the Punjab, Lahore-54590, Pakistan +72 University of Turin and INFN, (A)University of Turin, I-10125, Turin, Italy; (B)University +of Eastern Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy +73 Uppsala University, Box 516, SE-75120 Uppsala, Sweden +74 Wuhan University, Wuhan 430072, People’s Republic of China +75 Xinyang Normal University, Xinyang 464000, People’s Republic of China +76 Yantai University, Yantai 264005, People’s Republic of China + +4 +77 Yunnan University, Kunming 650500, People’s Republic of China +78 Zhejiang University, Hangzhou 310027, People’s Republic of China +79 Zhengzhou University, Zhengzhou 450001, People’s Republic of China +a Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia +b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia +c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia +d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany +e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry +of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute +of Nuclear and Particle Physics, Shanghai 200240, People’s Republic of China +f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute +of Modern Physics, Fudan University, Shanghai 200443, People’s Republic of China +g Also at State Key Laboratory of Nuclear Physics and Technology, +Peking University, Beijing 100871, People’s Republic of China +h Also at School of Physics and Electronics, Hunan University, Changsha 410082, China +i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of +Quantum Matter, South China Normal University, Guangzhou 510006, China +j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People’s Republic of China +k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People’s Republic of China +l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan +Using e+e− annihilation data corresponding to an integrated luminosity of 2.93 fb−1 taken at +a center-of-mass energy of 3.773 GeV with the BESIII detector, we report the first measurements +of the branching fractions of the inclusive decays D0 → π+π+π−X and D+ → π+π+π−X, where +pions from K0 +S decays have been excluded from the π+π+π− system and X denotes any possible +particle combination. +The branching fractions of D0(D+) → π+π+π−X are determined to be +B(D0 → π+π+π−X) = (17.60 ± 0.11 ± 0.22)% and B(D+ → π+π+π−X) = (15.25 ± 0.09 ± 0.18)%, +where the first uncertainties are statistical and the second systematic. +I. +INTRODUCTION +In recent years, tests of lepton flavor universality +(LFU) have become a very hot topic in heavy flavor +physics. +The world averages of the ratios R(D) = +B(B → Dτντ)/B(B → Dℓνℓ) and R(D∗) = B(B → +D∗τντ)/B(B → D∗ℓνℓ), with ℓ = e or µ, deviate from +the Standard Model (SM) predictions by more than +1.4σ and 2.8σ, respectively [1]. Additionally, the LHCb +experiment reported the ratio of branching fractions, +R(D∗−) = B(B0 → D∗−τ +ντ)/B(B0 → D∗−µ+νµ), +based on 3 fb−1 of pp data taken at 7 and 8 TeV (Run +I) [2, 3], which had the smallest statistical uncertainty +at the time and was consistent with the SM prediction +within 1.1σ. However, the LHCb measurement is limited +by the knowledge of the normalization channel B(B0 → +D∗−π+π+π−). +Future data taken at the Belle II and +LHCb experiments will help to further improve the +accuracy of the branching fractions and tests of LFU. +In these tests, the analyses adopt the decay chain +of B0 +→ D∗−τ +ντ with τ + +→ π+π+π−¯ντ, where +the leading and sub-leading background sources are +from D+ +s +→ π+π+π−X and D0(D+) → π+π+π−X +(where +π±s +from +K0 +S +decays +have +been +excluded +from π+π+π− and X denotes any possible particle +combination). +Unfortunately, information on inclusive +decays of charmed mesons into final states containing +π+π+π− is sparse. Measurements of the full and partial +decay branching fractions of the inclusive decays D+ +s → +π+π+π−X and D0(D+) → π+π+π−X offer important +inputs to precisely test LFU with semileptonic B decays. +Recently, the BESIII Collaboration reported the first +measurement of the branching fraction of the inclusive +decay D+ +s +→ π+π+π−X [4]. +The branching fraction +obtained is greater than the sum of the branching +fractions of the known exclusive D+ +s decays containing +π+π+π− by around 25%, thereby implying that many +exclusive +D+ +s +decays +containing +π+π+π− +are +still +unmeasured. +The sums of the branching fractions of +the known exclusive D0 and D+ decays containing +π+π+π− [5–8], as summarized in Appendix A, are +(16.05 ± 0.47)% and (14.74 ± 0.53)%, respectively. The +measurements of the branching fractions of the inclusive +decays D0(D+) can offer a check on the known exclusive +D0(D+) decays containing π+π+π−. +A measurable +difference between the branching fractions of inclusive +and exclusive decays would indicate that either some +exclusive decays are not measured or that some known +exclusive decays are overestimated. +In this paper, we report the first measurements of the +branching fractions of D0 → π+π+π−X and D+ → +π+π+π−X by analyzing 2.93 fb−1 of e+e− collision +data [9] taken at a center-of-mass energy of 3.773 GeV +with the BESIII detector. Throughout this paper, charge + +5 +conjugate decays are always implied. +II. +BESIII DETECTOR AND MONTE CARLO +SIMULATION +The BESIII detector is a magnetic spectrometer [10] +located +at +the +Beijing +Electron +Positron +Collider +(BEPCII) [11]. +The cylindrical core of the BESIII +detector consists of a helium-based multilayer drift cham- +ber (MDC), a plastic scintillator time-of-flight system +(TOF), and a CsI (Tl) electromagnetic calorimeter +(EMC), which are all enclosed in a superconducting +solenoidal magnet providing a 1.0 T magnetic field [12]. +The solenoid is supported by an octagonal flux-return +yoke with resistive plate counter muon identifier modules +interleaved with steel. +The solid angle coverage for +detecting charged particles is 93% over 4π. The charged- +particle momentum resolution at 1 GeV/c is 0.5%, +and the resolution of the specific ionization energy +loss (dE/dx) is 6% for the electrons from Bhabha +scattering. The EMC measures photon energies with a +resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) +region. The time resolution of the TOF barrel part is +68 ps, while that of the end cap part is 110 ps. More +details about the design and performance of the BESIII +detector are given in Ref. [10]. +Simulated samples produced with geant4-based [13] +Monte +Carlo +(MC) +software, +which +includes +the +geometric description of the BESIII detector and the +detector response, are used to determine the detection +efficiency and to estimate background contributions. The +simulations include the beam energy spread and initial +state radiation in the e+e− annihilations modeled with +the generator kkmc [14]. +The inclusive MC samples +consist of the production of D ¯D pairs with quantum +coherence (QC) for neutral D modes, the non-D ¯D decays +of the ψ(3770), the initial state radiation production +of the J/ψ and ψ(3686) states, and the continuum +processes. +The known decay modes are modeled with +evtgen [15] using the branching fractions taken from +the PDG [5], and the remaining unknown decays of the +charmonium states are modeled by lundcharm [16, 17]. +Final state radiation is incorporated using photos [18]. +III. +METHOD +As the peak of the ψ(3770) resonance lies just above +the D ¯D threshold, it decays predominately into D ¯D +meson pairs. We take this advantage by using a double- +tag (DT) method, which was first developed by the +MARKIII Collaboration [19, 20] to determine absolute +branching fractions. The single-tag (ST) ¯D0(D−) mesons +are selected by using the two hadronic decay modes ¯D0 → +K+π− and D− → K+π−π−, respectively, which have a +relatively large branching fraction and low background. +Events where D0(D+) decaying into signal particles +can be selected in the presence of ST ¯D0(D−) mesons +are called DT events. To compensate for the differences +of the 3π invariant mass M3π distributions between data +and MC simulation and to consider the signal migration +among different (M3π) intervals, we determine the partial +branching fractions of the D0(D+) → π+π+π−X decays +in bins of M3π at the production level. The number of +produced DT events and the numbers of observed DT +events are related in bins through a detector response +matrix that accounts for detector efficiency and detector +resolution, +N i +obs = +Nintervals +� +j=1 +ϵijN j +prod, +(1) +where N i +obs is the number of signal events observed in +the i-th M3π interval, N j +prod is the number of signal +events produced in the j-th M3π interval, and ϵij is the +efficiency matrix describing the detection efficiency and +migration effect across each M3π interval. The statistical +uncertainties of ϵij due to the limited size of the signal +MC simulation sample are considered as a source of +systematic uncertainties, as discussed in VI. +The number of the inclusive decays D0(D+) +→ +π+π+π−X produced in the i-th M3π interval is obtained +by solving Eq. (1) for N i +prod, which gives +N i +prod = +Nintervals +� +j=1 +(ϵ−1)ijN j +obs. +(2) +The statistical uncertainty of N i +prod is given by +(σstat(N i +prod))2 = +Nintervals +� +j=1 +(ϵ−1)2 +ij(σstat(N j +obs))2, +(3) +where σstat(N j +obs) is the statistical uncertainty of N j +obs. +The partial branching fraction of the i-th M3π interval +is determined by +dBsig = +N i +prod +NST/ϵtag +, +(4) +where NST/ϵtag is the efficiency corrected yield of the +ST ¯D0(D−) mesons. The partial branching fractions are +summed to obtain the total branching fraction Bsig. +Since the measurement of the branching fraction of +D0 → π+π+π−X is affected by quantum coherence (QC) +in the D0 ¯D0 system, the branching fraction of D0 → +π+π+π−X measured with the tag mode ¯D0 → K+π− +needs to be corrected by +dBcorr +sig = f corr +QC × dBsig, +(5) + +6 +where +f corr +QC = +1 +1 − Cf(2fCP + − 1), +(6) +Bsig is the branching fraction to be measured, and Cf +denotes the strong-phase factor calculated by +Cf = 2rR cos δ +1 + r2 . +(7) +In Eq. (7), r is the ratio between the doubly-Cabibbo- +suppressed and Cabibbo-favored amplitudes for D → +K±π∓, δ is the strong phase difference between the two +amplitudes and R = 1 is the coherence factor for D → +K±π∓ [21, 22]. Table 1 summarizes the parameters of r +and δ for D → K±π∓, which give Cf = (−11.3+0.4 +−0.9)% +for D → K±π∓. +Table 1. +Input parameters for the QC correction. +Parameter +Value +rKπ +0.0586±0.0002 [23] +δKπ +(194.7+8.4 +−17.0)◦ [23] +In Eq. (6), fCP + is the fraction of the CP-even +(+) component. +According to Refs. [24, 25], fCP + is +calculated by +fCP + = +N + +N + + N − +with +N ± = M ± +measured +S± +, +S± = S± +measured +1 − η±yD +. +Here, N ± is the ratio of DT and ST yields with the +CP even and odd tags CP∓ tags, M ± +measured denotes +the number of DT candidates for a signal channel versus +CP∓ tags, and S± +measured is the number of ST candidates +for CP± decay modes. Finally, η± = ±1 for CP± mode +and yD = (0.62±0.08)% is the mixing parameter of D0 ¯D0 +taken from the latest average by the PDG [5]. +IV. +THE ST ANALYSIS +The charged kaons and pions are selected and identified +with the same criteria as in Refs. [26, 27]. +For each +charged track, the polar angle (θ) is required to be within +the MDC acceptance |cos(θ)| < 0.93, where θ is defined +with respect to the z axis, which is the symmetry axis +of the MDC. The distance of the charged track’s closest +approach relative to the interaction point is required to +be within 10 cm along the z axis and within 1 cm in the +plane perpendicular to the z axis. Particle identification +(PID) for charged tracks combines the measurements of +dE/dx in the MDC and the flight time in the TOF +to form probabilities L(h)(h = K, π) for each hadron +(h) hypothesis. +Charged tracks are assigned as kaons +or pions if their probabilities satisfy one of the two +hypotheses, L(K) > L(π) or L(π) > L(K), respectively. +In the selection of ¯D0 → K+π− candidates, background +contributions from cosmic rays and Bhabha events are +rejected with the following requirements. First, the two +charged tracks must have a TOF time difference less than +5 ns and they must not be consistent with being a muon +pair or an electron–positron pair. Second, there must be +at least one EMC shower with an energy greater than 50 +MeV or at least one additional charged track detected in +the MDC. +To distinguish the ST ¯D mesons from combinatorial +background, we define the two kinematic variables of +energy difference ∆E and the beam-constrained mass +MBC as +∆E ≡ E ¯ +D − Ebeam, +(8) +and +MBC ≡ +� +E2 +beam/c4 − |⃗p ¯ +D|2/c2. +(9) +Here, Ebeam is the beam energy, and E ¯ +D and ⃗p ¯ +D are the +energy and momentum of the ¯D candidate in the rest +frame of the e+e− system. For each ST mode, if there +are multiple candidates in an event, the one with the +smallest |∆E| is kept. The ST ¯D0 and D− candidates are +required to satisfy |∆E| < 25 MeV and |∆E| < 20 MeV, +respectively, which corresponds to about ±3σ of the +fitted resolution. +To +determine +the +yields +of +ST +¯D0 +and +D− +mesons, +maximum likelihood fits are performed on +the corresponding MBC distributions of the accepted +ST candidates. +In the fits, +the signal shape of +¯D0 or D− is modeled by an MC-simulated shape +convolved with a double-Gaussian function, which is a +sum of two Gaussian functions with free parameters, +describing the resolution difference between data and +MC simulation due to two asymmetrical tails of signal. +The combinatorial background shape is described by an +ARGUS function [28]. +The resultant fits to the MBC +distributions are shown in Fig. 1. The yields of ST ¯D0 +and D−mesons are 548031 ± 775 and 812109 ± 1896, +respectively, where the uncertainties are statistical only. +The efficiencies of reconstructing the ST ¯D mesons are +estimated to be (67.70 ± 0.08)% and (51.58 ± 0.04)% for +neutral ¯D0 decay and charged D− decay, respectively, +by analyzing the inclusive MC sample with the same +procedure as that for data. + +7 +) +2 +c + (GeV/ +BC +M +1.84 +1.86 +1.88 +) +2 +c +Events / (0.5 MeV/ +2 +10 +3 +10 +4 +10 +5 +10 +) +2 +c + (GeV/ +BC +M +1.84 +1.86 +1.88 +2 +10 +3 +10 +4 +10 +5 +10 +Fig. 1. Fits to the MBC distributions of the ST candidates for +¯D0 → K+π− (left) and D− → K+π−π− (right), where the +points with error bars are data, the blue solid curves are the fit +results, and the red dashed curves are the fitted combinatorial +backgrounds. +V. +THE DT ANALYSIS +A. +Selection of D0(D+) → π+π+π−X +The candidates for D0(D+) → π+π+π−X are selected +in the presence of the ST ¯D mesons. We require that +there are at least three charged pions which have not +been used in the ST selection. If there are more than one +π− or two π+ mesons reconstructed on the signal side, +only the π+/π− candidates with the highest momentum +are kept. +To reject background components from D0(D+) → +π+K0 +S(→ π+π−)X decays (K0 +S BKG1), the invariant +mass of any π+π− combination from the three selected +pions +is +required +to +fulfill +|Mπ+π− − 0.4977| +> +0.030 +GeV/c2. +In addition, another K0 +S background +(K0 +S BKG2), where one of the π mesons comes from +the chosen three pions and another assumed π meson +comes from a remaining opposite charged track without +PID, is rejected. The K0 +S candidate is selected through +the following selection criteria: First, the π+π− pair is +constrained to originate from a common vertex. Second, +the invariant mass of the π+π− pair is in the range +of |Mπ+π− − 0.498| > 0.012 +GeV/c2. +Third, the +decay length of K0 +S candidates is greater than two +standard deviations of the vertex resolution away from +the interaction point. +To further suppress the remaining background contri- +butions of K0 +S BKG2 and D0 → π+π−K0 +S(→ π+π−) (K0 +S +BKG3), the recoil masses of the ¯D0π+π− combinations +are required to be |Mπ+π− −0.498| > 0.080 GeV/c2. For +background contributions from D0 → π+π−π0K0 +S (K0 +S +BKG3), a similar requirement is applied on ¯D0π+π−π0 +combinations if a good π0 is found. The π0 candidates +are reconstructed via the π0 +→ γγ decay and the +opening angle between the photon candidate and the +nearest charged track is required to be greater than +10◦. Any photon pair with an invariant mass between +(0.115, 0.150) GeV/c2 is regarded as a π0 candidate, and +a kinematic fit is imposed on the photon pair to constrain +its invariant mass to the known π0 mass [5]. +) +3 +10 +× +) ( +2c +Events / (17 MeV/ +) +2c + (GeV/ +π +3 +M +) +2c + (GeV/ +miss +M +0.5 +1.0 +Data + signal +-π ++ +π ++ +π +- +K +Other signal + BKG +K +Mis-ID + BKG +µ +Mis-ID +Mis-ID e BKG + BKG +0 +S +K +Wrong tag BKG +2 +4 +6 +1 +2 +0.5 +1.0 +1.5 +2.0 +Data + signal +-π ++ +π ++ +π +0 +S +K + signal +-π ++ +π ++ +π +0 +L +K +Other signal + BKG +K +Mis-ID + BKG +µ +Mis-ID +Mis-ID e BKG + BKG +0 +S +K +Wrong tag BKG +2 +4 +6 +8 +0.0 +0.5 +1.0 +Fig. 2. +Comparisons of the M3π (left) and Mmiss (right) +distributions of the DT candidates for D0 → π+π+π−X (top) +and D+ → π+π+π−X (bottom), where the points with error +bars are data and the color filled histograms are the inclusive +MC sample. To ensure a ¯D in the ST side, events must satisfy +the requirements mentioned in the text and an additional +requirement of |MBC − MD| < 0.005 GeV/c2, where MD is +the known D mass [5]. +) +3 +10 +× +) ( +c +Events / (12 MeV/ +)c + (GeV/ +1 ++ +π +p +)c + (GeV/ +2 ++ +π +p +)c + (GeV/ +- +π +p +0.5 +1.0 +1.5 +Data + signal +-π ++ +π ++ +π +- +K +Other signal + BKG +K +Mis-ID + BKG +µ +Mis-ID +Mis-ID e BKG + BKG +0 +S +K +Wrong tag BKG +1 +2 +0.5 +1.0 +1 +2 +0.0 +0.5 +1.0 +Data + signal +-π ++ +π ++ +π +0 +S +K + signal +-π ++ +π ++ +π +0 +L +K +Other signal + BKG +K +Mis-ID + BKG +µ +Mis-ID +Mis-ID e BKG + BKG +0 +S +K +Wrong tag BGK +1 +2 +0.0 +0.5 +1.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +Fig. 3. Comparisons of the momentum distributions of the +selected pions of the DT candidates for D0 → π+π+π−X +(top) and D+ → π+π+π−X (bottom), where the points with +error bars are data and the color filled histograms are the +inclusive MC sample. To ensure a ¯D in the ST side, events +must satisfy the requirements mentioned in the text and an +additional requirement of |MBC−MD| < 0.005 GeV/c2, where +MD is the known D mass [5], and π+ +1 and π+ +2 denote the higher +and lower momentum π+s, respectively. +Figure 2 shows the comparison of the distributions of +M3π and Mmiss of the selected charged pions for the +accepted DT candidates between data and the inclusive +MC sample. Throughout this paper, M3π is the invariant +mass of the selected π+π+π− combination and Mmiss is + +8 +the missing mass of the ¯Dπ+π+π− combination given by +M 2 +miss = (2Ebeam − E ¯ +D − E3π)2/c4 − |−⃗p ¯ +D − ⃗p3π|2 /c2, +(10) +where E3π and ⃗p3π are the total energy and momentum +of the selected π+π+π− combination of the signal side +in the e+e− center-of-mass frame. Small inconsistencies +between data and the inclusive MC sample around +(0.9, 1.2) GeV/c2 in the M3π distributions are mainly +due to imperfect simulations of multi-body hadronic +decays with low branching fractions. +For the D0 → +π+π+π−X and D+ → π+π+π−X decays, the largest +signal components are from D0 → K−π+π+π− and +D+ +→ +¯K0π+π+π− decays, +respectively, +and they +form peaks around the known +¯K mass in the Mmiss +distributions as expected. +For D+ → π+π+π−X, the +peak around the known D+ mass in the M3π distribution +is from D+ → π+π+π−; the peaks around zero and +0.135 GeV/c2 in the Mmiss distribution are from D+ → +π+π+π− and D+ → π+π+π−π0, respectively; the peak +around 0.51 GeV/c2 in the Mmiss distribution is mainly +from D+ → π+π+π−η. Comparisons of the distributions +of momenta of the selected three charged pions are in +good agreement, as shown in Fig. 3. +Background analysis based on the inclusive MC sample +shows that there are still some remaining background +components, even after imposing all the aforementioned +background rejection requirements. One is from events +with wrongly tagged ¯D decays (i.e. ST decays are not +¯D0 → K+π− or D− → K+π−π−) and the non-D ¯D +process, labeled as “wrong tag”. +Another background +component results from events with correctly tagged ¯D +decays, but incorporating particle mis-identifications of +K → π (Mis-ID K BKG), µ → π (Mis-ID µ BKG) and +e → π (Mis-ID e BKG) as well as the remaining K0 +S +background (K0 +S BKG). The background components are +also shown in Figs. 2 and 3. +B. +DT signal yields +To minimize a possible efficiency dependence of various +D decay modes and offer finer information for the LFU +tests in the semileptonic B decays, the partial branching +fractions of D0 → π+π+π−X and D+ → π+π+π−X are +measured in nine and ten M3π bins, respectively. +For +D0 → π+π+π−X and D+ → π+π+π−X, the lower +boundaries of the intervals are chosen as [0.40, 0.55, +0.70, 0.85, 1.00, 1.15, 1.30, 1.45, 1.60, 1.75] GeV/c2 +and [0.40, 0.55, 0.70, 0.85, 1.00, 1.15, 1.30, 1.45, 1.60, +1.75, 2.00] GeV/c2, respectively. For D+ → π+π+π−X, +the interval [1.75, 2.00] GeV/c2 is added to specifically +consider the hadronic decay D+ → π+π+π−. +The signal yields in each M3π bin are determined +by fits to the MBC distributions of the ST side when +the candidates for D0 +→ +π+π+π−X and D+ +→ +π+π+π−X are found. +Figures 4(a) and 4(b) show +the MBC distributions of the accepted candidates for +D0 → π+π+π−X and D+ → π+π+π−X. The fits to +these MBC distributions are similar to those of the ST +side. +Because the “wrong tag” background events do +not form peaking background in the MBC distribution +of the tag side, an ARGUS function is used to describe +the “wrong tag” background. In these fits, however, the +parameters of the Gaussian functions and the ARGUS +functions are fixed to the values from the MBC fits of +the ST side. In addition, since the Mis-ID K BKG, Mis- +ID µ BKG, Mis-ID e BKG, and K0 +S BKG components +can peak in the distribution of MBC, the contributions +of those backgrounds must be subtracted. +The yields +and shapes of the Mis-ID K BKG, Mis-ID µ BKG, Mis- +ID e BKG, and K0 +S BKG components are fixed based +on the fits to the MBC distributions of the background +events from the inclusive MC sample, after taking into +account the differences of the rates of mis-identifying +various particles as charged pions between data and MC +simulation. The fixed background yields and the signal +yields in various M3π intervals for D0 → π+π+π−X and +D+ → π+π+π−X are summarized in Tables 2 and 3, +respectively. +The efficiency matrix ϵij = N ij +reco/N j +prod is determined +based on signal MC events of all known exclusive D0(+) +decays that contain a π+π+π− combination, where N ij +reco +is the number of signal MC events generated in the j- +th M3π interval and reconstructed in the i-th interval. +The matrix elements of ϵij for D0 → π+π+π−X and +D+ → π+π+π−X are summarized in Tables 4 and 5, +respectively. +C. +Branching fractions +With the obtained yields of the inclusive decays +D0(D+) → π+π+π−X produced in the i-th M3π interval, +the partial decay branching fraction is determined by +Eq. (4). +The measurement of the branching fraction of D0 → +π+π+π−X is affected by the QC effect in neutral D +decays, as discussed in Sec. III. In this work, the QC +factors in the various M3π intervals are estimated by +using the CP-even (+) tag of D0 → K+K− and CP- +odd (−) tag of D0 → K0 +Sπ0 with a similar analysis +procedure as the one in the data analysis procedure +presented above. +The quantities used and the results +are summarized in Table 6. +The produced signal yields and the obtained partial +branching fractions of D0 → π+π+π−X and D+ → +π+π+π−X in different M3π intervals are presented in +Tables 7 and 8, respectively. +VI. +SYSTEMATIC UNCERTAINTIES +Benefiting from the DT method, the branching fraction +measurements are insensitive to the selection criteria of +the ST ¯D candidates. The systematic uncertainties in + +9 +) +2 +10 +× +) ( +2c +Events / (0.5 MeV/ +) +2c + (GeV/ +BC +M +) +2c + (GeV/ +BC +M +) +2c + (GeV/ +BC +M +0.0 +0.5 +1.0 +1.5 [0.40,0.55) +0 +2 +4 +6 +8 [0.55,0.70) +0 +5 +10 +1.84 +1.86 +1.88 +[0.70,0.85) +0 +5 +10 +15 +20 [0.85,1.00) +0 +10 +20 +30 [1.00,1.15) +0 +10 +20 +1.84 +1.86 +1.88 +[1.15,1.30) +0 +2 +4 +6 [1.30,1.45) +0.0 +0.5 +1.0 +1.5 [1.45,1.60) +0.0 +0.2 +0.4 +1.84 +1.86 +1.88 +[1.60,1.75] +(a) +) +2 +10 +× +) ( +2c +Events / (0.5 MeV/ +) +2c + (GeV/ +BC +M +) +2c + (GeV/ +BC +M +) +2c + (GeV/ +BC +M +0.0 +0.5 +1.0 +[0.40,0.55) +0 +2 +4 +6 [0.55,0.70) +0 +5 +10 +[0.70,0.85) +0 +10 +20 +[0.85,1.00) +0 +10 +20 +30 +[1.00,1.15) +0 +10 +20 +1.84 +1.86 +1.88 +[1.15,1.30) +0 +5 +10 +[1.30,1.45) +0 +2 +4 +[1.45,1.60) +0 +2 +4 +1.84 +1.86 +1.88 +[1.60,1.75) +0 +2 +4 +1.84 +1.86 +1.88 +[1.75,2.00] +(b) +Fig. 4. Fits to the MBC distributions of the tag side when the candidates for (a) D0 → π+π+π−X and (b) D+ → π+π+π−X +are found in various reconstructed M3π intervals (in unit of GeV/c2) indicated on each plot. The points with error bars are +data, the blue solid curves are the fit results, the pink dashed curves are the peaking background summing over Mis-ID K +BKG, Mis-ID µ BKG, Mis-ID e BKG, and K0 +S BKG, and the black dashed curves are the fitted combinatorial backgrounds. +Table 2. Signal yields (Nobs) of D0 → π+π+π−X observed from data in various reconstructed M3π intervals. The numbers +of background events are estimated by the inclusive MC sample, whose integrated luminosity is four times that of data. +The uncertainties of NMis−ID K BKG, NMis−ID µ BKG, NMis−ID e BKG, and NK0 +S BKG include statistical uncertainties and the +uncertainties of individual differences of the particle mis-identification rates between data and MC simulation. The uncertainties +of Nobs are statistical only. +M3π (GeV/c2) NMis−ID K BKG NMis−ID µ BKG NMis−ID e BKG +NK0 +S BKG +Nobs +[0.40, 0.55) +29.1 ± 6.9 +0.0 ± 0.0 +4.7 ± 0.9 +45.5 ± 3.5 +524.4 ± 24.8 +[0.55, 0.70) +78.1 ± 15.8 +3.1 ± 1.0 +26.7 ± 2.3 +221.0 ± 7.8 +2878.7 ± 57.3 +[0.70, 0.85) +81.5 ± 14.7 +21.2 ± 2.6 +32.8 ± 2.8 +428.1 ± 10.8 +4017.2 ± 69.6 +[0.85, 1.00) +85.9 ± 14.3 +73.8 ± 4.8 +43.2 ± 3.4 +721.4 ± 14.1 +7824.8 ± 96.0 +[1.00, 1.15) +58.8 ± 9.4 +127.6 ± 6.3 +35.3 ± 3.2 +766.6 ± 14.5 11221.3 ± 111.9 +[1.15, 1.30) +13.8 ± 2.9 +99.3 ± 5.4 +21.3 ± 2.5 +670.2 ± 13.5 +8540.3 ± 98.8 +[1.30, 1.45) +15.0 ± 2.9 +49.4 ± 3.8 +11.2 ± 1.9 +485.0 ± 11.5 +1966.5 ± 51.8 +[1.45, 1.60) +6.4 ± 1.6 +17.6 ± 2.2 +6.2 ± 1.4 +135.3 ± 6.1 +417.7 ± 25.3 +[1.60, 1.75] +2.8 ± 1.0 +0.6 ± 0.4 +0.0 ± 0.0 +21.1 ± 2.4 +145.9 ± 14.3 +the measurements of the branching fractions of D0 → +π+π+π−X and D+ → π+π+π−X are discussed below. +Both the ST and DT yields are determined from +the fits to the individual MBC distributions. +The +fits to the DT candidates are performed by using the +same fit strategy as the fits to the ST candidates, +with the parameters of the smeared Gaussian function +and the ARGUS background function derived from the +corresponding fits to the ST candidates. +In this case, +the fitted DT yields are correlated to those of the fitted +ST yields. Therefore, the systematic uncertainties in the +yields of the ST ¯D mesons are canceled in the branching +fraction measurements. +The tracking and PID efficiencies of π± are studied +with the DT hadronic D ¯D events. +The averaged +data/MC differences of π± tracking and PID efficiencies, +weighted by the corresponding momentum spectra of +signal MC events, are 0.62% and 0.17%, respectively. +After +correcting +the +MC +efficiencies +to +data +by +these averaged data/MC differences, +the systematic +uncertainties of tracking and PID efficiencies for the +three charged pions are assigned as 0.80% and 0.50% for + +10 +Table 3. Signal yields (Nobs) of D+ → π+π+π−X observed from data in various reconstructed M3π intervals. The numbers +of background events are estimated by the inclusive MC sample, whose integrated luminosity is four times that of data. +The uncertainties of NMis−ID K BKG, NMis−ID µ BKG, NMis−ID e BKG, and NK0 +S BKG include statistical uncertainties and the +uncertainties of individual differences of the particle mis-identification rates between data and MC simulation. The uncertainties +of Nobs are statistical only. +M3π (GeV/c2) NMis−ID K BKG NMis−ID µ BKG NMis−ID e BKG NK0 +S BKG +Nobs +[0.40, 0.55) +21.6 ± 4.5 +4.5 ± 0.9 +6.0 ± 0.9 +9.7 ± 1.2 +483.3 ± 24.5 +[0.55, 0.70) +126.1 ± 20.6 +52.4 ± 3.2 +29.6 ± 1.9 +76.2 ± 3.5 +2703.8 ± 58.4 +[0.70, 0.85) +254.3 ± 37.4 +119.6 ± 4.9 +42.4 ± 2.5 +127.7 ± 4.5 +4766.7 ± 77.6 +[0.85, 1.00) +506.6 ± 66.7 +244.4 ± 7.1 +57.4 ± 3.1 +151.6 ± 4.9 +9788.2 ± 109.6 +[1.00, 1.15) +535.4 ± 61.0 +346.7 ± 8.2 +75.2 ± 3.6 +131.3 ± 4.6 14979.8 ± 132.3 +[1.15, 1.30) +365.1 ± 35.8 +365.4 ± 8.2 +36.1 ± 2.6 +123.6 ± 4.5 11718.5 ± 117.0 +[1.30, 1.45) +292.4 ± 25.8 +296.1 ± 7.2 +16.0 ± 1.7 +107.3 ± 4.2 +4636.0 ± 76.8 +[1.45, 1.60) +320.4 ± 24.6 +195.2 ± 5.8 +6.6 ± 1.1 +91.0 ± 3.8 +1864.6 ± 52.8 +[1.60, 1.75) +557.3 ± 34.8 +84.4 ± 3.7 +3.0 ± 0.7 +47.5 ± 2.8 +1228.0 ± 46.0 +[1.75, 2.00] +157.4 ± 12.1 +15.5 ± 1.6 +0.6 ± 0.3 +57.0 ± 3.0 +1559.3 ± 44.1 +Table 4. +Efficiency matrix ϵij (in percent) for D0 → +π+π+π−X, where i denotes the reconstructed M3π interval +and j denotes the produced M3π interval. +The relative +statistical uncertainties of the diagonal efficiencies of the +matrix are no more than 0.7%. +ϵij +1 +2 +3 +4 +5 +6 +7 +8 +9 +1 27.74 0.79 0.08 0.03 0.00 0.01 0.03 0.00 0.00 +2 2.82 27.99 1.01 0.24 0.06 0.02 0.03 0.08 0.00 +3 0.70 0.95 25.70 0.90 0.16 0.05 0.12 0.34 0.27 +4 0.69 0.62 0.96 32.65 1.27 0.25 0.20 0.65 0.54 +5 0.44 0.45 0.33 1.25 42.40 2.07 0.78 0.98 0.54 +6 0.13 0.09 0.11 0.20 1.07 47.82 3.21 0.81 0.00 +7 0.00 0.05 0.03 0.06 0.14 0.90 40.78 1.74 0.54 +8 0.00 0.00 0.00 0.01 0.01 0.05 0.28 22.28 0.80 +9 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.34 20.57 +Table 5. +Efficiency matrix ϵij (in percent) for D+ → +π+π+π−X, where i denotes the reconstructed M3π interval +and j denotes the produced M3π interval. +The relative +statistical uncertainties of the diagonal efficiencies of the +matrix are no more than 0.5%. +ϵij +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 22.13 0.48 0.11 0.04 0.03 0.03 0.02 0.01 0.00 0.00 +2 2.18 24.02 0.99 0.25 0.18 0.13 0.11 0.08 0.01 0.00 +3 0.99 1.71 22.44 1.30 0.41 0.25 0.29 0.10 0.00 0.01 +4 0.90 1.24 1.90 30.31 2.14 0.83 0.66 0.13 0.04 0.01 +5 0.81 1.01 1.49 2.53 39.52 3.35 1.33 0.27 0.03 0.01 +6 0.47 0.71 1.03 1.65 3.39 47.30 3.87 0.35 0.04 0.00 +7 0.46 0.50 0.75 1.27 1.99 2.33 50.08 2.14 0.34 0.03 +8 0.15 0.22 0.23 0.36 0.38 0.18 0.76 54.16 2.29 0.19 +9 0.00 0.00 0.00 0.00 0.01 0.01 0.03 1.28 56.79 1.18 +10 0.02 0.00 0.00 0.00 0.00 0.00 0.01 0.05 0.71 55.00 +D0 → π+π+π−X and D+ → π+π+π−X, respectively. +The detection efficiencies of D0(+) +→ +π+π+π−X +are obtained from the signal MC sample including all +known decays with three charged pions. +The relevant +systematic uncertainties are estimated by varying the +input branching fractions of exclusive decays within +±1σ. The maximum changes of the detection efficiencies, +0.13% and 0.56%, are assigned as the corresponding +systematic uncertainties for D0 → π+π+π−X and D+ → +π+π+π−X, respectively. +The uncertainties due to the limited signal DT MC +samples are calculated by [29] +Csys +ij += ( +1 +NST +)2 � +αβ +N α +obsN β +obsCov(ϵ−1 +iα , ϵ−1 +jβ ), +(11) +where the covariances of the inverse efficiency matrix +elements are given by +Cov(ϵ−1 +αβ, ϵ−1 +ab ) = +� +ij +(ϵ−1 +αi ϵ−1 +ai )[σ(ϵij)]2(ϵ−1 +jβ ϵ−1 +jb ). +(12) +The corresponding systematic uncertainties are assigned +to be 0.31% and 0.23% for D0 → π+π+π−X and D+ → +π+π+π−X, respectively. +The efficiencies of mis-identifying e±, µ± and K± as +π± are studied with the e+e− → γe+e− events, the +e+e− → γµ+µ− events, and the DT hadron D ¯D events, +respectively. The averaged data/MC differences of the +efficiencies of mis-identifying e± (µ±) as π±, weighted by +the two-dimensional (momentum and cos θ) distributions +of signal MC events, are 12% and 5%, respectively. The +averaged data/MC differences of the efficiencies of mis- +identifying K± as π±, weighted by the corresponding +momentum spectra of signal MC events, are no more than +53%. After correcting these mis-identification efficiencies +to data by these averaged data/MC differences, the +associated systematic uncertainties are estimated by + +11 +Table 6. Quantities of S± +measured, M ± +measured, Cf, fCP + and f corr +QC for D0 → π+π+π−X in various reconstructed M3π intervals. +i +S− +measured +S+ +measured +Cf (%) +M − +measured +M + +measured +fCP + +f corr +QC +1 +70999±304 59238±442 −11.3+0.4 +−0.9 +171.8 ± 41.6 +161.9 ± 46.0 +0.46 ± 0.14 1.01 ± 0.03 +2 +1163.4 ± 90.3 +857.1 ± 84.6 +0.53 ± 0.05 0.99 ± 0.01 +3 +1729.7 ± 224.1 2124.3 ± 481.8 0.40 ± 0.08 1.02 ± 0.02 +4 +2697.6 ± 149.4 2620.2 ± 295.8 0.46 ± 0.04 1.01 ± 0.01 +5 +3241.0 ± 132.0 2798.1 ± 161.7 0.49 ± 0.03 1.00 ± 0.01 +6 +1776.2 ± 79.6 +2108.6 ± 89.8 +0.41 ± 0.02 1.02 ± 0.01 +7 +360.7 ± 52.2 +547.9 ± 72.6 +0.35 ± 0.10 1.04 ± 0.03 +8 +250.2 ± 50.5 +186.0 ± 45.6 +0.52 ± 0.18 0.99 ± 0.04 +9 +121.8 ± 33.8 +21.5 ± 8.1 +0.82 ± 0.60 0.93 ± 0.12 +Table 7. +The produced signal yields and the obtained +partial branching fractions of D0 → π+π+π−X in different +reconstructed M3π intervals. +dBCP +sig +is the partial decay +branching fraction corrected by the QC factor, where dBcorr +sig = +f corr +QC × dBsig according to Eq. (5). +i +Nprod +dBsig +dBcorr +sig +(%) +1 +1541.3 ± 89.9 +0.28 ± 0.02 +0.28 ± 0.02 +2 +9349.1 ± 206.0 1.71 ± 0.04 +1.70 ± 0.04 +3 +14235.8 ± 271.8 2.60 ± 0.05 +2.66 ± 0.05 +4 +22130.5 ± 295.0 4.04 ± 0.05 +4.08 ± 0.05 +5 +24638.2 ± 264.9 4.50 ± 0.05 +4.51 ± 0.05 +6 +16850.4 ± 207.4 3.07 ± 0.04 +3.14 ± 0.04 +7 +4228.6 ± 127.5 0.77 ± 0.02 +0.80 ± 0.02 +8 +1730.9 ± 113.7 0.32 ± 0.02 +0.31 ± 0.02 +9 +676.1 ± 69.6 +0.12 ± 0.01 +0.11 ± 0.01 +Total 95381.0 ± 598.9 +− +17.60 ± 0.11 +Table 8. +The produced signal yields and the obtained +partial branching fractions of D+ → π+π+π−X in different +reconstructed M3π intervals. +i +Nprod +dBsig (%) +1 +1747.1 ± 111.1 +0.22 ± 0.01 +2 +9683.3 ± 245.1 +1.19 ± 0.03 +3 +17890.3 ± 349.6 +2.20 ± 0.04 +4 +27671.6 ± 366.3 +3.41 ± 0.05 +5 +33224.6 ± 340.2 +4.09 ± 0.04 +6 +20383.9 ± 251.5 +2.51 ± 0.03 +7 +5772.7 ± 155.4 +0.71 ± 0.02 +8 +2661.8 ± 97.8 +0.33 ± 0.01 +9 +2032.0 ± 81.1 +0.25 ± 0.01 +10 +2803.0 ± 80.2 +0.35 ± 0.01 +Total 123870.2 ± 744.7 15.25 ± 0.09 +varying +the +fixed +background +yields +within +their +individual uncertainties. Their effects on the measured +branching fractions are 0.13% and 0.18% for D0 → +π+π+π−X and D+ → π+π+π−X, respectively. +The yields of background with correct +¯D tag but +wrong D signal are estimated using the inclusive MC +sample. The relevant systematic uncertainty is evaluated +by varying the world average branching fractions of the +top five background components (which account for more +than 52% of all background contributions) of the decays +in D0(+) → π+π+π−X within ±1σ. +The changes of +the branching fractions are assigned as the corresponding +systematic uncertainties, which are 0.09% and 0.20% for +D0 → π+π+π−X and D+ → π+π+π−X, respectively. +The K0 +S veto is considered in three aspects. +The +first systematic uncertainty comes from the requirements +of rejecting (K0 +S BKG1). +The associated systematic +uncertainty is estimated by altering the nominal veto +window of |Mπ+π− − 0.498| > 0.030 +GeV/c2 by ±5 +MeV/c2, which corresponds to about 1σ of the fitted +K0 +S mass resolution. +The largest change of the re- +measured branching fraction is taken as the systematic +uncertainties, which are 0.23% and 0.06% for D0 → +π+π+π−X and D+ → π+π+π−X, respectively. +The +second systematic uncertainty comes from the require- +ments of rejecting (K0 +S BKG2), and it is estimated by +varying the background yields by ±1.5% in the MBC fit. +Here, the 1.5% corresponds to the data/MC difference +of the K0 +S reconstruction efficiencies between data and +MC simulation, estimated with the control sample of +J/ψ → K∗(892)∓K± and J/ψ → φK0 +SK±π∓ [30]. +The changes of the re-measured branching fractions are +assigned as the systematic uncertainties, which are 0.11% +and 0.17% for D0 → π+π+π−X and D+ → π+π+π−X, +respectively. +The third systematic uncertainty comes +from the requirements of rejecting (K0 +S BKG3), and it +is estimated by altering the nominal veto window of +|Mπ+π− − 0.498| > 0.080 GeV/c2 by ±5 MeV/c2. The +largest change of the re-measured branching fraction +is taken as the systematic uncertainty, which is 0.28% +for D0 → π+π+π−X. +Adding these three items in +quadrature yields the systematic uncertainties due to K0 +S +veto to be 0.38% and 0.18% for D0 → π+π+π−X and +D+ → π+π+π−X, respectively. +The systematic uncertainties due to M3π divisions +are estimated by increasing or decreasing the interval +size by 50%. The larger differences of the re-measured +branching fractions to the nominal results are assigned + +12 +as the systematic uncertainties, which are 0.34% and +0.20% for D0 → π+π+π−X and D+ → π+π+π−X, +respectively. +The systematic uncertainty due to the correction factor +of QC effect in D0 → π+π+π−X decays is determined by +the residual uncertainty of f corr +QC , which are summarized in +Table 6. After weighting by the corresponding numbers +of the inclusive decays D0 → π+π+π−X produced in +each i-th M3π interval, 0.42% is taken as the systematic +uncertainty. +Assuming all systematic uncertainties to be uncorre- +lated and adding them in quadrature, we obtain the +total systematic uncertainties in the measurements of +the branching fractions of D0 → π+π+π−X and D+ → +π+π+π−X to be 1.23% and 1.18%, respectively. Table 9 +summarizes the systematic uncertainties discussed above. +VII. +SUMMARY +By analyzing 2.93 fb−1 of e+e− collision data taken +at √s = 3.773 GeV, we have measured the branching +fractions of the inclusive decays D0 → π+π+π−X and +D+ → π+π+π−X for the first time. The results are +B(D0 → π+π+π−X) = (17.60 ± 0.11 ± 0.22)%, +and +B(D+ → π+π+π−X) = (15.25 ± 0.09 ± 0.18)%, +where the first uncertainties are statistical and the second +are systematic. +They are consistent with the sums of +the branching fractions of the known decay modes as +summarized in Appendix A within about ±3σ. +The +partial branching fraction in the interval [1.75, 2.00] +GeV/c2 is consistent with the branching fraction of the +exclusive hadronic decay of D+ → π+π+π−, which is +(0.327±0.018)% according to the PDG [5]. These results +indicate that there is little room for possible missing +D0(D+) decays containing π+π+π−. The measured total +and partial branching fractions of D0(D+) → π+π+π−X +are important inputs for LFU tests for the semileptonic +B decays. +VIII. +ACKNOWLEDGEMENT +The BESIII collaboration thanks the staff of BEPCII +and the IHEP computing center for their strong support. +This work is supported in part by National Key R&D +Program of China under Grants Nos. 2020YFA0406400, +2020YFA0406300; National Natural Science Foundation +of +China +(NSFC) +under +Grants +Nos. +11875170, +12035009, +11635010, +11735014, +11835012, +11935015, +11935016, 11935018, 11961141012, 12022510, 12025502, +12035013, 12061131003, 12192260, 12192261, 12192262, +12192263, 12192264, 12192265; the Chinese Academy of +Sciences (CAS) Large-Scale Scientific Facility Program; +the CAS Center for Excellence in Particle Physics +(CCEPP); Joint Large-Scale Scientific Facility Funds +of the NSFC and CAS under Grant No. +U1832207; +CAS Key Research Program of Frontier Sciences under +Grants Nos. +QYZDJ-SSW-SLH003, +QYZDJ-SSW- +SLH040; 100 Talents Program of CAS; The Institute +of Nuclear and Particle Physics (INPAC) and Shanghai +Key Laboratory for Particle Physics and Cosmology; +ERC under Grant No. +758462; +European Union’s +Horizon 2020 research and innovation programme under +Marie Sklodowska-Curie grant agreement under Grant +No. 894790; German Research Foundation DFG under +Grants Nos. +443159800, +455635585, +Collaborative +Research Center CRC 1044, FOR5327, GRK 2149; +Istituto Nazionale di Fisica Nucleare, Italy; Ministry of +Development of Turkey under Grant No. +DPT2006K- +120470; National Science and Technology fund; National +Science Research and Innovation Fund (NSRF) via +the Program Management Unit for Human Resources +& Institutional Development, Research and Innovation +under Grant No. B16F640076; Olle Engkvist Foundation +under Grant No. +200-0605; STFC (United Kingdom); +Suranaree University of Technology (SUT), Thailand +Science Research and Innovation (TSRI), and National +Science Research and Innovation Fund (NSRF) under +Grant No. 160355; The Royal Society, UK under Grants +Nos. +DH140054, DH160214; The Swedish Research +Council; U. S. Department of Energy under Grant No. +DE-FG02-05ER41374. +Appendix A: Branching fractions of the known +exclusive D0(D+) decays involving π+π+π− +Tables 1 and 2 show the intermediate and final +states that contribute to the inclusive decays D0 → +π+π+π−X and D+ → π+π+π−X as well as the known +branching fractions. +The known branching fractions +of D0 +→ +π+π+π−X and D+ +→ +π+π+π−X are +B(D0 → π+π+π−X) = (16.05 ± 0.47)% and B(D+ → +π+π+π−X) = (14.74 ± 0.53)%, respectively. +[1] Y. Amhis et al. (Heavy Flavor Averaging Group), +Eur. +Phys. +J. +C +81, +226 +(2021); +Updated +re- +sults available at +https://hflav-eos.web.cern.ch/hflav- +eos/semi/spring21/html/RDsDsstar/RDRDs.html +[2] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. +120, 171802 (2018). +[3] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 97, +072013 (2018). + +13 +Table 9. Relative systematic uncertainties (in percent) in the measurements of the branching fractions of D0 → π+π+π−X +and D+ → π+π+π−X. +Source +D0 → π+π+π−X D+ → π+π+π−X +π± tracking +0.80 +0.80 +π± PID +0.50 +0.50 +Efficiency estimate +0.13 +0.56 +MC statistics +0.31 +0.23 +Mis-identification efficiencies +0.13 +0.18 +Background estimate +0.09 +0.20 +K0 +S vetoes +0.38 +0.18 +QC correction factor +0.42 +- +Binning scheme +0.34 +0.20 +Total +1.23 +1.18 +Table 1. +The initial and final states contributing to the inclusive decay D0 → π+π+π−X, along with the known branching +fractions. The branching fractions of the hadronic D decays containing ¯K0 have been obtained by scaling the known branching +fractions of K0 +S by a factor of two. Any π+ or π− from K0 +S decays have not been included. The contributions of some decays +containing η, η′, ω, and φ have been excluded to avoid overlaps among various decays. +Initial state +BInitial(%) +Final state +BFinal(%) +Reference +Note +K−π+π+π− +8.220 ± 0.140 +K−π+π+π− +8.168 ± 0.145 +[5] +None ω +K−π+ω +3.392 ± 0.096 +K−π+π+π−X +3.088 ± 0.087 +[6] +Scaled by B(ω → π+π−X) ∼ 91.0% +K−π+π+π−π0 +4.300 ± 0.400 +K−π+π+π−π0 +0.845 ± 0.409 +[5] +None η, η′, and ω +π+π+π−π+ +0.755 ± 0.020 +π+π+π−π+ +0.755 ± 0.020 +[5] +... +K−π+η′ +0.643 ± 0.034 +K−π+π+π−X +0.525 ± 0.028 +[5] +Scaled by B(η′ → π+π−X) ∼ 81.7% +K−π+η +1.880 ± 0.050 +K−π+π+π−X +0.514 ± 0.013 +[5] +Scaled by B(η → π+π−X) ∼ 27.4% +K∗−ρ0π+ +0.320 ± 0.060 +K∗−π+π−π+ +0.320 ± 0.060 +[5] +Scaled by B(ρ0 → π+π−) ∼ 100% +π+π+π−π−π0π0 0.442 ± 0.029 π+π+π−π−π0π0 0.317 ± 0.030 +[7] +None η, η′ and ω +π+π+π−π−π0 +0.420 ± 0.050 +π+π+π−π−π0 +0.275 ± 0.053 +[5] +None η, η′, and ω +K0η′ +1.898 ± 0.045 K0π+π−π+π−X 0.226 ± 0.005 +[5] +Scaled by B(η′ → π+π−η) × B(η → π+π−X) ∼ 11.9% +¯ +K0ρ0π+π− +0.220 ± 0.099 +¯ +K0π+π+π−π− +0.220 ± 0.099 +[5] +Scaled by B(ρ0 → π+π−) ∼ 100% +K−π+π0η +0.449 ± 0.027 K−π+π0π+π−X 0.123 ± 0.007 +[5] +Scaled by B(η → π+π−X) ∼ 27.4% +π+π−ω +0.133 ± 0.020 +π+π−π+π−X +0.121 ± 0.018 +[5] +Scaled by B(ω → π+π−X) ∼ 91.0% +K0π+π−π+π− +< 0.120 +K0π+π−π+π− +0.12 +[5] +... +Others +0.543 ± 0.041 +π+π+π−X +0.434 ± 0.033 +[5, 7] +Dominated by π+π−ηX, π+π+π−η′X, ηη′ and ωη ∼ 80% +Sum +... +... +16.05 ± 0.47 +... +... +[4] M. +Ablikim +et +al. +(BESIII +Collaboration), +arXiv:2212.13072. +[5] P. 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The branching fractions of the hadronic D decays containing ¯K0 have been obtained by scaling the known branching +fractions of K0 +S by a factor of two. Any π+ or π− from K0 +S decays have not been included. The contributions of some decays +containing η, η′, ω, and φ have been excluded to avoid overlaps among various decays. +Initial state +BInitial(%) +Final state +BFinal(%) +Reference +Note +¯ +K0π+π+π− +6.200 ± 0.127 +¯ +K0π+π+π− +6.178 ± 0.130 +[5] +None ω +¯ +K0π+ω +1.414 ± 0.071 +¯ +K0π+π+π−X +1.287 ± 0.053 +[5] +Scaled by B(ω → π+π−X) ∼ 91.0% +¯ +K0π+π+π−π0 +3.056 ± 0.102 +¯ +K0π+π+π−π0 +1.198 ± 0.117 +[8] +None η, η′, and ω +π+π+π−π0 +1.160 ± 0.080 +π+π+π−π0 +0.954 ± 0.083 +[5] +None η, η′, ω and φ +¯ +K0π+η +2.620 ± 0.071 +¯ +K0π+π+π−X +0.718 ± 0.019 +[5] +Scaled by B(η → π+π−X) ∼ 27.4% +K−π+π+π+π− +0.570 ± 0.050 +K−π+π+π+π− +0.570 ± 0.050 +[5] +... +π+η′ +0.497 ± 0.019 +π+π+π−X +0.406 ± 0.016 +[5] +Scaled by B(η′ → π+π−X) ∼ 81.7% +π+π0φ +2.3 ± 1.0 +π+π0π+π−X +0.362 ± 0.157 +[5] +Scaled by B(φ → π+π−X) ∼ 15.6% +π+π0ω +0.39 ± 0.09 +π+π0π+π−X +0.355 ± 0.082 +[5] +Scaled by B(ω → π+π−X) ∼ 91.0% +π+π+π−π0π0 +1.07 ± 0.41 +π+π+π−π0π0 +0.322 ± 0.445 +[5] +None η, η′, ω and φ +¯ +K0π+η′ +0.380 ± 0.030 +¯ +K0π+π+π−X +0.311 ± 0.025 +[5] +Scaled by B(η′ → π+π−X) ∼ 81.7% +π+π+π− +0.327 ± 0.018 +π+π+π− +0.305 ± 0.018 +[5] +None ω +π+π+π−π0η +0.388 ± 0.032 +π+π+π−π0η +0.251 ± 0.040 +[7] +None η and η′ +π+π+π−π0π0π0 +0.347 ± 0.031 +π+π+π−π0π0π0 +0.250 ± 0.032 +[7] +None η +π+π+π+π−π− +0.166 ± 0.016 +π+π+π+π−π− +0.166 ± 0.016 +[5] +... +π+ηη +0.296 ± 0.026 +π+ηπ+π−X +0.162 ± 0.014 +[5] +Scaled by B(η → π+π−X) × 2 ∼ 54.8% +π+π+π+π−π−π0 0.238 ± 0.022 π+π+π+π−π−π0 0.160 ± 0.023 +[7] +None η +π+π0η′ +0.16 ± 0.05 +π+π0π+π−X +0.131 ± 0.041 +[5] +Scaled by B(η′ → π+π−X) ∼ 81.7% +π+π+π−η +0.341 ± 0.021 +π+π+π−η +0.125 ± 0.023 +[5] +None η′ +π+η +0.377 ± 0.009 +π+π+π−X +0.103 ± 0.003 +[5] +Scaled by B(η → π+π−X) ∼ 27.4% +Others +0.532 ± 0.034 +π+π+π−X +0.426 ± 0.027 +[5, 7] +Dominated by π+η( ¯ +K0π0, π0, π0π0, π0π0π0) and π+φ ∼ 80% +Sum +... +... +14.74 ± 0.53 +... +... +[26] M. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hong12,f, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hou1,61, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hou61, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huang69,56, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huang57, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huang30,61, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huang48, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huang1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hussain71, N H¨usken27,34, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Imoehl27, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Irshad69,56, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jeong10A, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ji1, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ji20, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ji1,61, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ji1,56, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ji48, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jia69,56, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiang45,g, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiang38, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiang17, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiang1,56,61, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiang61, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiao48, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jiao23, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jin41, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jin64, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Jing1,61, T.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li1,61, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li20, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li54,i, Hui Li42, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li59, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li57, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li48, Ke Li1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J Li1,61, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li1, Lei Li3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li42, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li37,j,k, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li12, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' 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Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li16, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Li57, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liang41, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liang33, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liang1,61, 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L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liao48, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Libby26, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Limphirat58, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lin30,61, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lin1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu74, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu33, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu19,69, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu51, Fang Liu1, Feng Liu6, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu54,i, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu37,j,k, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu70, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu1,61, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu39, Ke Liu22, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu69,56, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu42, Lu Liu42, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu12,f, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu1, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu61, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu69,56, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu12,f, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu42, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu69,56, X.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Liu48, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lou1,56,61, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu57, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu23, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu1,56, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu7, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu1,56, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lu1,61, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Luo40, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Luo78, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Luo12,f, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Luo1,56, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lyu61, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lyu42, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma39, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1,61, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma48, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1,61, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1,61, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma61, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma1,56, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ma45,g, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Maas19, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Maggiora72A,72C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Maldaner4, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Malde67, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Mangoni28B, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Mao45,g, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Mao1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Marcello72A,72C, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Meng64, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Messchendorp14,62, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Mezzadri29A, H.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nisar11,l, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Niu 48, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Olsen61, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ouyang1,56,61, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Pacetti28B,28C, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Pogodin35, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Prasad69,56, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Prasad32, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Qi1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Qi69,56, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rodin62, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rolo72C, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rong1,61, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rosner19, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ruan42, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shan69,56, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shangguan53, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shao1,61, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shao69,56, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shen12,f, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shen1,61, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shen61, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shen1,61, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi61, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi69,56, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi1, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi53, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi1,61, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shi1,56, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Song20, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Song57, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Song33,1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Song45,g, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sosio72A,72C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Spataro72A,72C, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Stieler34, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Su61, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun74, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun61, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun20, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun59, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun74, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun1,61, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun1,61, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun33, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun9, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun69,56, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun1, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sun48, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tan69,56, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tang52, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tang1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tang57, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tang74, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y Tao70, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tao25,h, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tat67, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Teng69,56, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tian74, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Uman60B, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wang69,56, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wang1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wang61, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Xu64, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Xu53, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Xu76, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Xu41, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yan12,f, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yan12,f, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yan69,56, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yan79, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q Yan1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yang49,e, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yang33, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yang1, Tao Yang1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang75, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang40, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang1,56,61, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' X.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang57, Lei Zhang41, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang1, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang38,79, Shuihan Zhang1,61, Shulei Zhang25,h, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D.' metadata={'source': 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Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang33, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang42, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhang74, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhao1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhao38, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhao1,61, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhao1,56, Lei Zhao69,56, Ling Zhao1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 2 Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 3 Beijing Institute of Petrochemical Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Beijing 102617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 4 Bochum Ruhr-University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D-44780 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Republic of China 10 Chung-Ang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 06974,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Republic of Korea 11 COMSATS University Islamabad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lahore Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Defence Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Off Raiwind Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 54000 Lahore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Pakistan 12 Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 13 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Budker Institute of Nuclear Physics SB RAS (BINP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Novosibirsk 630090,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Russia 14 GSI Helmholtzcentre for Heavy Ion Research GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D-64291 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Germany 15 Guangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Guilin 541004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 16 Guangxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nanning 530004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 17 Hangzhou Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hangzhou 310036,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 18 Hebei University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Baoding 071002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 19 Helmholtz 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 22 Henan University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhengzhou 450001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 23 Huangshan College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Huangshan 245000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 24 Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Changsha 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 25 Hunan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Changsha 410082,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 26 Indian Institute of Technology Madras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Chennai 600036,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' India 27 Indiana University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Indiana 47405,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' USA 28 INFN Laboratori Nazionali di Frascati ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (A)INFN Laboratori Nazionali di Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' I-00044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Italy;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Johann-Joachim-Becher-Weg 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D-55099 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Germany 35 Joint Institute for Nuclear Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 141980 Dubna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Moscow region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Russia 36 Justus-Liebig-Universitaet Giessen,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 38 Liaoning Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Dalian 116029,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 39 Liaoning University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shenyang 110036,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 40 Nanjing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nanjing 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 41 Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 42 Nankai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Tianjin 300071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 43 National Centre for Nuclear Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 50 Shanxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Linfen 041004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 51 Shanxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Taiyuan 030006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 52 Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Chengdu 610064,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 53 Soochow University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Suzhou 215006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 54 South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 55 Southeast University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nanjing 211100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 56 State Key Laboratory of Particle Detection and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 57 Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Guangzhou 510275,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 58 Suranaree University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' University Avenue 111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Nakhon Ratchasima 30000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Thailand 59 Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 62 University of Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' NL-9747 AA Groningen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The Netherlands 63 University of Hawaii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hawaii 96822,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' USA 64 University of Jinan,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wilhelm-Klemm-Strasse 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 48149 Muenster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Germany 67 University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Keble Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Oxford OX13RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' United Kingdom 68 University of Science and Technology Liaoning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Anshan 114051,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 69 University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 70 University of South China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hengyang 421001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 71 University of the Punjab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lahore-54590,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Pakistan 72 University of Turin and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (A)University of Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' I-10125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (C)INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' I-10125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Italy 73 Uppsala University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Box 516,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' SE-75120 Uppsala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Sweden 74 Wuhan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wuhan 430072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 75 Xinyang Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Xinyang 464000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 76 Yantai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Yantai 264005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 4 77 Yunnan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Kunming 650500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 78 Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hangzhou 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China 79 Zhengzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Zhengzhou 450001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China a Also at the Moscow Institute of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Moscow 141700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Russia b Also at the Novosibirsk State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Novosibirsk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 630090,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Russia c Also at the NRC ”Kurchatov Institute”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' PNPI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 188300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Gatchina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Russia d Also at Goethe University Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 60323 Frankfurt am Main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Germany e Also at Key Laboratory for Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Astrophysics and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ministry of Education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shanghai Key Laboratory for Particle Physics and Cosmology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Institute of Nuclear and Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Shanghai 200443,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China g Also at State Key Laboratory of Nuclear Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China h Also at School of Physics and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Hunan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Changsha 410082,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' China i Also at Guangdong Provincial Key Laboratory of Nuclear Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Institute of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' China j Also at Frontiers Science Center for Rare Isotopes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China k Also at Lanzhou Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' People’s Republic of China l Also at the Department of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' IBA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Karachi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Pakistan Using e+e− annihilation data corresponding to an integrated luminosity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='93 fb−1 taken at a center-of-mass energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='773 GeV with the BESIII detector, we report the first measurements of the branching fractions of the inclusive decays D0 → π+π+π−X and D+ → π+π+π−X, where pions from K0 S decays have been excluded from the π+π+π− system and X denotes any possible particle combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The branching fractions of D0(D+) → π+π+π−X are determined to be B(D0 → π+π+π−X) = (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='22)% and B(D+ → π+π+π−X) = (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18)%, where the first uncertainties are statistical and the second systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' INTRODUCTION In recent years, tests of lepton flavor universality (LFU) have become a very hot topic in heavy flavor physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The world averages of the ratios R(D) = B(B → Dτντ)/B(B → Dℓνℓ) and R(D∗) = B(B → D∗τντ)/B(B → D∗ℓνℓ), with ℓ = e or µ, deviate from the Standard Model (SM) predictions by more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4σ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8σ, respectively [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Additionally, the LHCb experiment reported the ratio of branching fractions, R(D∗−) = B(B0 → D∗−τ +ντ)/B(B0 → D∗−µ+νµ), based on 3 fb−1 of pp data taken at 7 and 8 TeV (Run I) [2, 3], which had the smallest statistical uncertainty at the time and was consistent with the SM prediction within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' However, the LHCb measurement is limited by the knowledge of the normalization channel B(B0 → D∗−π+π+π−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Future data taken at the Belle II and LHCb experiments will help to further improve the accuracy of the branching fractions and tests of LFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In these tests, the analyses adopt the decay chain of B0 → D∗−τ +ντ with τ + → π+π+π−¯ντ, where the leading and sub-leading background sources are from D+ s → π+π+π−X and D0(D+) → π+π+π−X (where π±s from K0 S decays have been excluded from π+π+π− and X denotes any possible particle combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Unfortunately, information on inclusive decays of charmed mesons into final states containing π+π+π− is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Measurements of the full and partial decay branching fractions of the inclusive decays D+ s → π+π+π−X and D0(D+) → π+π+π−X offer important inputs to precisely test LFU with semileptonic B decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Recently, the BESIII Collaboration reported the first measurement of the branching fraction of the inclusive decay D+ s → π+π+π−X [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The branching fraction obtained is greater than the sum of the branching fractions of the known exclusive D+ s decays containing π+π+π− by around 25%, thereby implying that many exclusive D+ s decays containing π+π+π− are still unmeasured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The sums of the branching fractions of the known exclusive D0 and D+ decays containing π+π+π− [5–8], as summarized in Appendix A, are (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='47)% and (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='53)%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The measurements of the branching fractions of the inclusive decays D0(D+) can offer a check on the known exclusive D0(D+) decays containing π+π+π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' A measurable difference between the branching fractions of inclusive and exclusive decays would indicate that either some exclusive decays are not measured or that some known exclusive decays are overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In this paper, we report the first measurements of the branching fractions of D0 → π+π+π−X and D+ → π+π+π−X by analyzing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='93 fb−1 of e+e− collision data [9] taken at a center-of-mass energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='773 GeV with the BESIII detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Throughout this paper, charge 5 conjugate decays are always implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' BESIII DETECTOR AND MONTE CARLO SIMULATION The BESIII detector is a magnetic spectrometer [10] located at the Beijing Electron Positron Collider (BEPCII) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The cylindrical core of the BESIII detector consists of a helium-based multilayer drift cham- ber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI (Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 T magnetic field [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The solid angle coverage for detecting charged particles is 93% over 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The charged- particle momentum resolution at 1 GeV/c is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5%, and the resolution of the specific ionization energy loss (dE/dx) is 6% for the electrons from Bhabha scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The EMC measures photon energies with a resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5% (5%) at 1 GeV in the barrel (end cap) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The time resolution of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' More details about the design and performance of the BESIII detector are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Simulated samples produced with geant4-based [13] Monte Carlo (MC) software, which includes the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate background contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The simulations include the beam energy spread and initial state radiation in the e+e− annihilations modeled with the generator kkmc [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The inclusive MC samples consist of the production of D ¯D pairs with quantum coherence (QC) for neutral D modes, the non-D ¯D decays of the ψ(3770), the initial state radiation production of the J/ψ and ψ(3686) states, and the continuum processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The known decay modes are modeled with evtgen [15] using the branching fractions taken from the PDG [5], and the remaining unknown decays of the charmonium states are modeled by lundcharm [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Final state radiation is incorporated using photos [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' METHOD As the peak of the ψ(3770) resonance lies just above the D ¯D threshold, it decays predominately into D ¯D meson pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' We take this advantage by using a double- tag (DT) method, which was first developed by the MARKIII Collaboration [19, 20] to determine absolute branching fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The single-tag (ST) ¯D0(D−) mesons are selected by using the two hadronic decay modes ¯D0 → K+π− and D− → K+π−π−, respectively, which have a relatively large branching fraction and low background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Events where D0(D+) decaying into signal particles can be selected in the presence of ST ¯D0(D−) mesons are called DT events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To compensate for the differences of the 3π invariant mass M3π distributions between data and MC simulation and to consider the signal migration among different (M3π) intervals, we determine the partial branching fractions of the D0(D+) → π+π+π−X decays in bins of M3π at the production level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The number of produced DT events and the numbers of observed DT events are related in bins through a detector response matrix that accounts for detector efficiency and detector resolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' N i obs = Nintervals � j=1 ϵijN j prod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (1) where N i obs is the number of signal events observed in the i-th M3π interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' N j prod is the number of signal events produced in the j-th M3π interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' and ϵij is the efficiency matrix describing the detection efficiency and migration effect across each M3π interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The statistical uncertainties of ϵij due to the limited size of the signal MC simulation sample are considered as a source of systematic uncertainties, as discussed in VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The number of the inclusive decays D0(D+) → π+π+π−X produced in the i-th M3π interval is obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (1) for N i prod, which gives N i prod = Nintervals � j=1 (ϵ−1)ijN j obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (2) The statistical uncertainty of N i prod is given by (σstat(N i prod))2 = Nintervals � j=1 (ϵ−1)2 ij(σstat(N j obs))2, (3) where σstat(N j obs) is the statistical uncertainty of N j obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The partial branching fraction of the i-th M3π interval is determined by dBsig = N i prod NST/ϵtag , (4) where NST/ϵtag is the efficiency corrected yield of the ST ¯D0(D−) mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The partial branching fractions are summed to obtain the total branching fraction Bsig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Since the measurement of the branching fraction of D0 → π+π+π−X is affected by quantum coherence (QC) in the D0 ¯D0 system, the branching fraction of D0 → π+π+π−X measured with the tag mode ¯D0 → K+π− needs to be corrected by dBcorr sig = f corr QC × dBsig, (5) 6 where f corr QC = 1 1 − Cf(2fCP + − 1), (6) Bsig is the branching fraction to be measured, and Cf denotes the strong-phase factor calculated by Cf = 2rR cos δ 1 + r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (7) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (7), r is the ratio between the doubly-Cabibbo- suppressed and Cabibbo-favored amplitudes for D → K±π∓, δ is the strong phase difference between the two amplitudes and R = 1 is the coherence factor for D → K±π∓ [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Table 1 summarizes the parameters of r and δ for D → K±π∓, which give Cf = (−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9)% for D → K±π∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Input parameters for the QC correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Parameter Value rKπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0586±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0002 [23] δKπ (194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0)◦ [23] In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (6), fCP + is the fraction of the CP-even (+) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' According to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [24, 25], fCP + is calculated by fCP + = N + N + + N − with N ± = M ± measured S± , S± = S± measured 1 − η±yD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Here, N ± is the ratio of DT and ST yields with the CP even and odd tags CP∓ tags, M ± measured denotes the number of DT candidates for a signal channel versus CP∓ tags, and S± measured is the number of ST candidates for CP± decay modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Finally, η± = ±1 for CP± mode and yD = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08)% is the mixing parameter of D0 ¯D0 taken from the latest average by the PDG [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' THE ST ANALYSIS The charged kaons and pions are selected and identified with the same criteria as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For each charged track, the polar angle (θ) is required to be within the MDC acceptance |cos(θ)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='93, where θ is defined with respect to the z axis, which is the symmetry axis of the MDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The distance of the charged track’s closest approach relative to the interaction point is required to be within 10 cm along the z axis and within 1 cm in the plane perpendicular to the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Particle identification (PID) for charged tracks combines the measurements of dE/dx in the MDC and the flight time in the TOF to form probabilities L(h)(h = K, π) for each hadron (h) hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Charged tracks are assigned as kaons or pions if their probabilities satisfy one of the two hypotheses, L(K) > L(π) or L(π) > L(K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In the selection of ¯D0 → K+π− candidates, background contributions from cosmic rays and Bhabha events are rejected with the following requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' First, the two charged tracks must have a TOF time difference less than 5 ns and they must not be consistent with being a muon pair or an electron–positron pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Second, there must be at least one EMC shower with an energy greater than 50 MeV or at least one additional charged track detected in the MDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To distinguish the ST ¯D mesons from combinatorial background, we define the two kinematic variables of energy difference ∆E and the beam-constrained mass MBC as ∆E ≡ E ¯ D − Ebeam, (8) and MBC ≡ � E2 beam/c4 − |⃗p ¯ D|2/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (9) Here, Ebeam is the beam energy, and E ¯ D and ⃗p ¯ D are the energy and momentum of the ¯D candidate in the rest frame of the e+e− system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For each ST mode, if there are multiple candidates in an event, the one with the smallest |∆E| is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The ST ¯D0 and D− candidates are required to satisfy |∆E| < 25 MeV and |∆E| < 20 MeV, respectively, which corresponds to about ±3σ of the fitted resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To determine the yields of ST ¯D0 and D− mesons, maximum likelihood fits are performed on the corresponding MBC distributions of the accepted ST candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In the fits, the signal shape of ¯D0 or D− is modeled by an MC-simulated shape convolved with a double-Gaussian function, which is a sum of two Gaussian functions with free parameters, describing the resolution difference between data and MC simulation due to two asymmetrical tails of signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The combinatorial background shape is described by an ARGUS function [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The resultant fits to the MBC distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The yields of ST ¯D0 and D−mesons are 548031 ± 775 and 812109 ± 1896, respectively, where the uncertainties are statistical only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The efficiencies of reconstructing the ST ¯D mesons are estimated to be (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08)% and (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04)% for neutral ¯D0 decay and charged D− decay, respectively, by analyzing the inclusive MC sample with the same procedure as that for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 7 ) 2 c (GeV/ BC M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 ) 2 c Events / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 MeV/ 2 10 3 10 4 10 5 10 ) 2 c (GeV/ BC M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 2 10 3 10 4 10 5 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Fits to the MBC distributions of the ST candidates for ¯D0 → K+π− (left) and D− → K+π−π− (right), where the points with error bars are data, the blue solid curves are the fit results, and the red dashed curves are the fitted combinatorial backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' THE DT ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Selection of D0(D+) → π+π+π−X The candidates for D0(D+) → π+π+π−X are selected in the presence of the ST ¯D mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' We require that there are at least three charged pions which have not been used in the ST selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' If there are more than one π− or two π+ mesons reconstructed on the signal side, only the π+/π− candidates with the highest momentum are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To reject background components from D0(D+) → π+K0 S(→ π+π−)X decays (K0 S BKG1), the invariant mass of any π+π− combination from the three selected pions is required to fulfill |Mπ+π− − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4977| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='030 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In addition, another K0 S background (K0 S BKG2), where one of the π mesons comes from the chosen three pions and another assumed π meson comes from a remaining opposite charged track without PID, is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The K0 S candidate is selected through the following selection criteria: First, the π+π− pair is constrained to originate from a common vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Second, the invariant mass of the π+π− pair is in the range of |Mπ+π− − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='498| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='012 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Third, the decay length of K0 S candidates is greater than two standard deviations of the vertex resolution away from the interaction point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To further suppress the remaining background contri- butions of K0 S BKG2 and D0 → π+π−K0 S(→ π+π−) (K0 S BKG3), the recoil masses of the ¯D0π+π− combinations are required to be |Mπ+π− −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='498| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='080 GeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For background contributions from D0 → π+π−π0K0 S (K0 S BKG3), a similar requirement is applied on ¯D0π+π−π0 combinations if a good π0 is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The π0 candidates are reconstructed via the π0 → γγ decay and the opening angle between the photon candidate and the nearest charged track is required to be greater than 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Any photon pair with an invariant mass between (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='115, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='150) GeV/c2 is regarded as a π0 candidate, and a kinematic fit is imposed on the photon pair to constrain its invariant mass to the known π0 mass [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ) 3 10 × ) ( 2c Events / (17 MeV/ ) 2c (GeV/ π 3 M ) 2c (GeV/ miss M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 Data signal π + π + π K Other signal BKG K Mis-ID BKG µ Mis-ID Mis-ID e BKG BKG 0 S K Wrong tag BKG 2 4 6 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 Data signal π + π + π 0 S K signal π + π + π 0 L K Other signal BKG K Mis-ID BKG µ Mis-ID Mis-ID e BKG BKG 0 S K Wrong tag BKG 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Comparisons of the M3π (left) and Mmiss (right) distributions of the DT candidates for D0 → π+π+π−X (top) and D+ → π+π+π−X (bottom), where the points with error bars are data and the color filled histograms are the inclusive MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To ensure a ¯D in the ST side, events must satisfy the requirements mentioned in the text and an additional requirement of |MBC − MD| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='005 GeV/c2, where MD is the known D mass [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ) 3 10 × ) ( c Events / (12 MeV/ )c (GeV/ 1 + π p )c (GeV/ 2 + π p )c (GeV/ π p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 Data signal π + π + π K Other signal BKG K Mis-ID BKG µ Mis-ID Mis-ID e BKG BKG 0 S K Wrong tag BKG 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 Data signal π + π + π 0 S K signal π + π + π 0 L K Other signal BKG K Mis-ID BKG µ Mis-ID Mis-ID e BKG BKG 0 S K Wrong tag BGK 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Comparisons of the momentum distributions of the selected pions of the DT candidates for D0 → π+π+π−X (top) and D+ → π+π+π−X (bottom), where the points with error bars are data and the color filled histograms are the inclusive MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' To ensure a ¯D in the ST side, events must satisfy the requirements mentioned in the text and an additional requirement of |MBC−MD| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='005 GeV/c2, where MD is the known D mass [5], and π+ 1 and π+ 2 denote the higher and lower momentum π+s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Figure 2 shows the comparison of the distributions of M3π and Mmiss of the selected charged pions for the accepted DT candidates between data and the inclusive MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Throughout this paper, M3π is the invariant mass of the selected π+π+π− combination and Mmiss is 8 the missing mass of the ¯Dπ+π+π− combination given by M 2 miss = (2Ebeam − E ¯ D − E3π)2/c4 − |−⃗p ¯ D − ⃗p3π|2 /c2, (10) where E3π and ⃗p3π are the total energy and momentum of the selected π+π+π− combination of the signal side in the e+e− center-of-mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Small inconsistencies between data and the inclusive MC sample around (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2) GeV/c2 in the M3π distributions are mainly due to imperfect simulations of multi-body hadronic decays with low branching fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For the D0 → π+π+π−X and D+ → π+π+π−X decays, the largest signal components are from D0 → K−π+π+π− and D+ → ¯K0π+π+π− decays, respectively, and they form peaks around the known ¯K mass in the Mmiss distributions as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For D+ → π+π+π−X, the peak around the known D+ mass in the M3π distribution is from D+ → π+π+π−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' the peaks around zero and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='135 GeV/c2 in the Mmiss distribution are from D+ → π+π+π− and D+ → π+π+π−π0, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' the peak around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='51 GeV/c2 in the Mmiss distribution is mainly from D+ → π+π+π−η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Comparisons of the distributions of momenta of the selected three charged pions are in good agreement, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Background analysis based on the inclusive MC sample shows that there are still some remaining background components, even after imposing all the aforementioned background rejection requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' One is from events with wrongly tagged ¯D decays (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ST decays are not ¯D0 → K+π− or D− → K+π−π−) and the non-D ¯D process, labeled as “wrong tag”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Another background component results from events with correctly tagged ¯D decays, but incorporating particle mis-identifications of K → π (Mis-ID K BKG), µ → π (Mis-ID µ BKG) and e → π (Mis-ID e BKG) as well as the remaining K0 S background (K0 S BKG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The background components are also shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' DT signal yields To minimize a possible efficiency dependence of various D decay modes and offer finer information for the LFU tests in the semileptonic B decays, the partial branching fractions of D0 → π+π+π−X and D+ → π+π+π−X are measured in nine and ten M3π bins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For D0 → π+π+π−X and D+ → π+π+π−X, the lower boundaries of the intervals are chosen as [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75] GeV/c2 and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00] GeV/c2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' For D+ → π+π+π−X, the interval [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00] GeV/c2 is added to specifically consider the hadronic decay D+ → π+π+π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The signal yields in each M3π bin are determined by fits to the MBC distributions of the ST side when the candidates for D0 → π+π+π−X and D+ → π+π+π−X are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Figures 4(a) and 4(b) show the MBC distributions of the accepted candidates for D0 → π+π+π−X and D+ → π+π+π−X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The fits to these MBC distributions are similar to those of the ST side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Because the “wrong tag” background events do not form peaking background in the MBC distribution of the tag side, an ARGUS function is used to describe the “wrong tag” background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In these fits, however, the parameters of the Gaussian functions and the ARGUS functions are fixed to the values from the MBC fits of the ST side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In addition, since the Mis-ID K BKG, Mis- ID µ BKG, Mis-ID e BKG, and K0 S BKG components can peak in the distribution of MBC, the contributions of those backgrounds must be subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The yields and shapes of the Mis-ID K BKG, Mis-ID µ BKG, Mis- ID e BKG, and K0 S BKG components are fixed based on the fits to the MBC distributions of the background events from the inclusive MC sample, after taking into account the differences of the rates of mis-identifying various particles as charged pions between data and MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The fixed background yields and the signal yields in various M3π intervals for D0 → π+π+π−X and D+ → π+π+π−X are summarized in Tables 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The efficiency matrix ϵij = N ij reco/N j prod is determined based on signal MC events of all known exclusive D0(+) decays that contain a π+π+π− combination, where N ij reco is the number of signal MC events generated in the j- th M3π interval and reconstructed in the i-th interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The matrix elements of ϵij for D0 → π+π+π−X and D+ → π+π+π−X are summarized in Tables 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Branching fractions With the obtained yields of the inclusive decays D0(D+) → π+π+π−X produced in the i-th M3π interval, the partial decay branching fraction is determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The measurement of the branching fraction of D0 → π+π+π−X is affected by the QC effect in neutral D decays, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In this work, the QC factors in the various M3π intervals are estimated by using the CP-even (+) tag of D0 → K+K− and CP- odd (−) tag of D0 → K0 Sπ0 with a similar analysis procedure as the one in the data analysis procedure presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The quantities used and the results are summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The produced signal yields and the obtained partial branching fractions of D0 → π+π+π−X and D+ → π+π+π−X in different M3π intervals are presented in Tables 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' SYSTEMATIC UNCERTAINTIES Benefiting from the DT method, the branching fraction measurements are insensitive to the selection criteria of the ST ¯D candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The systematic uncertainties in 9 ) 2 10 × ) ( 2c Events / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 MeV/ ) 2c (GeV/ BC M ) 2c (GeV/ BC M ) 2c (GeV/ BC M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55) 0 2 4 6 8 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70) 0 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85) 0 5 10 15 20 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85,1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00) 0 10 20 30 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15) 0 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30) 0 5 10 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45) 0 2 4 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60) 0 2 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75) 0 2 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='88 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00] (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Fits to the MBC distributions of the tag side when the candidates for (a) D0 → π+π+π−X and (b) D+ → π+π+π−X are found in various reconstructed M3π intervals (in unit of GeV/c2) indicated on each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The points with error bars are data, the blue solid curves are the fit results, the pink dashed curves are the peaking background summing over Mis-ID K BKG, Mis-ID µ BKG, Mis-ID e BKG, and K0 S BKG, and the black dashed curves are the fitted combinatorial backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Signal yields (Nobs) of D0 → π+π+π−X observed from data in various reconstructed M3π intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The numbers of background events are estimated by the inclusive MC sample, whose integrated luminosity is four times that of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The uncertainties of NMis−ID K BKG, NMis−ID µ BKG, NMis−ID e BKG, and NK0 S BKG include statistical uncertainties and the uncertainties of individual differences of the particle mis-identification rates between data and MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The uncertainties of Nobs are statistical only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M3π (GeV/c2) NMis−ID K BKG NMis−ID µ BKG NMis−ID e BKG NK0 S BKG Nobs [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 2878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 4017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 7824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 11221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 8540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 the measurements of the branching fractions of D0 → π+π+π−X and D+ → π+π+π−X are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Both the ST and DT yields are determined from the fits to the individual MBC distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The fits to the DT candidates are performed by using the same fit strategy as the fits to the ST candidates, with the parameters of the smeared Gaussian function and the ARGUS background function derived from the corresponding fits to the ST candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' In this case, the fitted DT yields are correlated to those of the fitted ST yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Therefore, the systematic uncertainties in the yields of the ST ¯D mesons are canceled in the branching fraction measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The tracking and PID efficiencies of π± are studied with the DT hadronic D ¯D events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The averaged data/MC differences of π± tracking and PID efficiencies, weighted by the corresponding momentum spectra of signal MC events, are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='62% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='17%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' After correcting the MC efficiencies to data by these averaged data/MC differences, the systematic uncertainties of tracking and PID efficiencies for the three charged pions are assigned as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='80% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='50% for 10 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Signal yields (Nobs) of D+ → π+π+π−X observed from data in various reconstructed M3π intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The numbers of background events are estimated by the inclusive MC sample, whose integrated luminosity is four times that of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The uncertainties of NMis−ID K BKG, NMis−ID µ BKG, NMis−ID e BKG, and NK0 S BKG include statistical uncertainties and the uncertainties of individual differences of the particle mis-identification rates between data and MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The uncertainties of Nobs are statistical only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M3π (GeV/c2) NMis−ID K BKG NMis−ID µ BKG NMis−ID e BKG NK0 S BKG Nobs [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70) 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 2703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85) 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 4766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='85, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00) 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 9788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15) 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 14979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30) 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 11718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='30, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45) 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 4636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60) 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 1864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75) 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00] 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Efficiency matrix ϵij (in percent) for D0 → π+π+π−X, where i denotes the reconstructed M3π interval and j denotes the produced M3π interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The relative statistical uncertainties of the diagonal efficiencies of the matrix are no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ϵij 1 2 3 4 5 6 7 8 9 1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='95 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='27 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='96 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='54 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='54 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='07 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='34 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='57 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Efficiency matrix ϵij (in percent) for D+ → π+π+π−X, where i denotes the reconstructed M3π interval and j denotes the produced M3π interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The relative statistical uncertainties of the diagonal efficiencies of the matrix are no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ϵij 1 2 3 4 5 6 7 8 9 10 1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='71 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The detection efficiencies of D0(+) → π+π+π−X are obtained from the signal MC sample including all known decays with three charged pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The relevant systematic uncertainties are estimated by varying the input branching fractions of exclusive decays within ±1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The maximum changes of the detection efficiencies, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='56%, are assigned as the corresponding systematic uncertainties for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The uncertainties due to the limited signal DT MC samples are calculated by [29] Csys ij = ( 1 NST )2 � αβ N α obsN β obsCov(ϵ−1 iα , ϵ−1 jβ ), (11) where the covariances of the inverse efficiency matrix elements are given by Cov(ϵ−1 αβ, ϵ−1 ab ) = � ij (ϵ−1 αi ϵ−1 ai )[σ(ϵij)]2(ϵ−1 jβ ϵ−1 jb ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (12) The corresponding systematic uncertainties are assigned to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='31% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='23% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The efficiencies of mis-identifying e±, µ± and K± as π± are studied with the e+e− → γe+e− events, the e+e− → γµ+µ− events, and the DT hadron D ¯D events, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The averaged data/MC differences of the efficiencies of mis-identifying e± (µ±) as π±, weighted by the two-dimensional (momentum and cos θ) distributions of signal MC events, are 12% and 5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The averaged data/MC differences of the efficiencies of mis- identifying K± as π±, weighted by the corresponding momentum spectra of signal MC events, are no more than 53%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' After correcting these mis-identification efficiencies to data by these averaged data/MC differences, the associated systematic uncertainties are estimated by 11 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Quantities of S± measured, M ± measured, Cf, fCP + and f corr QC for D0 → π+π+π−X in various reconstructed M3π intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' i S− measured S+ measured Cf (%) M − measured M + measured fCP + f corr QC 1 70999±304 59238±442 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 2 1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 3 1729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 2124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 4 2697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 2620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 5 3241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 2798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 6 1776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 7 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 8 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 9 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='12 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The produced signal yields and the obtained partial branching fractions of D0 → π+π+π−X in different reconstructed M3π intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' dBCP sig is the partial decay branching fraction corrected by the QC factor, where dBcorr sig = f corr QC × dBsig according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' i Nprod dBsig dBcorr sig (%) 1 1541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 2 9349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 3 14235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 4 22130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 ± 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 5 24638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 6 16850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 ± 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 7 4228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 8 1730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 9 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 Total 95381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The produced signal yields and the obtained partial branching fractions of D+ → π+π+π−X in different reconstructed M3π intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' i Nprod dBsig (%) 1 1747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 ± 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 2 9683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 3 17890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 4 27671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 5 33224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6 ± 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='04 6 20383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9 ± 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='03 7 5772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 ± 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='02 8 2661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 ± 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 9 2032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 10 2803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 ± 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='01 Total 123870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='2 ± 744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 varying the fixed background yields within their individual uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Their effects on the measured branching fractions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The yields of background with correct ¯D tag but wrong D signal are estimated using the inclusive MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The relevant systematic uncertainty is evaluated by varying the world average branching fractions of the top five background components (which account for more than 52% of all background contributions) of the decays in D0(+) → π+π+π−X within ±1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The changes of the branching fractions are assigned as the corresponding systematic uncertainties, which are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The K0 S veto is considered in three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The first systematic uncertainty comes from the requirements of rejecting (K0 S BKG1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The associated systematic uncertainty is estimated by altering the nominal veto window of |Mπ+π− − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='498| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='030 GeV/c2 by ±5 MeV/c2, which corresponds to about 1σ of the fitted K0 S mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The largest change of the re- measured branching fraction is taken as the systematic uncertainties, which are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='23% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='06% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The second systematic uncertainty comes from the require- ments of rejecting (K0 S BKG2), and it is estimated by varying the background yields by ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5% in the MBC fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Here, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='5% corresponds to the data/MC difference of the K0 S reconstruction efficiencies between data and MC simulation, estimated with the control sample of J/ψ → K∗(892)∓K± and J/ψ → φK0 SK±π∓ [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The changes of the re-measured branching fractions are assigned as the systematic uncertainties, which are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='17% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The third systematic uncertainty comes from the requirements of rejecting (K0 S BKG3), and it is estimated by altering the nominal veto window of |Mπ+π− − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='498| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='080 GeV/c2 by ±5 MeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The largest change of the re-measured branching fraction is taken as the systematic uncertainty, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='28% for D0 → π+π+π−X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Adding these three items in quadrature yields the systematic uncertainties due to K0 S veto to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='38% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The systematic uncertainties due to M3π divisions are estimated by increasing or decreasing the interval size by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The larger differences of the re-measured branching fractions to the nominal results are assigned 12 as the systematic uncertainties, which are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='34% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20% for D0 → π+π+π−X and D+ → π+π+π−X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The systematic uncertainty due to the correction factor of QC effect in D0 → π+π+π−X decays is determined by the residual uncertainty of f corr QC , which are summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' After weighting by the corresponding numbers of the inclusive decays D0 → π+π+π−X produced in each i-th M3π interval, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='42% is taken as the systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Assuming all systematic uncertainties to be uncorre- lated and adding them in quadrature, we obtain the total systematic uncertainties in the measurements of the branching fractions of D0 → π+π+π−X and D+ → π+π+π−X to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='23% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Table 9 summarizes the systematic uncertainties discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' SUMMARY By analyzing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='93 fb−1 of e+e− collision data taken at √s = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='773 GeV, we have measured the branching fractions of the inclusive decays D0 → π+π+π−X and D+ → π+π+π−X for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The results are B(D0 → π+π+π−X) = (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='22)%, and B(D+ → π+π+π−X) = (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18)%, where the first uncertainties are statistical and the second are systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' They are consistent with the sums of the branching fractions of the known decay modes as summarized in Appendix A within about ±3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The partial branching fraction in the interval [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='00] GeV/c2 is consistent with the branching fraction of the exclusive hadronic decay of D+ → π+π+π−, which is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='327±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='018)% according to the PDG [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' These results indicate that there is little room for possible missing D0(D+) decays containing π+π+π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The measured total and partial branching fractions of D0(D+) → π+π+π−X are important inputs for LFU tests for the semileptonic B decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ACKNOWLEDGEMENT The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' This work is supported in part by National Key R&D Program of China under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 2020YFA0406400, 2020YFA0406300;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' National Natural Science Foundation of China (NSFC) under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 11875170, 12035009, 11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' the CAS Center for Excellence in Particle Physics (CCEPP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' U1832207;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' CAS Key Research Program of Frontier Sciences under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' QYZDJ-SSW-SLH003, QYZDJ-SSW- SLH040;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 100 Talents Program of CAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' ERC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 758462;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 894790;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' German Research Foundation DFG under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 443159800, 455635585, Collaborative Research Center CRC 1044, FOR5327, GRK 2149;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Istituto Nazionale di Fisica Nucleare, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ministry of Development of Turkey under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' DPT2006K- 120470;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' National Science and Technology fund;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B16F640076;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Olle Engkvist Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 200-0605;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' STFC (United Kingdom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Suranaree University of Technology (SUT), Thailand Science Research and Innovation (TSRI), and National Science Research and Innovation Fund (NSRF) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 160355;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The Royal Society, UK under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' DH140054, DH160214;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The Swedish Research Council;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Department of Energy under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' DE-FG02-05ER41374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Appendix A: Branching fractions of the known exclusive D0(D+) decays involving π+π+π− Tables 1 and 2 show the intermediate and final states that contribute to the inclusive decays D0 → π+π+π−X and D+ → π+π+π−X as well as the known branching fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The known branching fractions of D0 → π+π+π−X and D+ → π+π+π−X are B(D0 → π+π+π−X) = (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='47)% and B(D+ → π+π+π−X) = (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='53)%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Amhis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (Heavy Flavor Averaging Group), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' C 81, 226 (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Updated re- sults available at https://hflav-eos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='ch/hflav- eos/semi/spring21/html/RDsDsstar/RDRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='html [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (LHCb Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 120, 171802 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Aaij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (LHCb Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D 97, 072013 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 13 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Relative systematic uncertainties (in percent) in the measurements of the branching fractions of D0 → π+π+π−X and D+ → π+π+π−X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Source D0 → π+π+π−X D+ → π+π+π−X π± tracking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='80 π± PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='50 Efficiency estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='56 MC statistics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='23 Mis-identification efficiencies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18 Background estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20 K0 S vetoes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18 QC correction factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='42 Binning scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='20 Total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='18 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The initial and final states contributing to the inclusive decay D0 → π+π+π−X, along with the known branching fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The branching fractions of the hadronic D decays containing ¯K0 have been obtained by scaling the known branching fractions of K0 S by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Any π+ or π− from K0 S decays have not been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The contributions of some decays containing η, η′, ω, and φ have been excluded to avoid overlaps among various decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Initial state BInitial(%) Final state BFinal(%) Reference Note K−π+π+π− 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='220 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='140 K−π+π+π− 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='168 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='145 [5] None ω K−π+ω 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='392 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='096 K−π+π+π−X 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='087 [6] Scaled by B(ω → π+π−X) ∼ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0% K−π+π+π−π0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='400 K−π+π+π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='845 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='409 [5] None η, η′, and ω π+π+π−π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='020 π+π+π−π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='020 [5] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' K−π+η′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='643 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='034 K−π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='028 [5] Scaled by B(η′ → π+π−X) ∼ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7% K−π+η 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='880 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='050 K−π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='013 [5] Scaled by B(η → π+π−X) ∼ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4% K∗−ρ0π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='320 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='060 K∗−π+π−π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='320 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='060 [5] Scaled by B(ρ0 → π+π−) ∼ 100% π+π+π−π−π0π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='442 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='029 π+π+π−π−π0π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='030 [7] None η, η′ and ω π+π+π−π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='420 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='050 π+π+π−π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='275 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='053 [5] None η, η′, and ω K0η′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='898 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='045 K0π+π−π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='226 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='005 [5] Scaled by B(η′ → π+π−η) × B(η → π+π−X) ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='9% ¯ K0ρ0π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='220 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='099 ¯ K0π+π+π−π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='220 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='099 [5] Scaled by B(ρ0 → π+π−) ∼ 100% K−π+π0η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='449 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='027 K−π+π0π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='123 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='007 [5] Scaled by B(η → π+π−X) ∼ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4% π+π−ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='133 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='020 π+π−π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='121 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='018 [5] Scaled by B(ω → π+π−X) ∼ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0% K0π+π−π+π− < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='120 K0π+π−π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='12 [5] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Others 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='543 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='041 π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='434 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='033 [5, 7] Dominated by π+π−ηX, π+π+π−η′X, ηη′ and ωη ∼ 80% Sum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='47 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ablikim et al.' metadata={'source': 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Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B 303, 163 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Baltrusaitis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (Mark III Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='Rev.' metadata={'source': 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Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 60, 89 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='Gershon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Libby, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Wilkinson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B 750, 338 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B 757, 520 (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 765, 402(E) (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [23] Heavy Flavor Averaging Group (HFLAV), (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='slac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='edu/xorg/hflav/charm/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' B 757, 520 (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 765, 402(E) (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ablikim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (BESIII Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D 100, 072006 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 14 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The initial and final states contributing to the inclusive decay D+ → π+π+π−X, along with the known branching fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The branching fractions of the hadronic D decays containing ¯K0 have been obtained by scaling the known branching fractions of K0 S by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Any π+ or π− from K0 S decays have not been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' The contributions of some decays containing η, η′, ω, and φ have been excluded to avoid overlaps among various decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Initial state BInitial(%) Final state BFinal(%) Reference Note ¯ K0π+π+π− 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='200 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='127 ¯ K0π+π+π− 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='178 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='130 [5] None ω ¯ K0π+ω 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='414 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='071 ¯ K0π+π+π−X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='053 [5] Scaled by B(ω → π+π−X) ∼ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0% ¯ K0π+π+π−π0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='056 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='102 ¯ K0π+π+π−π0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='198 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='117 [8] None η, η′, and ω π+π+π−π0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='080 π+π+π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='954 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='083 [5] None η, η′, ω and φ ¯ K0π+η 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='620 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='071 ¯ K0π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='718 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='019 [5] Scaled by B(η → π+π−X) ∼ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4% K−π+π+π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='570 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='050 K−π+π+π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='570 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='050 [5] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' π+η′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='019 π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='406 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='016 [5] Scaled by B(η′ → π+π−X) ∼ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7% π+π0φ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0 π+π0π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='362 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='157 [5] Scaled by B(φ → π+π−X) ∼ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='6% π+π0ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='09 π+π0π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='355 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='082 [5] Scaled by B(ω → π+π−X) ∼ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='0% π+π+π−π0π0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='41 π+π+π−π0π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='322 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='445 [5] None η, η′, ω and φ ¯ K0π+η′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='380 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='030 ¯ K0π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='311 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='025 [5] Scaled by B(η′ → π+π−X) ∼ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7% π+π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='327 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='018 π+π+π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='305 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='018 [5] None ω π+π+π−π0η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='388 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='032 π+π+π−π0η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='040 [7] None η and η′ π+π+π−π0π0π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='347 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='031 π+π+π−π0π0π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='250 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='032 [7] None η π+π+π+π−π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='166 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='016 π+π+π+π−π− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='166 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='016 [5] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' π+ηη 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='026 π+ηπ+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='162 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='014 [5] Scaled by B(η → π+π−X) × 2 ∼ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='8% π+π+π+π−π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='238 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='022 π+π+π+π−π−π0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='023 [7] None η π+π0η′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='05 π+π0π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='041 [5] Scaled by B(η′ → π+π−X) ∼ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='7% π+π+π−η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='341 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='021 π+π+π−η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='023 [5] None η′ π+η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='377 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='009 π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='103 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='003 [5] Scaled by B(η → π+π−X) ∼ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='4% Others 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='532 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='034 π+π+π−X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='426 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='027 [5, 7] Dominated by π+η( ¯ K0π0, π0, π0π0, π0π0π0) and π+φ ∼ 80% Sum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='53 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Ablikim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' (BESIII Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} +page_content=' 92, 112008 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE1T4oBgHgl3EQffATW/content/2301.03214v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..78f946fffda66501d1da025ba2db2fb4c7867462 --- /dev/null +++ b/htE0T4oBgHgl3EQfpgEn/content/tmp_files/2301.02539v1.pdf.txt @@ -0,0 +1,545 @@ +On the coalitional decomposition of parameters of interest +Marouane Il Idrissia,b,c,e, Nicolas Bousqueta,b,d, Fabrice Gamboac, Bertrand Ioossa,b,c, Jean-Michel +Loubesc +aEDF Lab Chatou, 6 Quai Watier, 78401 Chatou, France +bSINCLAIR AI Lab., Saclay, France +cInstitut de Math´ematiques de Toulouse, 31062 Toulouse, France +dSorbonne Universit´e, LPSM, 4 place Jussieu, Paris, France +eCorresponding Author - Email: marouane.il-idrissi@edf.fr +Abstract +Understanding the behavior of a black-box model with probabilistic inputs can be based on the decom- +position of a parameter of interest (e.g., its variance) into contributions attributed to each coalition +of inputs (i.e., subsets of inputs). In this paper, we produce conditions for obtaining unambiguous +and interpretable decompositions of very general parameters of interest. This allows to recover known +decompositions, holding under weaker assumptions than stated in the literature. +Keywords: +interpretability, sensitivity analysis, combinatorics, probability theory, statistics +1. Introduction and preliminaries +The decomposition of a parameter of interest, also known as a quantity of interest (QoI) in the +uncertainty quantification framework, with respect to (w.r.t.) coalitions of covariables is crucial in +both the field of sensitivity analysis of numerical models and in explainable artificial intelligence [12]. +These decompositions allow to distribute shares of QoI to the inputs of an input-output black-box +model. Depending on the QoI, they both allow to better understand the behavior of such models, and +to perform post-hoc interpretability [1]. +For instance, the well-known Hoeffding-Sobol’ decomposition is a particular instance of output +variance decomposition, which has been used for both settings [7, 8, 5]. It relies on a unique decompo- +sition of an input-output model in L2. Nevetheless, it requires independent covariables [11], but allows +to quantify the influence (in terms of percentages of output variance) of each inputs of a black-box +model, as well as interaction influence due to coalitions of inputs. +In this paper, the concept of “coalitional decomposition of QoI” is developped, generalizing the idea +of the Hoeffding-Sobol’ variance decomposition to other types of QoIs, leveraging results from the field +of combinatorics. In particular, Rota’s extension of the M¨obius inversion formula to partially ordered +sets [14]). Necessary conditions are presented in order to define coalitional decompositions of abstract +QoIs. It is shown, among other QoI decompositions proposed in the litterature, that the Hoeffding- +Sobol’ decomposition still holds without the need for independent inputs, but its interpretation as +interaction effects holds only when input independence is assumed. Furthermore, a quite general point +of view is adopted, allowing to define decompositions for a large variety of QoIs. +1.1. Notations and tools +1.1.1. Inputs, model and outputs +Let (Ω, F, P) be some probability space. Let, for i = 1, . . . , d, d ∈ N∗, (Ei, B(Ei)) be abstract +polish measurable space, i.e., Ei is a separable completely metrizable topological space, and B(Ei) +denotes its associated Borel σ-algebra. Let D = {1, . . . , d} and denote by P (D) its power-set (i.e., the +arXiv:2301.02539v1 [math.ST] 6 Jan 2023 + +set of all possible subsets of D, including ∅). For any A ⊆ D, denote the marginal measurable spaces +(EA, EA), where +EA =× +i∈A +Ei, +EA = +� +i∈A +B(Ei) = B +� +× +i∈A +Ei +� +, +Set also (E, E) := (ED, ED). Let X = (X1, . . . , Xd)⊤ be an E-valued random vector (i.e., a measurable +function X : Ω → E), referred to as the inputs. Let PX be the distribution of the inputs. Define the +marginal distributions, for each A ⊂ D, as: +PXA = +� +EA +dPX, +where XA = (Xi)i∈A is the coalition of inputs whose indices are in A (i.e., the subset XA of X). +Further, A denotes the complementary set of A in D (i.e., A = D \ A). Additionally, for every A ⊂ D, +the conditional distributions PXA|XA are assumed to be regular, and if not uniquely defined, they are +chosen to be regular (see [4], Chap. 4). +Let G : E → Z be an measurable function. Here Z denotes an abstract polish space. G(X) is the +Z-valued random variable, resulting from the composition of G with X. In the following, the function +G is referred to as a model, meanwhile G(X) is referred to as the output of the model. Denote P(E) +the set of all probability distributions on (E, E). M(E) denotes the set of Z-valued models, i.e., every +Z-valued, measurable functions. +Remark 1. In essence, the random inputs X, and the output G(X) are not restricted to be real-valued, +but can be defined on more complex measurable spaces (e.g., images, functions, stochastic processes). +A particular subset of M(E) is of interest in the present work whenever Z = R: L2(PX, R). It is the +set of measurable, R-valued functions which are square-integrable against PX. Recall that L2(PX; R) +is a Hilbert space with the inner product: +∀f, g ∈ L2(PX; R), +⟨f, g⟩L2 = +� +E +f(x)g(x)dPX(x), +and associated norm: +∀f ∈ L2(PX; R), +∥f∥2 +L2 = +� +E +f 2(x)dPX(x). +Denote, for any A ⊂ D, L2 (PXA; R) the Hilbert subspaces of L2(PX; R), of square integrable, EA- +measurable functions. In other words, any f ∈ L2(PXA, R) is a square-integrable function f : EA → R: +elements of L2(PXA, R) only take |A| := card(A) inputs. Whenever Z = Rk for a positive integer k, +one can also define the set L2 � +PX; Rk� +accordingly (see [9]). +1.1.2. Some elements of combinatorics and abstract algebra +A partially ordered set (poset) is defined as a pair (S, ≤) where S is a non-empty set, and ≤ is a +partial order binary relation on elements of S. A poset (S, ≤) is said to be locally finite if, for any +x, z ∈ S, the sets {y ∈ S : x ≤ y ≤ z} (also called segments of S) are finite. +A commutative ring with identity, is a triplet (A, +, ×) where A is a non-empty set, and where + and +× are addition and multiplication operators respectively, which are both associative and commutative +on A, × is distributive w.r.t. + on A, A contains both an additive and multiplicative identity, but +only an additive inverse. A commutative ring with identity that admits a multiplicative inverse is +generally called a field. In the following, abstract commutative rings with identity are denoted A, +and are assumed to be endowed with the usual addition and multiplication operator, unless stated +otherwise. For instance, R is a commutative ring with identity (it is in fact, a field). +Denote IA(S) the incidence algebra of a locally finite poset (S, ≤) over a commutative ring with +identity A, i.e., the set of functions f : S×S → A such that f(x, y) = 0 if x ̸≤ y (see [18], Definition 1.2.1 +2 + +p.10). (IA(S), +, ∗) forms an A-algebra with the usual pointwise addition + and the usual convolution +∗, i.e., for any f, g ∈ IA(S), and any x, z ∈ S such that the segment {y ∈ S : x ≤ y ≤ z} is non-empty, +(f ∗ g)(x, z) = +� +x≤y≤z +f(x, y)g(y, z). +The zeta function ζ ∈ IA(S) is convolutional identity of the incidence algebra, and is defined as, +∀x, y ∈ S: +ζ(x, y) = +� +1 +if x = y, +0 +otherwise. +The M¨obius function, denoted µ ∈ IA(S), in the case of locally finite posets S, is defined as the inverse +of the zeta function for the convolution operator defined on the incidence algebra of S, and can be +computed recursively, for any x, y ∈ S with x ≤ y, as [13] +µ(x, y) = +� +� +� +1 +if x = y +− +� +x≤z 0, the decomposition is thus fractional. +This result is analogue to the Hoeffding-Sobol’ functional analysis-of-variance (FANOVA) [11, 17]. +Traditionally, this decomposition is the result of a functional decomposition of the model G when it is +assumed to be in L2(PX; R), into orthogonal elements, requiring the inputs to be independent. How- +ever, as shown above, this decomposition holds even when the inputs are endowed with a dependence +structure. However, one can notice that input independence allow the decomposition to be fractional, +and hence, in-fine, lets the ratios (i.e., as in (3)) to be interpreted as a percentage of the output’s +variance attributed to each input coalition. +3.2. Covariance decomposition +Now, let G : E → R2 be a model with a bivariate output. Denote G = +� +G1 +G2 +� +and assume that +G ∈ L2(PX; R2). Let +φPX(G) = ⟨G1 − E [G1(X)] , G2 − E [G2(X)]⟩L2 += Cov (G1(X), G2(X)) , +in other words, the QoI is the covariance between the two random outputs of the model. +Proposition 2 (Covariance decomposition). Let, ∀A ∈ P (D): +fA(XA) = +�E [G1(X) | XA] +E [G2(X) | XA] +� +, +and, +ϕA = φPX(fA) = ⟨E [G1(X) | XA] − E [G1(X)] , E [G2(X) | XA] − E [G2(X)]⟩L2 += Cov (E [G1(X) | XA] , E [G2(X) | XA]) +Then, φPX(G) admits the following gradual M¨obius decomposition: +Cov (G1(X), G2(X)) = +� +A∈P(D) +ψA, +where, ∀A ∈ P (D), +ψA = +� +B⊆A +(−1)|A|−|B|Cov (E [G1(X) | XB] , E [G2(X) | XB]) . +7 + +Proof of Proposition 2. Notice that since G ∈ L2(PX; R2), ∀A ∈ P (D), the quantities +Cov (E [G1(X) | XA] , E [G2(X) | XA]) +are well defined, and that ϕD = Cov (G1(X), G2(X)). Applying Lemma 1 then leads to the gradual +decomposition. +Whenever G : E → Rk, for k ∈ N∗, the two previous results can be generalized using a covariance +matrix decomposition (see [9]). Let Dk be the set of (k × k) symmetric semi-definite (positive or +negative) matrices with non-zero entries on the diagonal, and where elements on the diagonal have the +same sign. Note that the triplet (Dk, +, ◦) where + denotes the usual element-wise matrix addition +and ◦ denotes the element-wise (Hadamard) multiplication, forms a commutative ring with identity +(if all the entries were non-zero, it would be a field since the Hadamard inverse would always be +well-defined). Let Σ be the covariance matrix of the output G(X) = (G1(X), . . . , Gk(X))⊤, defined +element-wise, for i, j = 1, . . . , k: +Σij = Cov (Gi(X), Gj(X)) . +Σ is necessarily semi-definite positive (since it is a covariance matrix) and is in Dk under the assumption +that each element of the output is not constant almost surely. It is then a Dk-valued QoI, and can be +decomposed as follows: +Proposition 3 (Covariance matrix decomposition). Let, ∀A ∈ P (D), the matrices ΣA ∈ Dk be defined +element-wise as: +ΣA +i,j = Cov (E [Gi(X) | XA] , E [Gj(X) | XA]) , +i, j = 1, . . . , k. +Then, Σ admits the following gradual M¨obius decomposition: +Σ = +� +A∈P(D) +ψA, +where, ∀A ∈ P (D), +ψA = +� +B⊆A +(−1)|A|−|B|ΣB. +Proof of Proposition 3. Notice that since G ∈ L2(PX; Rk), ΣA is well-defined ∀A ∈ P (D). Moreover, +notice that ΣD = Σ. Applying Lemma 1 then leads to the decomposition. +One can notice that, in that setting, decomposing Σ amounts to performing the variance decom- +position of Proposition 1 on the diagonal elements, and the covariance decomposition of Proposition 2 +on the other elements. +3.3. Mean maximum-mean discrepancy decomposition +Aside from moment-based quantities, more complicated QoIs can also be decomposed. Such quan- +tities can be based on kernel embedding of the model G. One can refer to [6] for additional details. +For the sake of completeness, some elements are recalled here. +Let G ∈ M(E), be a Z-valued model with inputs X ∼ PX ∈ P(E). Denote PY the distribution the +random output G(X). Moreover, for any A ∈ P (D), let the conditional distribution of G(X) given +XA be denoted by PY |XA. Let k : Z × Z → R be a kernel associated with a reproducing kernel Hilbert +space (RKHS) H [2]. Let: +µG(t) = +� +Z +k (z, t) dPY (z) = +� +E +k (G(z), t) dPX(z) = E [k (G(X), t)] +8 + +denotes the kernel mean embedding of G(X). Moreover, denote: +µG|X(t) = E [k (G(X), t) | X] = k (G(X), t) . +The maximum-mean discrepancy between PY and PY |X is given by: +MMD2(PY , PY |X) = +��µG − µG|X +��2 +H += E [µG(G(X))] + µG|X(G(X)) − 2E [k (G(X), G(X))] +One is interested in the QoI defined as the mean MMD, i.e., +SMMD := E +� +MMD2(PY , PY |X) +� += E [µG(G(X))] − E [k (G(X), G(X))] +Proposition 4. Let X ∼ PX be E-valued random inputs of a model G : E → Z. Let k : Z × Z → R +be the reproducing kernel of a RKHS H. Assume that k is such that, ∀A ∈ P (D): +SMMD +A +:= EXA +� +MMD2(PY , PY |XA) +� +< ∞. +Then, SMMD admits the following M¨obius decomposition: +SMMD = +� +A∈P(D) +ψA, +where, ∀A ∈ P (D), +ψA = +� +B⊆A +(−1)|A|−|B|SMMD +B +Proof. By assumption, SMMD +A +is well-defined ∀A ∈ P (D). Moreover, notice that SMMD +D += SMMD. +Applying Lemma 1 then leads to the decomposition. +This decomposition, analogous to the one presented in [6], not only holds when the inputs are +independent, but also when they are endowed with a dependence structure. +4. Discussion +Traditionally, in the field of global sensitivity analysis, QoI decompositions are defined using a +“model-centric” approach. It can be summarized as follows: find a suitable coalitional decomposition +of the model G in L2, such that φPX becomes an additive map when applied to G. For instance, if the +QoI is the variance of the output, orthogonality of the ψA is often desired (as defined in Definition 2). +The new viewpoint provided by this communication adopts an “input-centric” approach: first define +a suitable ϕA (as in (2)), such that it accurately represents the effect of XA, and then define a +suitable decomposition using the reverse implication of the M¨obius inversion formula. This approach +is analogous to the field of cooperative game theory [3], where ϕ represents the value function of a +cooperative game, and ψA are none other than its Harsanyi dividends [10]. The understanding and +possible combination of both approaches to find theoretically suitable candidates for ϕ is the subject +of ongoing research. +References +[1] A. Barredo Arrieta, N. D´ıaz-Rodr´ıguez, J. Del Ser, A. Bennetot, S. Tabik, A. Barbado, S. Gar- +cia, S. Gil-Lopez, D. Molina, R. Benjamins, R. Chatila, and F. Herrera. Explainable Artificial +Intelligence (XAI): Concepts, taxonomies, opportunities and challenges toward responsible AI. +Information Fusion, 58:82–115, June 2020. +9 + +[2] Alain Berlinet and Christine Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability +and Statistics. Springer US, Boston, MA, 2004. +[3] J. M. Bilbao. Cooperative Games on Combinatorial Structures, volume 26 of Theory and Decision +Library. Springer US, Boston, MA, 2000. +[4] L. Breiman. Probability. Society for Industrial and Applied Mathematics, 1992. +[5] C. B´enesse, F. Gamboa, J-M. Loubes, and T. Boissin. Fairness seen as global sensitivity analysis. +Machine Learning, 2022. +[6] S. Da Veiga. Kernel-based ANOVA decomposition and Shapley effects - Application to global +sensitivity analysis. working paper or preprint, 2021. +[7] S. Da Veiga, F. Gamboa, B. Iooss, and C. Prieur. Basics and Trends in Sensitivity Analysis. +Theory and Practice in R. SIAM, 2021. +[8] T. Fel, R. Cadene, M. Chalvidal, M. Cord, D. Vigouroux, and T. Serre. Look at the Variance! +Efficient Black-box Explanations with Sobol-based Sensitivity Analysis. In Advances in Neural +Information Processing Systems, volume 34, pages 26005–26014, 2021. +[9] F. Gamboa, A. Janon, T. Klein, and A. Lagnoux. Sensitivity indices for multivariate outputs. +Comptes Rendus Mathematique, 351(7):307–310, 2013. +[10] J. C. Harsanyi. A Simplified Bargaining Model for the n-Person Cooperative Game. International +Economic Review, 4(2):194–220, 1963. +[11] W. Hoeffding. A class of statistics with asymptotically normal distribution. Annals of Mathemat- +ical Statistics, 19(3):293–325, 1948. +[12] B. Iooss, R. Kenett, and P. Secchi. Different Views of Interpretability. In Interpretability for +Industry 4.0 : Statistical and Machine Learning Approaches, pages 1–20. Springer International +Publishing, Cham, 2022. +[13] J. Kock. From M¨obius inversion to renormalisation. Communications in Number Theory and +Physics, 14(1):171–198, 2020. +[14] J. P. S. Kung, G-C. Rota, and C. Hung Yan. Combinatorics: the Rota way. Cambridge University +Press, New York, 2012. OCLC: 1226672593. +[15] A.F. M¨obius. +¨Uber eine besondere art von umkehrung der reihen. Journal f¨ur die reine und +angewandte Mathematik, 9:105–123, 1832. +[16] G-C. Rota. On the foundations of combinatorial theory I. Theory of M¨obius Functions. Zeitschrift +f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2(4):340–368, 1964. +[17] I.M Sobol’. Global sensitivity indices for nonlinear mathematical models and their monte carlo +estimates. Mathematics and Computers in Simulation, 55(1):271–280, 2001. +[18] Eugene Spiegel and Christopher J. O’Donnell. Incidence algebras. Number 206 in Monographs +and textbooks in pure and applied mathematics. M. Dekker, New York, 1997. +10 + diff --git a/htE0T4oBgHgl3EQfpgEn/content/tmp_files/load_file.txt b/htE0T4oBgHgl3EQfpgEn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6166fd54e38bab221904f22997d572d88e039848 --- /dev/null +++ b/htE0T4oBgHgl3EQfpgEn/content/tmp_files/load_file.txt @@ -0,0 +1,416 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf,len=415 +page_content='On the coalitional decomposition of parameters of interest Marouane Il Idrissia,b,c,e, Nicolas Bousqueta,b,d, Fabrice Gamboac, Bertrand Ioossa,b,c, Jean-Michel Loubesc aEDF Lab Chatou, 6 Quai Watier, 78401 Chatou, France bSINCLAIR AI Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', Saclay, France cInstitut de Math´ematiques de Toulouse, 31062 Toulouse, France dSorbonne Universit´e, LPSM, 4 place Jussieu, Paris, France eCorresponding Author - Email: marouane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='il-idrissi@edf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='fr Abstract Understanding the behavior of a black-box model with probabilistic inputs can be based on the decom- position of a parameter of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', its variance) into contributions attributed to each coalition of inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', subsets of inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In this paper, we produce conditions for obtaining unambiguous and interpretable decompositions of very general parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' This allows to recover known decompositions, holding under weaker assumptions than stated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Keywords: interpretability, sensitivity analysis, combinatorics, probability theory, statistics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Introduction and preliminaries The decomposition of a parameter of interest, also known as a quantity of interest (QoI) in the uncertainty quantification framework, with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=') coalitions of covariables is crucial in both the field of sensitivity analysis of numerical models and in explainable artificial intelligence [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' These decompositions allow to distribute shares of QoI to the inputs of an input-output black-box model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Depending on the QoI, they both allow to better understand the behavior of such models, and to perform post-hoc interpretability [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' For instance, the well-known Hoeffding-Sobol’ decomposition is a particular instance of output variance decomposition, which has been used for both settings [7, 8, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' It relies on a unique decompo- sition of an input-output model in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Nevetheless, it requires independent covariables [11], but allows to quantify the influence (in terms of percentages of output variance) of each inputs of a black-box model, as well as interaction influence due to coalitions of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In this paper, the concept of “coalitional decomposition of QoI” is developped, generalizing the idea of the Hoeffding-Sobol’ variance decomposition to other types of QoIs, leveraging results from the field of combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In particular, Rota’s extension of the M¨obius inversion formula to partially ordered sets [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Necessary conditions are presented in order to define coalitional decompositions of abstract QoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' It is shown, among other QoI decompositions proposed in the litterature, that the Hoeffding- Sobol’ decomposition still holds without the need for independent inputs, but its interpretation as interaction effects holds only when input independence is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Furthermore, a quite general point of view is adopted, allowing to define decompositions for a large variety of QoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Notations and tools 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Inputs, model and outputs Let (Ω, F, P) be some probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Let, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' , d, d ∈ N∗, (Ei, B(Ei)) be abstract polish measurable space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', Ei is a separable completely metrizable topological space, and B(Ei) denotes its associated Borel σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Let D = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' , d} and denote by P (D) its power-set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='02539v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='ST] 6 Jan 2023 set of all possible subsets of D, including ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' For any A ⊆ D, denote the marginal measurable spaces (EA, EA), where EA =× i∈A Ei, EA = � i∈A B(Ei) = B � × i∈A Ei � , Set also (E, E) := (ED, ED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Let X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' , Xd)⊤ be an E-valued random vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', a measurable function X : Ω → E), referred to as the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Let PX be the distribution of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Define the marginal distributions, for each A ⊂ D, as: PXA = � EA dPX, where XA = (Xi)i∈A is the coalition of inputs whose indices are in A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', the subset XA of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Further, A denotes the complementary set of A in D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', A = D \\ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Additionally, for every A ⊂ D, the conditional distributions PXA|XA are assumed to be regular, and if not uniquely defined, they are chosen to be regular (see [4], Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Let G : E → Z be an measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Here Z denotes an abstract polish space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' G(X) is the Z-valued random variable, resulting from the composition of G with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In the following, the function G is referred to as a model, meanwhile G(X) is referred to as the output of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Denote P(E) the set of all probability distributions on (E, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' M(E) denotes the set of Z-valued models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', every Z-valued, measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In essence, the random inputs X, and the output G(X) are not restricted to be real-valued, but can be defined on more complex measurable spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', images, functions, stochastic processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' A particular subset of M(E) is of interest in the present work whenever Z = R: L2(PX, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' It is the set of measurable, R-valued functions which are square-integrable against PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Recall that L2(PX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' R) is a Hilbert space with the inner product: ∀f, g ∈ L2(PX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' R), ⟨f, g⟩L2 = � E f(x)g(x)dPX(x), and associated norm: ∀f ∈ L2(PX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' R), ∥f∥2 L2 = � E f 2(x)dPX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Denote, for any A ⊂ D, L2 (PXA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' R) the Hilbert subspaces of L2(PX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' R), of square integrable, EA- measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In other words, any f ∈ L2(PXA, R) is a square-integrable function f : EA → R: elements of L2(PXA, R) only take |A| := card(A) inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Whenever Z = Rk for a positive integer k, one can also define the set L2 � PX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Rk� accordingly (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Some elements of combinatorics and abstract algebra A partially ordered set (poset) is defined as a pair (S, ≤) where S is a non-empty set, and ≤ is a partial order binary relation on elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' A poset (S, ≤) is said to be locally finite if, for any x, z ∈ S, the sets {y ∈ S : x ≤ y ≤ z} (also called segments of S) are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' A commutative ring with identity, is a triplet (A, +, ×) where A is a non-empty set, and where + and × are addition and multiplication operators respectively, which are both associative and commutative on A, × is distributive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' + on A, A contains both an additive and multiplicative identity, but only an additive inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' A commutative ring with identity that admits a multiplicative inverse is generally called a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' In the following, abstract commutative rings with identity are denoted A, and are assumed to be endowed with the usual addition and multiplication operator, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' For instance, R is a commutative ring with identity (it is in fact, a field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' Denote IA(S) the incidence algebra of a locally finite poset (S, ≤) over a commutative ring with identity A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', the set of functions f : S×S → A such that f(x, y) = 0 if x ̸≤ y (see [18], Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='1 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' (IA(S), +, ∗) forms an A-algebra with the usual pointwise addition + and the usual convolution ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=', for any f, g ∈ IA(S), and any x, z ∈ S such that the segment {y ∈ S : x ≤ y ≤ z} is non-empty, (f ∗ g)(x, z) = � x≤y≤z f(x, y)g(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' The zeta function ζ ∈ IA(S) is convolutional identity of the incidence algebra, and is defined as, ∀x, y ∈ S: ζ(x, y) = � 1 if x = y, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQfpgEn/content/2301.02539v1.pdf'} +page_content=' The M¨obius function, denoted µ ∈ IA(S), in the case of locally finite posets S, is defined as the inverse of the zeta function for the convolution operator defined on the incidence algebra of S, and can be computed recursively, for any x, y ∈ S with x ≤ y, as [13] µ(x, y) = � � � 1 if x = y − � x≤z> 12 +Target Page +36 bits +Region (20 bits) +Offset (10 bits) +Page (16 bits) +PC (48 bits) +00 +Fig. 6: PDede BTB Organization. +Valid +1 bit +Tag +12 bits +Valid +1 bit +Tag +12 bits +Offset +10 bits +Delta bit +1 bit +Rep_policy +3 bits +Rep_policy +3 bits +Branch type +2 bits +Branch type +2 bits +Offset +10 bits +Page-BTB +Pointer +Region-BTB +Pointer +Same-page branch entry +Different-page branch entry +Fig. 7: Different- and same-page PDede entry composition. +complexities lead to increased access latency and power re- +quirements. In addition, the state-of-the-art BTB designs are +suboptimal in utilizing the available storage budget. +Indirection: As Figures 5 and 6 show, the access to Main- +BTB only provides a part of the target address, i.e., page +offset. The other parts have to be retrieved from Page-BTB and +Region-BTB. Also, Page- and Region-BTB cannot be accessed +in parallel with Main-BTB because the Main-BTB access pro- +vide the pointers to them. As a result, the sequential Main-BTB +and Page-/Region-BTB accesses increase the overall BTB +access latency. This additional latency either enforces a two- +cycle BTB lookup or necessitates a longer clock period. Both +of these alternatives are detrimental to performance. PDede +does avoid this indirection penalty to some extent because the +same-page branches do not need to access Page-/Region-BTB +rather they get their page and region number from the branch +PC itself. However, the different-page branches, i.e., where +branch PC and target address lie in different pages, do need +to pay the indirection penalty. +Associative searches: +On allocating a new BTB entry, all +BTB partitions (Main-BTB, Page-BTB, and Region-BTB) may +need to be updated. The replacement policy chooses the entry +to be replaced in Main-BTB. However, Page- and Region-BTB +need to be searched to check if the page/region number for the +incoming target is already present or not. As the page/region +number can be present anywhere in the Page/Region-BTB, +ITTAGE [48] uses fully-associative searches which increase +the power requirements especially when the number of entries +grows. PDede (partially) solves this limitation by restricting +the number of entries where a page number can reside for a +5 + +Multiplexer +Meta- +data +Target +46 bits +BTB-XC +BTB-X +Metadata +RAS +Branch PC +Concatenation +Hit/miss +Hit/miss +Full Target +Target offset +Tag: 12 +Rep_policy: 3 +Type: 2 +Valid: 1 +Target offsets +Metadata +Way 1 +4 bits +Way 0 +0 bits +Way 2 +5 bits +Way 6 +19 bits +Way 7 +25 bits +Way 4 +9 bits +Way 3 +7 bits +Way 5 +11 bits +Way +1 +Way +3 +Way +0 +Way +4 +Way +7 +Way +2 +Way +5 +Way +6 +Fig. 8: BTB-X organization and entry/set composition. +given branch target to 16. However, limiting the number of +entries increases the likelihood of conflict misses. +Suboptimal storage utilization: Though R-BTB and PDede +significantly improve storage utilization over conventional +BTB, they still miss plenty of opportunity. This is because, +in case of R-BTB, it uses a fixed size target representation +for all branches, i.e., a 10-bit offset plus a fixed sized page- +BTB pointer. In contrast, our analysis of Section III shows a +large variance in target offset sizes that naturally makes the +single sized organization of R-BTB storage inefficient as it +needs to be sized for the largest target offset. For example, +as shown by Figure 4, 54% of target offsets fit in six or +fewer bits; however, R-BTB needs to use all 10+ bits for these +branches, thus resulting in a high storage under-utilization. +PDede provides slightly better storage utilization than R-BTB +as it has differently sized entries for Same-page and Different- +page branches. However, BTB entries of only two different +sizes, i.e. Same-page and Different-page, are not enough to +capture the large offset size variance (Figure 4) observed in +server applications. +V. BTB-X +BTB-X is a simple and storage-effective BTB organization. +Building on the insights gained in Section III, it stores target +offsets, instead of full targets, to minimize storage require- +ments while also accounting for the large variance in target +offset sizes. Its microarchitecture and entry/set composition +are shown in Figure 8. +A. BTB-X Organization +The offset field in BTB-X needs to accommodate the uneven +distribution of target offset sizes as observed in Section III. In +principle, this field should be sized such that the BTB can store +the largest target offset. However, as the largest offset can be +nearly as large as the full target, sizing the offset field this +way would nearly eliminate the potential storage saving from +storing offsets. An attractive alternative is to size the offset +field such that it can store the majority of offsets. Looking at +Figure 4, an offset field of 25-bits would capture more than +99% of branch target offsets as they requires 25-bits or fewer. +However, there are two major drawbacks to this scheme. First, +it still leads to poor storage utilization. This is because, as +Figure 4 implies, 54% of branches would waste more than +three quarters of offset storage capacity as they require only 6- +bits or fewer for their offsets. Another 25% of branches would +waste nearly half of the offset storage as their offsets fit in 7- +12 bits. Second, all branches that need more than 25 bits for +their offsets can not fit in the BTB and will always cause BTB +misses. Though, as there are very few such branches (¡ 1%), +their impact is likely to be small. +To minimize the storage under-utilization, we size different +ways of a set associative BTB-X to hold different sized target +offsets. A branch is allocated to a way whose offset field is at +least as large as the number of bit required to store the target +offset. We use an 8-way set associative BTB-X and leverage +the data in Figure 4 to appropriately size the offset field of +each way such that each way covers about 12.5% dynamically +executed branches. Figure 4 shows that, on average, 0-, 4-, 5-, +7-, 9-, 11-, 19-, and 25-bit offsets cover about 20%, 36%, 46%, +61%, 72%, 79%, 90%, and 99% dynamic branches. Therefore, +we size the 8-ways of BTB-X ways to hold 0-, 4-, 5-, 7- +, 9-, 11-, 19-, and 25-bit target offsets respectively. Notice +that about 20% of dynamic branches that require 0-bits for +their offset are return instructions that read their target from +RAS, as discussed in Section II. Therefore, way-0 of BTB-X +does not feature any storage for target offsets. Though return +instruction do not get their target from BTB, they still need to +be allocated BTB entries so that the branch prediction unit can +identify them and pick their target from RAS while generating +instruction stream to be fetched. +BTB-X covers 99% of the dynamically executed branches +and we employ a very small conventional direct-mapped +BTB, called BTB-XC, that stores full target addresses for +6 + +the remaining 1% branches. Reserving a way in BTB-X +for such branches would unnecessarily increase the storage +requirements as these branches require much fewer entries than +the number of sets in BTB-X. Indeed, based on our analysis, +we size BTB-XC to store 64x fewer entries than BTB-X, i,e, +8x fewer entries than the number of sets in BTB-X. +B. Accessing BTB-X +BTB-X Lookup: +A BTB-X lookup is very similar to a +conventional BTB lookup as shown in Figure 8. It is accessed +with the index bits of a PC and all eight ways are looked up in +parallel. Also, BTB-XC is looked up in parallel with BTB-X. +The main difference between a conventional BTB lookup and +BTB-X lookup is that a BTB-X lookup provides target offset, +rather than full target address, if the lookup hits in way-1 +to way-7. Thus, target offset needs to be concatenated with +branch PC to get the full target address. The number of bits +to be concatenated from branch PC depends on the BTB-X +way in which the lookup hits. For example, a hit in way-1 +provides 4 lower order bits of target while the rest needs to be +concatenated from branch PC. Further, a hit in way-0 implies +that the full target is in RAS, while a hit in BTB-XC provides +the full target address. +BTB-X Allocation: As with any existing BTB organization, +BTB-X entries are allocated (or updated) as branch instruc- +tions retire. The number of bits required to represent a branch +target offset determines the way(s) where a branch can be +allocated an entry. For example, return instructions can be +allocated entries in any of the ways, based on replacement +policy’s decision, as they have no offset and can fit in all +ways. Other branches have fewer ways where they can be +allocated entries that are determined by the minimum number +of bits required to store their offsets. For example, if a branch +requires 20 bits for its target offset, it cannot be allocated in +way-0 to way-6. +BTB-X uses a slightly modified least recently used (LRU) +replacement policy. Concretely, we modify it to compare the +LRU counters of only the entries that can accommodate the +target offset and replace the one that is least recently used +among them. All other aspect of LRU, such as counter updates, +stay exactly the same as in baseline policy. +VI. EVALUATION +A. Methodology +We use ChampSim [4] to evaluate the efficacy of BTB-X +on server and client workload traces provided by Qualcomm +for the first Instruction Prefetching Championship (IPC-1) [1]. +We warm up microarchitectural structures for 50M instructions +and collect statistics over the next 50M. The microarchitectural +parameters for the modeled processor, resembling Intel Sunny +Cove [5], are listed in Table II. +We improved two important aspects of Champsim to eval- +uate the baseline, state-of-the-art, and proposed BTB orga- +nizations. First, being a trace-driven simulator, Champsim +detects branches by consulting the information available in +the traces, rather than looking up a BTB. This essentially +TABLE II: Microarchitectural parameters +Parameter +Value +Fetch +6-wide, 128-instruction FTQ +Branch Predictor +Hashed Perceptron +Return address stack +64 entries +Scheduler +128 entries +Re-order buffer +352 entries +Load queue +128 entries +Store queue +72 entries +L1-I +32 KB, 8-way, +4 cycle latency, 8 MSHRs +L1-D +48 KB, 12-way, +5 cycle latency, 16 MSHRs +L2 +512 KB, 8-way, +14/15 cycle latency, 32 MSHRs +LLC +2MB, 16-way, +34/35 cycle latency, 64 MSHRs +TABLE III: BTB-X storage requirements. The numbers in +parentheses are for BTB-XC. +Entries +Sets +Set size +Storage +256(4) +32(4) +224(64)-bits +0.9KB +512(8) +64(8) +224(64)-bits +1.8KB +1K(16) +128(16) +224(64)-bits +3.6KB +2K(32) +256(32) +224(64)-bits +7.25KB +4K(64) +512(64) +224(64)-bits +14.5KB +8K(128) +1024(128) +224(64)-bits +29KB +16K(256) +2048(256) +224(64)-bits +58KB +translates to Champsim using an ideal BTB. Therefore, we +first implement a realistic conventional BTB (Conv-BTB), +presented in Section II, in Champsim. Second, Champsim re- +solves all branches in execute stage, i.e., branch mispredictions +are detected and the fetch is resteered to correct path only +when a mispredicted branch instruction reaches the execute +stage. Such branch resolution overestimates the misprediction +penalty of unconditional direct branches. This is because such +branches can be resolved in the decode stage (hence, fetch can +be resteered sooner) as they are always taken, thus the PC of +the next instruction can be compared to the target encoded +in the branch instruction to detect mispredictions. Further, +taken conditional branches that miss in BTB but are correctly +predicted by the direction predictor, can also be resolved in +the decode stage. To do so, the fetch stage passes the direction +prediction for all instructions, despite BTB hit/miss, to decode +stage. If decode identifies a branch that missed in the BTB +but predicted taken by the direction predictor, it resteers the +fetch to the target encoded in the branch instruction, thus +reducing BTB miss penalty. Given that the direction predictors +are highly accurate, this optimization reduces average BTB +miss penalty. Overall, we improve branch resolution so that +the unconditional direct branches and the taken conditional +branches that miss in BTB are resolved in the decode stage. +Finally, BTB is updated at commit stage by only the taken +branches (both conditional and unconditional). +B. Storage breakdown +We first assess the number of branches different BTB orga- +nizations (Conv-BTB, PDede, and BTB-X) can accommodate +7 + +TABLE IV: Number of branches in different BTB organizations at various storage budgets. +Storage +BTB-X + BTB-XC +PDede +Conv-BTB +0.9KB +1.8KB +3.6KB +7.25KB +14.5KB +29KB +58KB +Branches +256 + 4 +512 + 8 +1K + 16 +2K + 32 +4K + 64 +8K + 128 +16K + 256 +Page-BTB budget +Main-BTB budget +Entry Size +Branches +0.078KB +0.817KB +32-bits +210 +0.156KB +1.645KB +32.5-bits +415 +0.312KB +3.3KB +33-bits +820 +0.625KB +6.6KB +33.5-bits +1617 +1.25KB +13.2KB +34-bits +3190 +2.5KB +26.5KB +34.5-bits +6292 +5KB +53KB +35-bits +12405 +Entry Size +Branches +64-bits +116 +64-bits +232 +64-bits +464 +64-bits +928 +64-bits +1856 +64-bits +3712 +64-bits +7424 +in a given storage budget compared to each other. We use +storage budgets required for storing 256, 512, 1K, 2K, 4K, +8K, and 16K branches in BTB-X as presented in Table III. +Our calculations assume a 48-bit virtual address space and +BTB-X entry compositions presented in Figure 8. To double +the number of entries in BTB-X, we double the number of +sets while keeping the associativity same. Notice that Table III +presents set size instead of entry size. This is because BTB-X +features different sized entries in different ways; however, the +set size remains constant. +Table IV presents the number of branches the different BTB +organizations can track at different storage budgets. PDede +distributes the overall BTB storage budget among its Main- +BTB, Page-BTB, and Region-BTB. We follow the distribution +used by its inventors [49] to allocate the budget among +different PDede BTBs as shown in Table IV. Accordingly, for +29KB storage budget, we configure PDede to use 1K Page- +BTB entries and about 6K Main-BTB entries. While halving +the storage budget to lower values, we halve the number of +entries in the Main-BTB as well as the Page-BTB. Halving +the number of Page-BTB entries reduces the number of bits +required to store Page-BTB pointer in the Main-BTB. Thus, +the Main-BTB entry size reduces with the reduction in storage +budget. Further, we use four Region-BTB entries across all +storage budgets, so Region-BTB requires a fixed storage of +0.0107KB. Also recall that PDede reserves half of the ways +in a set for same-page branches while the other half can store +both same-page and different-page branches. Therefore, its +entries are of two different sizes. The PDede entry size shown +in Table IV is the average of two sizes. +As the table shows BTB-X stores significantly more +branches than any other BTB organizations. Concretely, it +stores 2.24x more branches than a conventional BTB organiza- +tion. Compared to PDede, BTB-X stores 1.24x more branches +at 0.9KB storage budget and 1.34x more branches at 58KB +storage budget. BTB-X’s advantage over PDede increases with +storage budget because PDede entries require more bits at +higher budgets to accommodate larger Page-BTB pointers. +C. BTB MPKI +To understand the advantage of higher BTB-X branch +density, we measure misses per 1000 instructions (MPKI) +that different BTB organizations incur on client and server +workloads. Since BTB misses for not-taken branches do not +hurt performance, we only consider the BTB misses for taken +branches. For this analysis, we assume a BTB storage budget +of 14.5KB that corresponds to 4160-, 3190-, and 1856-entries +in BTB-X, PDede and Conv-BTB respectively. The results are +presented in Figure 9. +As the figure shows, server workloads experience signifi- +cantly higher MPKI compared to client workload due their +massive instruction and branch footprints. The figure also +shows that BTB-X provides a much lower MPKI compared +to both conventional BTB and PDede especially on server +workload. Concretely, on average, conventional BTB incurs +25 MPKI on server workload as it stores the least amount +of branches among the three organization for a given storage +budget. PDede is able to lower the MPKI to 13.7 while BTB- +X brings it further down to 9.5. The advantage of BTB-X over +other organizations is particularly evident on very high MPKI +workloads, i.e., server 023 to server 035, where it provides +much lower MPKI compared to conventional BTB and PDede. +D. Performance +To assess how the reduced MPKI translates to performance, +we compare the performance of the three BTB organizations +on client and server workloads. Recall from Section II that +a larger BTB delivers two distinct benefits: 1) it reduces +the incidence of pipeline flushes by detecting branches in +the upcoming control flow and 2) it facilitates instruction +prefetching when coupled with FDIP. Thus, we compare the +performance gains achieved by the three BTB organizations +by evaluating them with FDIP. +Figure 10 presents the performance gains obtained on server +and client traces. The results are normalized to the perfor- +mance of the Conv-BTB without any instruction prefetching. +The figure shows three bars for each workload. The first +bar presents performance gain achieved by Conv-BTB when +copuled with FDIP. The second and third bars present the +performance gains achieved by PDede and BTB-X respec- +tively. The PDede and BTB-X bars divide the performance +gain into contributions from fewer pipeline flushes and from +better instruction prefetching stemming from capturing more +branches in the BTB. +Looking at the overall performance gain with instruction +prefetcher (FDIP), the figure shows that BTB-X provides +a geometric mean gain of 39% over baseline on server +workloads. In comparison, PDede and Conv-BTB deliver a +performance gain of only 33% and 24% on these workloads. +Looking at individual workloads, BTB-X comprehensively +outperforms PDede and Conv-BTB on server 023 to sever 32. +For example, on server 032, BTB-X provides 83% speedup +over baseline whereas PDede and Conv-BTB achieve only +60% and 32% performance gain. This is because the branch +8 + +0 +10 +20 +30 +40 +50 +60 +70 +client_001 +client_002 +client_003 +client_004 +client_005 +client_006 +client_007 +client_008 +server_001 +server_002 +server_003 +server_004 +server_009 +server_010 +server_011 +server_012 +server_013 +server_014 +server_015 +server_016 +server_017 +server_018 +server_019 +server_020 +server_021 +server_022 +server_023 +server_024 +server_025 +server_026 +server_027 +server_028 +server_029 +server_030 +server_031 +server_032 +server_033 +server_034 +server_035 +server_036 +server_037 +server_038 +server_039 +Client +Server +Client workloads +Server workloads +Avg +BTB misses per 1000 instructions +Conv-BTB +PDede +BTB-X +Fig. 9: BTB MPKI experienced by different BTB organizations. +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +client_001 +client_002 +client_003 +client_004 +client_005 +client_006 +client_007 +client_008 +server_001 +server_002 +server_003 +server_004 +server_009 +server_010 +server_011 +server_012 +server_013 +server_014 +server_015 +server_016 +server_017 +server_018 +server_019 +server_020 +server_021 +server_022 +server_023 +server_024 +server_025 +server_026 +server_027 +server_028 +server_029 +server_030 +server_031 +server_032 +server_033 +server_034 +server_035 +server_036 +server_037 +server_038 +server_039 +Client +Server +Client workloads +Server workloads +Gmean +Performance gain +Gain from fewer flushes +Gain from L1-I prefetching +Fig. 10: Performance gain obtained by conventional BTB (with FDIP), PDede and BTB-X (with and without FDIP) over the +conventional BTB without FDIP. The three bars for each workload correspond to Conv-BTB, PDede, and BTB-X respectively. +working set of these workloads starts to fit in BTB-X due to its +higher branch capacity. As a result, BTB MPKI lowers which +not only reduces pipeline flushes and but also keeps FDIP on +correct prefetch path for longer intervals. +Looking at the results without instruction prefetcher, Fig- +ure 10 shows that BTB-X provides 13% performance gain +over the baseline Conv-BTB whereas PDede is achieves 8% +gain. On individual workloads, BTB-X achieves significantly +high gain over Conv-BTB and PDede on workloads from +server 23 to server 32 even without FDIP. Figure 10 also +shows the FDIP by itself performs better with more number of +BTB entries. For example, on server 32 FDIP with Conv-BTB +provides 32% performance gain. With PDede, the performance +gain from prefetching increases to 42% and with BTB-X it +further increases to 51%. +These results show that by accommodating more branches +in a given storage budget, BTB-X not only reduces pipeline +flushes but also improves instruction prefetching, both lead to +better performance. +TABLE V: Energy requirements of different BTB designs. +BTB +Access Type +Energy +#Accesses +Energy +(Per access) +(Total) +Conv-BTB +Read +13.2pJ +1.60E+08 +2122µJ +Write +25.2pJ +4.36E+06 +110µJ +Total Energy +2232µJ +PDede +Main-BTB Read +8.4pJ +1.24E+08 +1047µJ +Main-BTB Write +12.5pJ +5.74E+05 +7µJ +Page-BTB Read +0.9pJ +2.01E+06 +2µJ +Page-BTB Write +0.8pJ +2.04E+04 +0.02µJ +Page-BTB Search +6.2pJ +2.14E+05 +2µJ +Total Energy +1058µJ +BTB-X +Read +8.5pJ +1.16E+08 +994µJ +Write +11.4pJ +4.03E+05 +5µJ +Total Energy +999µJ +Finally, Figure 10 shows that all three BTB organizations +perform similar on client workloads. This is because their +branch working sets mostly fit in the baseline Conv-BTB and +the additional entries in PDede and BTB-X do not bring much +performance benefit. +9 + +1.00 +1.10 +1.20 +1.30 +1.40 +1.50 +0.9KB +1.8KB +3.6KB +7.25KB 14.5KB +29KB +58KB +Performance gain +Conv-BTB +Pdede +BTB-X +(a) Server workloads +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +0.9KB +1.8KB +3.6KB +7.25KB 14.5KB +29KB +58KB +Performance gain +Conv-BTB +Pdede +BTB-X +(b) Client workloads +Fig. 11: Performance gains for conventional BTB, PDede, and BTB-X on (a) server and (b) client workloads over a conventional +BTB with 0.9KB storage budget. X-axis label is storage requirements of 256-, 512-, 1K-, 2K-, 4K-, 8K-, and 16K-entry BTB-X. +E. Energy and delay analysis +We use Cacti 7.0 [35] to analyze the energy requirements +and access latencies of Conv-BTB, PDede, and BTB-X at 22 +nm, which is the most recent technology node supported by +Cacti. For this analysis we assume the same storage budget, i.e. +14.5KB, as used in Section VI-D for the performance analysis. +Energy requirements: Table V shows the per access read +and write energy requirements of different BTB designs. As +the table shows, BTB-X and PDede’s Main-BTB incur very +similar per access read and write energy cost. However, in +addition to Main-BTB, PDede also needs to access Page- +BTB for different-page branches, i.e., the branches that have +their targets in a different page than the branches themselves. +Further, the Page-BTB needs to be searched on a BTB write +to check if the target page number is already in Page-BTB or +not. Consequently, PDede’s per access read and write energy +for different page branches reaches 9.3 pico Joules (pJ) and +19.5 pJ, respectively, compared to 8.5pJ and 11.4 pJ of BTB- +X. PDede also features a Region-BTB; however, its energy +requirements are negligible and, thus, not shown in Table V. +Finally, Conv-BTB’s per access energy cost is significantly +higher than BTB-X as its each read and write access requires +13.2pJ and 25.2pJ respectively. +Table V also shows the number of read/write accesses, aver- +aged across the workloads, and the total energy consumption. +Despite very similar per access energy cost, PDede’ Main- +BTB consumes considerably higher energy than BTB-X. This +is because PDede often goes on the wrong execution path +due to its higher MPKI. These additional wrong path BTB +accesses, reflected in higher BTB reads in Table V, result in +higher energy consumption. Further, PDede needs to handle +more BTB writes than BTB-X because it holds fewer branches, +which results in frequent replacements. Thus, the total energy +consumption of PDede reaches 1058µJ compared to 999µJ of +BTB-X. Finally, the energy requirements of Conv-BTB are +significantly higher, 2232µJ, than BTB-X because of higher +per access energy and higher number of total accesses. +Overall, this analysis shows that BTB-X not only delivers +better performance than PDede but also consumes less energy, +thus providing much better energy efficiency. +Access Latency: Our analysis shows that the Conv-BTB +requires about 0.36ns to complete an access. As discussed +in Section IV-B, PDede’s access latency is the sum of Main- +BTB and Page-BTB access latencies as these two structures +are accessed sequentially. Our analysis shows that the Main- +BTB and Page-BTB accesses require 0.34ns and 0.13ns, re- +spectively, thus resulting in an overall PDede access latency of +0.47ns which is considerably higher than Conv-BTB latency. +To address this, PDede employs multi-cycle BTB accesses: the +Main-BTB is accessed in the first cycle, and the Page-BTB is +accessed in the next cycle only if the branch is predicted to +be taken and it’s target is in a different page than the branch. +Thus, the same page branches need one cycle and the taken +different page branches need two cycles to get their target +address from PDede. Finally, our analysis shows that a BTB-X +access takes only 0.33ns. In summary, this analysis shows that +BTB-X provides better storage efficiency without any adverse +effects on the access latency. +F. Performance variation with BTB storage budget +To further understand the performance advantage of BTB-X +over PDede and Conv-BTB, we compare their performances +across different storage budgets. Figure 11 presents the per- +formance gains obtained on server and client workloads. The +results are normalized to the performance of Conv-BTB with +0.9KB storage budget. Instruction prefetching is enabled in all +designs including baseline. +As the figure shows, on server workloads, BTB-X pro- +vides significantly higher performance than the Conv-BTB +and PDede for equal storage budgets of up to 29KB and +14.5KB respectively. The performance advantage of BTB-X is +10 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0 +2 +4 +6 +8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 +Fraction of dynamic branches covered +Number of bits required for branch target offsets +IPC-1 traces avg +CVP-1 traces avg +Fig. 12: Target offset distribution in CVP-1 and IPC traces. +pronounced on server traces whose large instruction footprints +pressure the BTB and L1-I. For instance, BTB-X provides +35% performance gain over the baseline compared to 29% +and 20% of PDede and Conv-BTB respectively at 14.5KB +budget. At large BTB storage budgets, the branch working +sets of many workloads start to fit in the available BTB +capacity, at which point the performance gap between BTB-X +and the other two designs diminishes. Also, the performance +gap between the three BTB organizations levels off earlier on +client trace due to their smaller instruction working sets. +A key take-away from this figure is that BTB-X outperforms +the conventional BTB even when it is given just half the +storage budget of its conventional counterpart. For example, +in Figure 11a, the Conv-BTB improves performance by 20% +with a 14.5KB budget whereas BTB-X provides a 24% im- +provement with just 7.25KB. The reason for this phenomenon +is that BTB-X accommodates 2.24x more entries than Conv- +BTB of equal storage budget; thus, halving BTB-X’s budget +still gives a slight capacity advantage over Conv-BTB. +G. Analyzing target offset distribution in more workloads +We study the target offset distribution in 750+ Qualcomm +server traces that were provided for the first Championship +Value Prediction(CVP-1) [6]. The results, presented in Fig- +ure 12, show that their offset distribution is very similar to the +distribution in IPC-1 traces presented in Figure 4. This study +confirms that such an offset distribution is a consequence of +how the applications are written and the resulting control-flow +behavior. As discussed in Section III, such offset distribution +stems from the fact that the conditional branches dominate +dynamic branch working set and they tend to have short +offsets. This is because conditional branches guide the control- +flow inside functions, and software engineering principles +favor small functions. Consequently, short offsets dominate +the branch offset distribution. +In addition to CVP-1 traces, we analyze five more server ap- +plications - Wordpress [53], Mediawiki [52], and Drupal [51] +from Facebook’s HHVM OSS-performance benchmarks [8], +Kafka [50] from Java DaCapo [14], and Finagle-HTTP [7] +from Java Renaissance [42]. Further, these applications are +compiled to x86 (CVP-1 and IPC-1 traces are compiled to +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +2 +4 +6 +8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 +Fraction of dynamic branches covered +Number of bits required for branch target offsets +Wordpress +Mediawiki +Kafka +Finagle_http +Drupal +IPC-1 traces avg +Fig. 13: Target offset distribution in x86 compiled server +applications and Arm64 IPC traces. +Arm64) which also enables us to assess the impact of ISA on +target offset distribution. The results presented in Figure 13 +show that the offset distribution in these applications is also +very similar to that in IPC-1 traces. The only difference is +that x86 traces require slightly larger offsets (1 or 2-bits more) +to achieve a similar dynamic branch coverage as the Arm64 +(CVP-1 and IPC-1) traces. For example, 6-bit offsets cover +about 54% branches in Arm64 traces, whereas x86 offsets +need 8-bits to achieve 58% branch coverage. This is because +x86 offsets specify the distance between branch PC and target +in number of bytes because x86 instruction are variable size. +In contrast, Arm64 offsets specify this distance in number of +instructions because all instructions are 4-bytes, thus saving 2 +offset bits. +As BTB-X needs to store slightly larger offsets for x86 +than Arm64, we reassess its storage advantage over PDede +and Conv-BTB for x86 architectures. As each way in 8-way +BTB-X needs to cover about 12.5% of branches, we size its +ways to store offset of 0-, 5-, 6-, 7-, 9-, 12-, 20-, and 27-bits +based on the offset distribution in x86 applications shown in +Figure 13. Thus, each set needs 86-bits for offsets compared +to 80-bit in Arm64. Consequently, BTB-X’s storage advantage +is slightly lower for x86 than Arm64. However, BTB-X still +stores 2.18x more branches than Conv-BTB for x86 (2.24x +for Arm). Compared to PDede, BTB-X stores 1.21x more +branches (1.24x for Arm64) at 0.9KB storage budget and 1.31x +more branches (1.34x for Arm64) at 58KB storage budget. +(Section VI-B presents this analysis of Arm64 traces.) +VII. RELATED WORK +Mitigating BTB misses: +BTB was first disclosed by +Losq [36] and was further expanded by Lee et al [33]. Since +BTB lies on the critical path for instruction delivery, there has +been several proposals to increase its effectiveness. Instead of +accessing BTB with the PC of each individual instruction, Yeh +et al. [54] proposed to access it with basic-block address and +store not only the target but also the fall-through address in the +BTB. In case the branch is predicted to be not taken, the fall- +through address is used, after fetching the current basic-block, +as the next PC for both instruction fetch as well as for the +11 + +next BTB access. The advantage of such a BTB organization +over the conventional BTB organization is that it reduces +BTB bandwidth and power requirements as a single access +provides the next control flow divergence point, whereas the +conventional organization requires as many accesses as the +number of instructions until the next branch. Whereas the +initial proposal on basic-block-based BTB [54] stores full fall- +through address, the later work [44] proposed to store the +delta between two sequential basic-block addresses. Fagin [18] +proposed to use the BTB storage more effectively by storing +only the partial tags. To further amortize the tag storage cost, +some designs proposed to share a BTB entry among multiple +branches that reside in the same cache block [2], [27]. Though +these BTB designs aim to improve different aspects of BTB +management, they all share a common trait, i.e., they store full +target addresses. Thus, the key idea of BTB-X can be applied +to all of these BTB designs to reduce their target storage cost. +Prior work [10], [11], [23], [47]–[49] has also explored +mechanisms to reduce the storage cost of branch targets. +Seznec [47], [48] proposed to break the target address into +page number and offset; and store a pointer to the page +number, along with the page offset, in the BTB while the +page number itself is stored in a separate structure. It reduces +the storage cost as a pointer to page number is smaller than +the page number itself, and the page number for all the targets +in a page is stored only once. Hoogerbrugge [23] proposed to +size some of the entries in a set for storing small target offsets, +thus reducing BTB storage requirements. +The state-of-the-art BTB design, PDede [49], combines +these two ideas to address their individual limitations. Con- +cretely, Sezenc’s design is sub-optimal for same-page branches +as it unnecessarily stores (pointer to) their target page number +even though it is same as the page number of their branch +PCs. In contrast, Hoogerbrugge’s design is sub-optimal for +inter-page branches as it stores their full targets. Inspired from +Hoogerbrugge’s design, PDede sizes some entries in a set to +store same-page branch targets; and similar to Seznec’s design, +for inter-page branches, it stores pointers to page numbers +instead of page numbers themselves. PDede further reduces the +inter-page target storage cost by dividing the page number into +page- and region-number. However, as it is based on Seznec’s +design, it also has to pay the addition latency cost of indi- +rection between main-BTB and the page-/region-BTB. Micro +BTB [22], proposes a flexible BTB entry structure where each +entry can store either one branch, if its offset is large, or two +branches if their offsets are small. We show that all these +designs are sub-optimal in exploiting the storage optimization +opportunity presented by the uneven branch offset distribution. +BTB-X not only captures this opportunity but also avoids the +BTB indirection of the state-of-the-art. +Apart from optimizing BTB organization, prior work [15], +[16], [28], [31], [32] has also explored BTB prefilling/prefetch- +ing to mitigate BTB misses. The state-of-the-art in BTB +prefetching is a profile guided software prefetcher, called +Twig [28]. It analyzes an application’s execution profile to +identify critical BTB misses and then injects software prefetch +instructions. The prefetch instruction takes compressed branch +PC and target as operands and its execution fills this informa- +tion in BTB. These prefetching techniques are complementary +to BTB organization and, thus, can be used along with BTB-X. +Mitigating L1-I misses: +As L1-I misses continue to be a +major performance limiter in server applications [12], [25], +[46], prior work has proposed both hardware and software +mechanisms to mitigate L1-I misses. On the hardware side, +state-of-the-art temporal stream prefetchers [19], [20] record +the L1-I miss/access history and replay it to discover prefetch +candidates. While such prefetchers are highly effective, their +huge metadata storage cost renders them impractical despite +recent attempts to address this weakness [26], [27]. Fetch- +directed prefetchers use in-core structures (BTB and branch +direction predictor) to run ahead of the fetch unit to find +prefetch candidates. While the early work [43] focused on L1-I +prefetching only, the state-of-the-art fetch-directed prefetchers +[31], [32] also prefill into the BTB. +Several purely-software based approaches to instruction +prefetching and improving the L1-I capacity has also been +proposed [9], [13], [17], [29], [34], [37]–[40]. These methods +use data from application profiling to perform either compile- +time, link-time or post-link time optimizations. Since these +methods are software-only they will benefit from the increased +BTB capacity provided by the BTB-X organization. +VIII. CONCLUSION +The multi-megabyte instruction footprints of contemporary +server applications cause frequent BTB and L1-I misses, +which have become major performance limiters. Because BTB +capacity greatly affects front-end performance by dictating +pipeline flush rate and the efficacy of fetch-directed instruction +prefetching, commercial products allocate tens to hundreds of +KBs of storage to BTBs. We observe that the single largest +contributor to the BTB storage cost is the cost of storing +branch target. We further observe that BTB storage cost can +be drastically reduced by storing target offsets instead of full +or even compressed targets. This is because targets of most +branches lie relatively close to the branches themselves and +our analysis shows that more than 99% of offsets can be +represented with at most half the bits required to store the full +targets. Based on these observations, we propose a storage- +effective BTB organization, called BTB-X, that stores target +offsets in place of target address. Furthermore, BTB-X, an +8-way set associative BTB, uses differently sized ways with +each storing offsets of a different length, thus accounting +for the uneven distribution of offset lengths. Overall, BTB- +X is capable of storing about 2.24x more branches than a +conventional BTB and 1.3x more branches than a state-of- +the-art BTB organization within the same storage budget. +ACKNOWLEDGEMENTS +This work is partially supported through the Research +Council of Norway (NFR) grant 302279 to NTNU. +12 + +REFERENCES +[1] “1st Instruction Prefetching Championship,” https://research.ece.ncsu. +edu/ipc/. +[2] “AMD software optimization guide. 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MICRO +25. +Washington, DC, USA: IEEE Computer Society Press, 1992, p. +129–139. +APPENDIX A +ARTIFACT APPENDIX +A. Abstract +We implement BTB-X in Champsim simulator. Our artifacts +provide the following: 1) BTB-X implementation in Champ- +sim, 2) Link to workload traces, 3) Scripts for generating +configuration files, launching simulations, and collecting re- +sults, and 4) Excel file for plotting the most important results. +We identify three key results for artifact evaluation: a) Branch +Target Offset distribution (Figure 4), b) BTB MPKI reduction +(Figure 9), and c) Performance improvement (Figure 10). +B. Meta-information +• Compilation: +Tested with GCC 8.5.0. It should also +work with other recent GCC versions. +• Code/Workloads: +Download code/workloads from the +provided link. +• Experiments: Modify the provided scripts (as described +below) to run simulations. +• Metrics: IPC, BTB MPKI, Branch Target Offset distri- +bution. +• Time needed to run experiments: Less than 30 minutes +when running all traces in parallel. +• Plotting graphs: Excel file, BTBX artifact results.xlsx, +is provided to plot graphs. +C. Access to artifacts +• Code: Download BTB-X implementation from [3]. +• Workloads: The workloads can be downloaded from +https://drive.google.com/file/d/1qs8t8-YWc7lLoYbjbH +d3lf1xdoYBznf/view?usp=sharing +Place the workloads in /inc/ooo cpu.h. +• Generating +configuration +files: +Go +to +directory +/launch/scripts/. +In +the +script +file +createConfig.sh, +point +PATH TO CHAMPSIM +to +. Run this script (./createConfig.sh) to +generate config files needed by Champsim. +• Running simulations: +Running +all +workloads: +Go +to +the +directory +/launch/. +In +script +file +launch.sh, +replace the line (line +64) with the command to run experiments on your cluster. +A sample command is given that runs experiments on +our cluster. Running this script (./launch.sh) will run +simulations, and the stats will be stored in directory +/results 50M/. +Running a single workload: An example command to run +simulation for a single workload is provided at [3]. +14 + +F. Results +• Collecting results: Go to /collectStats/. +Run the script getResults.sh, and it will collect results +from all workloads and save them in a file all res. +• Plotting Results: +Download the all res file. Open the +provided excel file BTBX artifact results.xlsx. Click on +“Data” in MS-Excel top menu bar. Click on “Refresh All” +in “Queries and Connections” ribbon, go to the folder +where you stored all res and double click on all res. +Now “Offset Distribution”, “MPKI”, and “Performance” +sheets in the excel file should have plots for Figure 4, +Figure 9, and Figure 10 respectively. +15 + diff --git a/idE2T4oBgHgl3EQfcweU/content/tmp_files/load_file.txt b/idE2T4oBgHgl3EQfcweU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a624238ab2c6289f2c667485202bbf84d0e11222 --- /dev/null +++ b/idE2T4oBgHgl3EQfcweU/content/tmp_files/load_file.txt @@ -0,0 +1,1314 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf,len=1313 +page_content='A Storage-Effective BTB Organization for Servers Truls Asheim NTNU, Norway Boris Grot University of Edinburgh, UK Rakesh Kumar NTNU, Norway Abstract—Many contemporary applications feature multi- megabyte instruction footprints that overwhelm the capacity of branch target buffers (BTB) and instruction caches (L1-I), caus- ing frequent front-end stalls that inevitably hurt performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB capacity is crucial for performance as a sufficiently large BTB enables the front-end to accurately resolve the upcoming execution path and steer instruction fetch appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' More- over, it also enables highly effective fetch-directed instruction prefetching that can eliminate a large portion L1-I misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For these reasons, commercial processors allocate vast amounts of storage capacity to BTBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This work aims to reduce BTB storage requirements by optimizing the organization of BTB entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our key insight is that storing branch target offsets, instead of full or compressed targets, can drastically reduce BTB storage cost as the vast majority of dynamic branches have short offsets requiring just a handful of bits to encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Based on this insight, we size the ways of a set associative BTB to hold different number of target offset bits such that each way stores offsets within a particular range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Doing so enables a dramatic reduction in storage for target addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our final design, called BTB-X, uses an 8-way set associative BTB with differently sized ways that enables it to track about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more branches than a conventional BTB and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3x more branches than a storage-optimized state-of-the-art BTB organization, called PDede, with the same storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' INTRODUCTION Contemporary server applications feature massive instruc- tion footprints stemming from deeply layered software stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These footprints may far exceed the capacity of the branch target buffer (BTB) and instruction cache (L1-I), resulting in the so-called front-end bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB misses may lead to wrong-path execution, triggering a pipeline flush when misspeculation is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Such pipeline flushes not only throw away tens of cycles of work but also expose the fill latency of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Similarly, L1-I misses cause the core front-end to stall for tens of cycles while the miss is being served from lower-level caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB stands at the center of a high-performance core front end for three key reasons: it determines the instruction stream to be fetched, it identifies branches for the branch direction predictor, and it affects the L1-I hit rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Specifically, by identifying control flow divergences, the BTB ensures that the branch direction predictor can make predictions for upcoming conditional branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For predicted-taken and unconditional branches, the BTB supplies targets to which instruction fetch should be redirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Finally, the BTB together with the direction predictor enables an important class of instruc- tion prefetchers called fetch-directed instruction prefetchers (FDIP) [30]–[32], [43], which rely on the BTB to discover L1-I prefetch candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Considering the criticality of capturing the large branch working sets of modern workloads, commercial CPUs feature BTBs with colossal capacities, a trend also observed by [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' With each BTB entry potentially requiring 8 bytes or more (Section II), BTB storage costs can easily reach into tens and even hundreds of KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Indeed, the Samsung Exynos M6 mobile processor allocates a staggering 529KB of on-chip storage to BTBs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Not only the BTB storage cost is high, it is increasing at a rapid pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, the Samsung Exynos BTB storage budget increased nearly six fold (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB to 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB) from M2 to M6, over a period of about eight years [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' While such massive BTBs are effective at capturing branch working sets, they do so at staggering area costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To reduce the BTB storage cost, prior work [47]–[49] has focused on compressing the branch targets as they account for the majority of BTB storage budget as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Concretely, Seznec [47], [48] observes that all branch targets within a page share the same page number and BTB storage requirements can be significantly reduced by storing the page number only once per page instead of once per target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To ex- ploit this observation, he partitions the BTB in two structures, Main-BTB and Page-BTB, each storing different portions of branch targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The Main-BTB stores the page offset and a pointer to the Page-BTB entry that stores the page number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The state-of-the-art BTB organization, PDede [49], further observes that target addresses span significantly fewer regions than pages, where a region is a group of contiguous pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, it partitions the BTB even further and introduces a Region-BTB that lowers page number storage cost as the region number for all pages inside a region is stored only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' By storing page/region numbers only once for all branches in a page/region, these BTBs avoid information duplication, thus reducing storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Though, these designs significantly reduce BTB storage requirements, they introduce several complexities that increase their access latency and power requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' First, these designs introduce a level of indirection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', on a BTB access, Main-BTB is accessed first to get the pointers to the Page-BTB and Region-BTB and only then these BTBs can be accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This sequential access, Main-BTB followed by Page/Region- BTBs, increases the overall BTB access latency which either requires a two-cycle BTB lookup or a longer clock period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Both of these alternatives incur a performance penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Second, on allocating a new BTB entry, Page-BTB and Region-BTB need to be searched to check if the page/region number for the target address is already present or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the page/region number can be anywhere in Page/Region-BTB, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='03899v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='AR] 10 Jan 2023 Tag: 12 Type: 2 Target: 46 Valid: 1 Rep_policy: 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 1: Composition of an entry in conventional BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The numbers are the number of bits required to encode each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' a fully-associative associative search is required [48] which increases BTB power requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' An alternative is to restrict the locations where a page/region number can be stored in Page/Region-BTB [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' however, it increase the likelihood of conflict misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This work seeks to reduce BTB storage requirements with- out increasing BTB complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To that end, we propose to store target offsets, defined as delta between the address of the branch instruction and that of its target, instead of full or compressed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', page offset, page number, and region number) targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our key insight is that target offsets are unevenly distributed but tend to require significantly fewer bits to represent than full and even compressed target addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our analysis reveals that 54% of dynamic branches require only 6 bits or fewer for offset encoding, while a meager 1% of branches need 25 bits or more to store their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Based on this insight, we propose to store target offsets in the BTB rather than compressed or full target addresses, which can be up to 64 bits long depending on the size of virtual address space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To accommodate the varied distribution of target offsets, we size different ways of a set associative BTB to hold different number of offset bits such that each way stores only those branches whose target offsets can be encoded with a certain number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In doing so, we not only significantly reduce BTB storage requirements but also avoid the complexities, indirection and fully-associative searches of the state-of-the-art BTB designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This paper introduces BTB-X, a simple yet highly storage- effective BTB organization, that incarnates our idea of storing target offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X is a set associative BTB with its ways sized to store different sized target offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our evaluation shows that BTB-X can track about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more branches than a conventional BTB storing full targets and about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3x more branches than PDede, a state-of-the-art BTB organization, with the same storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Conversely, BTB-X can accommo- date the same number of branches as conventional BTB and PDede while requiring 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3x less storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This work makes the following key contributions: We show that storing branch target offsets, instead of full or compressed target addresses, can provide drastic BTB storage savings because about 54% of branches require only 6 bits or fewer to encode their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A further 22% of branches require between 7 and 10 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We show that the target offset sizes are unevenly dis- tributed with 0-6 bits, 7-10 bits, and 11-25 bits required to encode the offsets of 54%, 22% and 23% of branches respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, a single size offset field cannot provide storage optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We introduce BTB-X, a simple and highly storage- effective BTB organization, that stores target offsets instead of targets themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Furthermore, BTB-X ways are sized to hold different sized target offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We demonstrate that, with the same storage budget, BTB- X can accommodate about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3x more branches than a conventional BTB and PDede, a state-of-the-art BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our evaluation further shows that BTB-X outper- forms the conventional BTB even when provisioned with just half the storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BACKGROUND AND MOTIVATION Branch prediction unit predicts the program control flow and supplies a stream of instruction addresses/program counters (PCs) on the predicted path to the fetch unit which fetches the corresponding instructions to feed the rest of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As branch instructions disturb the otherwise sequential control flow, the branch prediction unit needs to identify them to predict the upcoming control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, whether an in- struction is a branch or not can only be determined after it has been fetched and decoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To avoid the latency of fetching and decoding instructions before generating next PCs, the branch prediction unit employs a special hardware structure, called branch target buffer (BTB), to identify branch instructions solely from their PCs before the instructions themselves are even fetched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Branch Target Buffer (BTB) Figure 1 presents the conventional BTB organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Each BTB entry consists of valid, tag, type, target, and rep policy fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 1 also shows the typical number of bits needed for these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The tag field usually stores only a partial tag, which is generated by hashing the full tag, to reduce storage cost while introducing minimal aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The number of bits for target field depends on the size of virtual address space and instruction set architecture (ISA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We assume a 48-bit virtual address space and ARMv8 ISA to calculate target field size in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As ARMv8 instructions are always 32-bits and 4-byte aligned, the least significant two bits of a PC are always zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, we only need 46-bits for the target field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The valid bit indicates whether the entry contains valid information or not, while rep policy bits choose one of the existing branches for eviction when a new branch is inserted in the BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To check whether a PC corresponds to a branch instruction, the BTB is indexed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' accessed, with the low order PC bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The high order PC bits are hashed, using the same function that is used to generate partial tags, and compared to the tag field of the indexed BTB entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A match indicates that the PC belongs to a branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A branch instruction simply implies the presence of a potential control flow divergence point in program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, whether or not the divergence actually happens depends on the type of branch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' call, return, conditional, or unconditional branch, which is stored in the type field of a BTB entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Call, return, and unconditional branches always cause control flow divergence as they are always taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 2 FTQ Instruction fetch unit Prefetch candidate Prefetch request Prefetch fill L1I LLC Branch prediciton unit Prefetch engine Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 2: FDIP microarchitecture Conditional branches, in contrast, are not always taken and a direction predictor is used to predict their direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' If a branch is predicted to be taken, the target field in the BTB entry provides the address for the next instruction, except for returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because the return address is call-site dependent and a given function can be called from different call sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, a return address stack (RAS) is typically employed to record return addresses at call-sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' On a function call, the call instruction pushes the return address to RAS, which is later popped by the corresponding return instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The cost of a BTB miss A BTB miss for a branch instruction means that the branch is undetected and the front-end continues to fetch instructions sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Whether or not the sequential path is the correct one depends on the actual direction of the missed branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Unless the missed branch is a conditional branch that is not taken, the sequential path is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' When the wrong path is eventually detected by the core, all the instructions after the branch that missed in the BTB are flushed, fetch is redirected to the branch target and pipeline is filled with correct-path instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB misses are thus highly deleterious to per- formance as they result in a loss of tens of cycles of work and expose the pipeline fill latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB’s role in instruction prefetching Fetch-directed instruction prefetchers are a class of powerful L1-I prefetchers that intrinsically rely on BTB to identify prefetch candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These prefetchers are highly effective and, when coupled with a sufficiently large BTB, outperform the winner of the recently-concluded Instruction Prefetching Championship [1] and approach the performance of an ideal L1-I, as reported by Ishii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Variants of these prefetch- ers have been adopted in commercial products, for example in IBM z15 [45], ARM Neoverse N1 [41] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 2 shows a canonical organization of a fetch-directed instruction prefetcher (FDIP) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As originally proposed, FDIP decouples the branch-prediction unit and the fetch en- gine via the fetch target queue (FTQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This decoupling allows the branch prediction unit to run ahead of the fetch engine and discover prefetch candidates by predicting the control flow far into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' With FDIP, each cycle, the branch prediction unit identifies and predicts branches to anticipate upcoming execution path and inserts corresponding instruction addresses TABLE I: BTB storage cost in Samsung Exynos processors CPU BTB Storage M1/M2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB M3 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8KB M4 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0KB M5 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8KB M6 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB into the FTQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Consequently, the FTQ contains a stream of anticipated instruction addresses to be fetched by the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The prefetch engine scans the FTQ to identify prefetch candidates and issue prefetch requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For FDIP to be effective, the BTB needs to accommodate the branch working set, otherwise frequent BTB misses will cause FDIP to prefetch the wrong path as FTQ will be filled with wrong path instruction addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is one of the key reasons why commercial processors deploy massive BTBs, as also observed by [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These massive BTBs incur astronomical storage overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also, not only the BTB storage overhead is high, it is increasing at a rapid pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, Table I, presents the BTB storage cost in several generations of Sam- sung Exynos processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the table shows, the BTB storage cost nearly doubled in each generation, except between M4 and M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Overall, the storage cost increased nearly six fold (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB to 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB) from M1 to M6, over a period of about eight years [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the instruction footprints of server applications continue to expand, a trend also reflected in Google Web Search workload whose instruction footprint is growing at annualized rate of 27% [25], the BTB sizes and their storage overheads are destined to increase in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, there is an urgent need to investigate storage-effective BTB organizations to combat the front-end bottleneck without necessitating pro- hibitive area budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BRANCH TARGET DISTANCE ANALYSIS The storage cost of branch targets accounts for a major fraction of BTB storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, in a conventional BTB, as depicted in Figure 1, the target field accounts for about 72% (46 of 64 bits) of the total BTB storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We analyze the number of bits required for branch target offsets to assess if storing the offsets, instead of the full or compressed targets, can reduce BTB storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We define the target offset as the n least signif- icant bits of target address, with n being the position of most significant bit that differs among branch PC and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As an example, for the branch PC and target shown in Figure 3, the most significant bit that differs among them is at position five, whereas all bits at positions higher than five are same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Bit position 48 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 9 8 7 6 5 4 3 2 1 Branch PC 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 1 0 1 1 0 1 0 0 0 Branch Target 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 1 0 1 1 1 1 0 0 0 Target Offset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 1 1 0 0 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 3: Branch target offset example 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Fraction of dynamic branches covered Number of bits required for branch target offsets Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 4: Distribution of branch target offsets in different workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, the target offset for this branch PC and target pair is ‘11000’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', the target bits from position 5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also, as our modelled ARM v8 ISA aligns instructions at 4-byte boundaries, the two least significant bits of a target are always zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, we only need to store ‘110’ as offset in the BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' An advantage of defining an offset as n lower order target bits, instead of the numerical distance between branch PC and target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', target - PC), is that the full target can be recovered by simply concatenating the shifted branch PC with the offset retrieved from BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In contrast, using numerical distance as offset would require a 48-bit adder to recover the full target from offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 4 plots the distribution of branch target offsets in the branch working sets of our workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The data includes both conditional and unconditional branches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' hence, it comprehen- sively covers the full branch working set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The X-axis shows the number of bits required to store offsets, while the Y-axis plots the fraction of dynamic branches covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the figure shows, short offsets dominate the distribution with 54% of branches requiring only six bits or fewer for their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A further 22% of branches only require between 7 and 10-bits to represent their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The reason why such a high fraction of offsets is short is that conditional branches dominate the dynamic branch working set, and they tend to have short offsets [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because conditional branches generally guide the control flow only inside a func- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' meanwhile, software engineering principles favor small functions, thus restricting conditional branch target offsets to short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Furthermore, as discussed in Section II, return instructions get their target from RAS, thus they do not need to store any target bits in BTBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, for the purpose of this analysis, we assume 0-bit offsets for return instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Perhaps surprisingly, Figure 4 also shows that very few branches require a large number of bits for their offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Indeed, a meagre 1% of branches requires more than 25 bits for their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The sum of these results indicates that reserving space for the full 46-bit target address results in an appalling under- utilization of BTB storage, since 99% of branches need at most half the number of bits needed to represent the full target address if offsets are used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We gain two key insights from this analysis: Key Insight 1: The targets of most branches lie relatively close in the virtual address space to the branch itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As a result, storing the distance to the target, in the form of an offset from the branch instruction can provide drastic storage savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Key Insight 2: The target offset sizes are unevenly distributed with 0-6 bits, 7-10 bits, and 11-25 bits required to encode the offsets of 54%, 22% and 23% of branches respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, a single size offset field cannot provide storage optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' STATE-OF-THE-ART BTBS AND THEIR LIMITATIONS Prior work has proposed several BTB organizations that aim to reduce the storage cost by compressing branch targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This section presents the most representative BTB organizations and analyzes their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Reduced BTB: Seznec [47] made a critical observation that all branch targets within a page share the same page number and only differ in page offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, storing full target addresses in a BTB results in massive duplication of page numbers and wastage of storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To eliminate this duplication, Seznec proposed Reduced BTB (R-BTB), a variant of which was also used in ITTAGE [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The key innovation of R-BTB is to store a pointer to the page number rather than storing the page number itself in BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 5 presents the logical organization of R-BTB and the composition of its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' R-BTB is composed of two partitions: Main-BTB and Page-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For each branch target, apart from page offset, Main-BTB stores a pointer to the page number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The page number itself is stored in Page-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' If two or more branches have their targets in the same page, their 4 Page number (36 bits) Offset (10 bits) Valid 1 bit Branch type 2 bits Rep_policy 3 bits Tag 12 bits Offset 10 bits Page-BTB Pointer Page Number 36 bits Page BTB Main BTB PC (48 bits) 00 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 5: Reduced BTB Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Main-BTB entries will hold pointer to the same Page-BTB entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the number of pages is significantly smaller than number of branch targets, fewer bits are needed to hold Page- BTB pointers than page numbers themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Consequently, by storing a page number only once in Page-BTB, R-BTB avoids duplication and reduces storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede: PDede [49] is the state-of-the-art BTB organization that comes with three different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 6 depicts the most storage effective and best performing PDede variant, called PDede-Multi Entry Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' It improves over R-BTB in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' First, it reduces the cost of storing page numbers in the Page-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede observes that server applications, due to their large instruction footprints, touch a large number of pages thus increasing Page-BTB storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' They further observe that, as different libraries get dynamically mapped to different locations in address space, the pages tend to form spatial regions, where a region consists of multiple contiguous pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Just like branch targets inside a page share the same page number, the page numbers inside a region share the same region number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To eliminate the duplication of region numbers, as shown in Figure 6 PDede introduces a Region-BTB which stores the region number while Main-BTB stores a pointer to it just like it stores a pointer to page BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Second, for the same-page branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', when the branch and its target are in the same page, PDede does not store page/region numbers as they can be recovered from branch PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede reserves half of the ways in a set associate BTB for same-page branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the ways reserved for same-page branches do not need to store Page-BTB and Region-BTB pointers, as shown in Figure 7, PDede achieves additional storage savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Limitations of the state-of-the-art: Though R-BTB and PDede achieve significant storage sav- ing by avoiding page and region number duplication, they increase BTB complexity by introducing a level of indirection and associative searches in Page- and Region-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These Valid 1 bit Branch type 2 bit Rep_policy 3 bit Tag 12 bits Offset 10 bits Delta bit 1 bit Region-BTB Pointer Page-BTB Pointer Rep_policy 4 bits Page number 16 bits Page BTB Rep_policy 2 bits Region 20 bits Region BTB Main BTB PC >> 12 Target Page 36 bits Region (20 bits) Offset (10 bits) Page (16 bits) PC (48 bits) 00 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 6: PDede BTB Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Valid 1 bit Tag 12 bits Valid 1 bit Tag 12 bits Offset 10 bits Delta bit 1 bit Rep_policy 3 bits Rep_policy 3 bits Branch type 2 bits Branch type 2 bits Offset 10 bits Page-BTB Pointer Region-BTB Pointer Same-page branch entry Different-page branch entry Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 7: Different- and same-page PDede entry composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' complexities lead to increased access latency and power re- quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In addition, the state-of-the-art BTB designs are suboptimal in utilizing the available storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Indirection: As Figures 5 and 6 show, the access to Main- BTB only provides a part of the target address, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', page offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The other parts have to be retrieved from Page-BTB and Region-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also, Page- and Region-BTB cannot be accessed in parallel with Main-BTB because the Main-BTB access pro- vide the pointers to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As a result, the sequential Main-BTB and Page-/Region-BTB accesses increase the overall BTB access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This additional latency either enforces a two- cycle BTB lookup or necessitates a longer clock period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Both of these alternatives are detrimental to performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede does avoid this indirection penalty to some extent because the same-page branches do not need to access Page-/Region-BTB rather they get their page and region number from the branch PC itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, the different-page branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', where branch PC and target address lie in different pages, do need to pay the indirection penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Associative searches: On allocating a new BTB entry, all BTB partitions (Main-BTB, Page-BTB, and Region-BTB) may need to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The replacement policy chooses the entry to be replaced in Main-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, Page- and Region-BTB need to be searched to check if the page/region number for the incoming target is already present or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the page/region number can be present anywhere in the Page/Region-BTB, ITTAGE [48] uses fully-associative searches which increase the power requirements especially when the number of entries grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede (partially) solves this limitation by restricting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='the number of entries where a page number can reside for a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Multiplexer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Meta- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='46 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='BTB-XC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='BTB-X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Metadata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='RAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Branch PC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Hit/miss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Hit/miss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Full Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Target offset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Tag: 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Rep_policy: 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Type: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Valid: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Target offsets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Metadata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='19 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='11 bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 8: BTB-X organization and entry/set composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' given branch target to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, limiting the number of entries increases the likelihood of conflict misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Suboptimal storage utilization: Though R-BTB and PDede significantly improve storage utilization over conventional BTB, they still miss plenty of opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because, in case of R-BTB, it uses a fixed size target representation for all branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', a 10-bit offset plus a fixed sized page- BTB pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In contrast, our analysis of Section III shows a large variance in target offset sizes that naturally makes the single sized organization of R-BTB storage inefficient as it needs to be sized for the largest target offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, as shown by Figure 4, 54% of target offsets fit in six or fewer bits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' however, R-BTB needs to use all 10+ bits for these branches, thus resulting in a high storage under-utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede provides slightly better storage utilization than R-BTB as it has differently sized entries for Same-page and Different- page branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, BTB entries of only two different sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Same-page and Different-page, are not enough to capture the large offset size variance (Figure 4) observed in server applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X BTB-X is a simple and storage-effective BTB organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Building on the insights gained in Section III, it stores target offsets, instead of full targets, to minimize storage require- ments while also accounting for the large variance in target offset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Its microarchitecture and entry/set composition are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X Organization The offset field in BTB-X needs to accommodate the uneven distribution of target offset sizes as observed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In principle, this field should be sized such that the BTB can store the largest target offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, as the largest offset can be nearly as large as the full target, sizing the offset field this way would nearly eliminate the potential storage saving from storing offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' An attractive alternative is to size the offset field such that it can store the majority of offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Looking at Figure 4, an offset field of 25-bits would capture more than 99% of branch target offsets as they requires 25-bits or fewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, there are two major drawbacks to this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' First, it still leads to poor storage utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because, as Figure 4 implies, 54% of branches would waste more than three quarters of offset storage capacity as they require only 6- bits or fewer for their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Another 25% of branches would waste nearly half of the offset storage as their offsets fit in 7- 12 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Second, all branches that need more than 25 bits for their offsets can not fit in the BTB and will always cause BTB misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Though, as there are very few such branches (¡ 1%), their impact is likely to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To minimize the storage under-utilization, we size different ways of a set associative BTB-X to hold different sized target offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A branch is allocated to a way whose offset field is at least as large as the number of bit required to store the target offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We use an 8-way set associative BTB-X and leverage the data in Figure 4 to appropriately size the offset field of each way such that each way covers about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5% dynamically executed branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 4 shows that, on average, 0-, 4-, 5-, 7-, 9-, 11-, 19-, and 25-bit offsets cover about 20%, 36%, 46%, 61%, 72%, 79%, 90%, and 99% dynamic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, we size the 8-ways of BTB-X ways to hold 0-, 4-, 5-, 7- , 9-, 11-, 19-, and 25-bit target offsets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Notice that about 20% of dynamic branches that require 0-bits for their offset are return instructions that read their target from RAS, as discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, way-0 of BTB-X does not feature any storage for target offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Though return instruction do not get their target from BTB, they still need to be allocated BTB entries so that the branch prediction unit can identify them and pick their target from RAS while generating instruction stream to be fetched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X covers 99% of the dynamically executed branches and we employ a very small conventional direct-mapped BTB, called BTB-XC, that stores full target addresses for 6 the remaining 1% branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Reserving a way in BTB-X for such branches would unnecessarily increase the storage requirements as these branches require much fewer entries than the number of sets in BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Indeed, based on our analysis, we size BTB-XC to store 64x fewer entries than BTB-X, i,e, 8x fewer entries than the number of sets in BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Accessing BTB-X BTB-X Lookup: A BTB-X lookup is very similar to a conventional BTB lookup as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' It is accessed with the index bits of a PC and all eight ways are looked up in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also, BTB-XC is looked up in parallel with BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The main difference between a conventional BTB lookup and BTB-X lookup is that a BTB-X lookup provides target offset, rather than full target address, if the lookup hits in way-1 to way-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, target offset needs to be concatenated with branch PC to get the full target address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The number of bits to be concatenated from branch PC depends on the BTB-X way in which the lookup hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, a hit in way-1 provides 4 lower order bits of target while the rest needs to be concatenated from branch PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, a hit in way-0 implies that the full target is in RAS, while a hit in BTB-XC provides the full target address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X Allocation: As with any existing BTB organization, BTB-X entries are allocated (or updated) as branch instruc- tions retire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The number of bits required to represent a branch target offset determines the way(s) where a branch can be allocated an entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, return instructions can be allocated entries in any of the ways, based on replacement policy’s decision, as they have no offset and can fit in all ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Other branches have fewer ways where they can be allocated entries that are determined by the minimum number of bits required to store their offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, if a branch requires 20 bits for its target offset, it cannot be allocated in way-0 to way-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X uses a slightly modified least recently used (LRU) replacement policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Concretely, we modify it to compare the LRU counters of only the entries that can accommodate the target offset and replace the one that is least recently used among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' All other aspect of LRU, such as counter updates, stay exactly the same as in baseline policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Methodology We use ChampSim [4] to evaluate the efficacy of BTB-X on server and client workload traces provided by Qualcomm for the first Instruction Prefetching Championship (IPC-1) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We warm up microarchitectural structures for 50M instructions and collect statistics over the next 50M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The microarchitectural parameters for the modeled processor, resembling Intel Sunny Cove [5], are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We improved two important aspects of Champsim to eval- uate the baseline, state-of-the-art, and proposed BTB orga- nizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' First, being a trace-driven simulator, Champsim detects branches by consulting the information available in the traces, rather than looking up a BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This essentially TABLE II: Microarchitectural parameters Parameter Value Fetch 6-wide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 128-instruction FTQ Branch Predictor Hashed Perceptron Return address stack 64 entries Scheduler 128 entries Re-order buffer 352 entries Load queue 128 entries Store queue 72 entries L1-I 32 KB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 8-way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 4 cycle latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 8 MSHRs L1-D 48 KB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 12-way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 5 cycle latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 16 MSHRs L2 512 KB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 8-way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 14/15 cycle latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 32 MSHRs LLC 2MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 16-way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 34/35 cycle latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 64 MSHRs TABLE III: BTB-X storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The numbers in parentheses are for BTB-XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Entries Sets Set size Storage 256(4) 32(4) 224(64)-bits 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB 512(8) 64(8) 224(64)-bits 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8KB 1K(16) 128(16) 224(64)-bits 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6KB 2K(32) 256(32) 224(64)-bits 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB 4K(64) 512(64) 224(64)-bits 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 8K(128) 1024(128) 224(64)-bits 29KB 16K(256) 2048(256) 224(64)-bits 58KB translates to Champsim using an ideal BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, we first implement a realistic conventional BTB (Conv-BTB), presented in Section II, in Champsim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Second, Champsim re- solves all branches in execute stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', branch mispredictions are detected and the fetch is resteered to correct path only when a mispredicted branch instruction reaches the execute stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Such branch resolution overestimates the misprediction penalty of unconditional direct branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because such branches can be resolved in the decode stage (hence, fetch can be resteered sooner) as they are always taken, thus the PC of the next instruction can be compared to the target encoded in the branch instruction to detect mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, taken conditional branches that miss in BTB but are correctly predicted by the direction predictor, can also be resolved in the decode stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To do so, the fetch stage passes the direction prediction for all instructions, despite BTB hit/miss, to decode stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' If decode identifies a branch that missed in the BTB but predicted taken by the direction predictor, it resteers the fetch to the target encoded in the branch instruction, thus reducing BTB miss penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Given that the direction predictors are highly accurate, this optimization reduces average BTB miss penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Overall, we improve branch resolution so that the unconditional direct branches and the taken conditional branches that miss in BTB are resolved in the decode stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Finally, BTB is updated at commit stage by only the taken branches (both conditional and unconditional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Storage breakdown We first assess the number of branches different BTB orga- nizations (Conv-BTB, PDede, and BTB-X) can accommodate 7 TABLE IV: Number of branches in different BTB organizations at various storage budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Storage BTB-X + BTB-XC PDede Conv-BTB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8KB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6KB 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 29KB 58KB Branches 256 + 4 512 + 8 1K + 16 2K + 32 4K + 64 8K + 128 16K + 256 Page-BTB budget Main-BTB budget Entry Size Branches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='078KB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='817KB 32-bits 210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='156KB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='645KB 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5-bits 415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='312KB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3KB 33-bits 820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='625KB 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6KB 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5-bits 1617 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2KB 34-bits 3190 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5-bits 6292 5KB 53KB 35-bits 12405 Entry Size Branches 64-bits 116 64-bits 232 64-bits 464 64-bits 928 64-bits 1856 64-bits 3712 64-bits 7424 in a given storage budget compared to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We use storage budgets required for storing 256, 512, 1K, 2K, 4K, 8K, and 16K branches in BTB-X as presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our calculations assume a 48-bit virtual address space and BTB-X entry compositions presented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To double the number of entries in BTB-X, we double the number of sets while keeping the associativity same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Notice that Table III presents set size instead of entry size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because BTB-X features different sized entries in different ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' however, the set size remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Table IV presents the number of branches the different BTB organizations can track at different storage budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede distributes the overall BTB storage budget among its Main- BTB, Page-BTB, and Region-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We follow the distribution used by its inventors [49] to allocate the budget among different PDede BTBs as shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Accordingly, for 29KB storage budget, we configure PDede to use 1K Page- BTB entries and about 6K Main-BTB entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' While halving the storage budget to lower values, we halve the number of entries in the Main-BTB as well as the Page-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Halving the number of Page-BTB entries reduces the number of bits required to store Page-BTB pointer in the Main-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, the Main-BTB entry size reduces with the reduction in storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, we use four Region-BTB entries across all storage budgets, so Region-BTB requires a fixed storage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0107KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also recall that PDede reserves half of the ways in a set for same-page branches while the other half can store both same-page and different-page branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Therefore, its entries are of two different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The PDede entry size shown in Table IV is the average of two sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the table shows BTB-X stores significantly more branches than any other BTB organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Concretely, it stores 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more branches than a conventional BTB organiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Compared to PDede, BTB-X stores 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more branches at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB storage budget and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='34x more branches at 58KB storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X’s advantage over PDede increases with storage budget because PDede entries require more bits at higher budgets to accommodate larger Page-BTB pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB MPKI To understand the advantage of higher BTB-X branch density, we measure misses per 1000 instructions (MPKI) that different BTB organizations incur on client and server workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Since BTB misses for not-taken branches do not hurt performance, we only consider the BTB misses for taken branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For this analysis, we assume a BTB storage budget of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB that corresponds to 4160-, 3190-, and 1856-entries in BTB-X, PDede and Conv-BTB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The results are presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the figure shows, server workloads experience signifi- cantly higher MPKI compared to client workload due their massive instruction and branch footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The figure also shows that BTB-X provides a much lower MPKI compared to both conventional BTB and PDede especially on server workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Concretely, on average, conventional BTB incurs 25 MPKI on server workload as it stores the least amount of branches among the three organization for a given storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede is able to lower the MPKI to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 while BTB- X brings it further down to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The advantage of BTB-X over other organizations is particularly evident on very high MPKI workloads, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', server 023 to server 035, where it provides much lower MPKI compared to conventional BTB and PDede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Performance To assess how the reduced MPKI translates to performance, we compare the performance of the three BTB organizations on client and server workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Recall from Section II that a larger BTB delivers two distinct benefits: 1) it reduces the incidence of pipeline flushes by detecting branches in the upcoming control flow and 2) it facilitates instruction prefetching when coupled with FDIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, we compare the performance gains achieved by the three BTB organizations by evaluating them with FDIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 10 presents the performance gains obtained on server and client traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The results are normalized to the perfor- mance of the Conv-BTB without any instruction prefetching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The figure shows three bars for each workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The first bar presents performance gain achieved by Conv-BTB when copuled with FDIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The second and third bars present the performance gains achieved by PDede and BTB-X respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The PDede and BTB-X bars divide the performance gain into contributions from fewer pipeline flushes and from better instruction prefetching stemming from capturing more branches in the BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Looking at the overall performance gain with instruction prefetcher (FDIP), the figure shows that BTB-X provides a geometric mean gain of 39% over baseline on server workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In comparison, PDede and Conv-BTB deliver a performance gain of only 33% and 24% on these workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Looking at individual workloads, BTB-X comprehensively outperforms PDede and Conv-BTB on server 023 to sever 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, on server 032, BTB-X provides 83% speedup over baseline whereas PDede and Conv-BTB achieve only 60% and 32% performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because the branch ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Client workloads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Server workloads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Avg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='BTB misses per 1000 instructions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Conv-BTB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='PDede ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Client workloads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Server workloads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Gmean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Performance gain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Gain from fewer flushes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Gain from L1-I prefetching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 10: Performance gain obtained by conventional BTB (with FDIP), PDede and BTB-X (with and without FDIP) over the conventional BTB without FDIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The three bars for each workload correspond to Conv-BTB, PDede, and BTB-X respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' working set of these workloads starts to fit in BTB-X due to its higher branch capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As a result, BTB MPKI lowers which not only reduces pipeline flushes and but also keeps FDIP on correct prefetch path for longer intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Looking at the results without instruction prefetcher, Fig- ure 10 shows that BTB-X provides 13% performance gain over the baseline Conv-BTB whereas PDede is achieves 8% gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' On individual workloads, BTB-X achieves significantly high gain over Conv-BTB and PDede on workloads from server 23 to server 32 even without FDIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 10 also shows the FDIP by itself performs better with more number of BTB entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, on server 32 FDIP with Conv-BTB provides 32% performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' With PDede, the performance gain from prefetching increases to 42% and with BTB-X it further increases to 51%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These results show that by accommodating more branches in a given storage budget, BTB-X not only reduces pipeline flushes but also improves instruction prefetching, both lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' TABLE V: Energy requirements of different BTB designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB Access Type Energy #Accesses Energy (Per access) (Total) Conv-BTB Read 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2pJ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='60E+08 2122µJ Write 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2pJ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='36E+06 110µJ Total Energy 2232µJ PDede Main-BTB Read 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4pJ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24E+08 1047µJ Main-BTB Write 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5pJ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='74E+05 7µJ Page-BTB Read 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9pJ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='01E+06 2µJ Page-BTB Write 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8pJ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='04E+04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='02µJ Page-BTB Search 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2pJ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='14E+05 2µJ Total Energy 1058µJ BTB-X Read 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5pJ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='16E+08 994µJ Write 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4pJ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='03E+05 5µJ Total Energy 999µJ Finally, Figure 10 shows that all three BTB organizations perform similar on client workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because their branch working sets mostly fit in the baseline Conv-BTB and the additional entries in PDede and BTB-X do not bring much performance benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='00 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6KB 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 29KB 58KB Performance gain Conv-BTB Pdede BTB-X (a) Server workloads 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8KB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6KB 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB 29KB 58KB Performance gain Conv-BTB Pdede BTB-X (b) Client workloads Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 11: Performance gains for conventional BTB, PDede, and BTB-X on (a) server and (b) client workloads over a conventional BTB with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' X-axis label is storage requirements of 256-, 512-, 1K-, 2K-, 4K-, 8K-, and 16K-entry BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Energy and delay analysis We use Cacti 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0 [35] to analyze the energy requirements and access latencies of Conv-BTB, PDede, and BTB-X at 22 nm, which is the most recent technology node supported by Cacti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For this analysis we assume the same storage budget, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB, as used in Section VI-D for the performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Energy requirements: Table V shows the per access read and write energy requirements of different BTB designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the table shows, BTB-X and PDede’s Main-BTB incur very similar per access read and write energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, in addition to Main-BTB, PDede also needs to access Page- BTB for different-page branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', the branches that have their targets in a different page than the branches themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, the Page-BTB needs to be searched on a BTB write to check if the target page number is already in Page-BTB or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Consequently, PDede’s per access read and write energy for different page branches reaches 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3 pico Joules (pJ) and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 pJ, respectively, compared to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5pJ and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 pJ of BTB- X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede also features a Region-BTB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' however, its energy requirements are negligible and, thus, not shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Finally, Conv-BTB’s per access energy cost is significantly higher than BTB-X as its each read and write access requires 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2pJ and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2pJ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Table V also shows the number of read/write accesses, aver- aged across the workloads, and the total energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Despite very similar per access energy cost, PDede’ Main- BTB consumes considerably higher energy than BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because PDede often goes on the wrong execution path due to its higher MPKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These additional wrong path BTB accesses, reflected in higher BTB reads in Table V, result in higher energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, PDede needs to handle more BTB writes than BTB-X because it holds fewer branches, which results in frequent replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, the total energy consumption of PDede reaches 1058µJ compared to 999µJ of BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Finally, the energy requirements of Conv-BTB are significantly higher, 2232µJ, than BTB-X because of higher per access energy and higher number of total accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Overall, this analysis shows that BTB-X not only delivers better performance than PDede but also consumes less energy, thus providing much better energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Access Latency: Our analysis shows that the Conv-BTB requires about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='36ns to complete an access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As discussed in Section IV-B, PDede’s access latency is the sum of Main- BTB and Page-BTB access latencies as these two structures are accessed sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our analysis shows that the Main- BTB and Page-BTB accesses require 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='34ns and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='13ns, re- spectively, thus resulting in an overall PDede access latency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='47ns which is considerably higher than Conv-BTB latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To address this, PDede employs multi-cycle BTB accesses: the Main-BTB is accessed in the first cycle, and the Page-BTB is accessed in the next cycle only if the branch is predicted to be taken and it’s target is in a different page than the branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, the same page branches need one cycle and the taken different page branches need two cycles to get their target address from PDede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Finally, our analysis shows that a BTB-X access takes only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='33ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In summary, this analysis shows that BTB-X provides better storage efficiency without any adverse effects on the access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Performance variation with BTB storage budget To further understand the performance advantage of BTB-X over PDede and Conv-BTB, we compare their performances across different storage budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Figure 11 presents the per- formance gains obtained on server and client workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The results are normalized to the performance of Conv-BTB with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Instruction prefetching is enabled in all designs including baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As the figure shows, on server workloads, BTB-X pro- vides significantly higher performance than the Conv-BTB and PDede for equal storage budgets of up to 29KB and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The performance advantage of BTB-X is 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Fraction of dynamic branches covered Number of bits required for branch target offsets IPC-1 traces avg CVP-1 traces avg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 12: Target offset distribution in CVP-1 and IPC traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' pronounced on server traces whose large instruction footprints pressure the BTB and L1-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For instance, BTB-X provides 35% performance gain over the baseline compared to 29% and 20% of PDede and Conv-BTB respectively at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' At large BTB storage budgets, the branch working sets of many workloads start to fit in the available BTB capacity, at which point the performance gap between BTB-X and the other two designs diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Also, the performance gap between the three BTB organizations levels off earlier on client trace due to their smaller instruction working sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' A key take-away from this figure is that BTB-X outperforms the conventional BTB even when it is given just half the storage budget of its conventional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, in Figure 11a, the Conv-BTB improves performance by 20% with a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5KB budget whereas BTB-X provides a 24% im- provement with just 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='25KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The reason for this phenomenon is that BTB-X accommodates 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more entries than Conv- BTB of equal storage budget;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' thus, halving BTB-X’s budget still gives a slight capacity advantage over Conv-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Analyzing target offset distribution in more workloads We study the target offset distribution in 750+ Qualcomm server traces that were provided for the first Championship Value Prediction(CVP-1) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The results, presented in Fig- ure 12, show that their offset distribution is very similar to the distribution in IPC-1 traces presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This study confirms that such an offset distribution is a consequence of how the applications are written and the resulting control-flow behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As discussed in Section III, such offset distribution stems from the fact that the conditional branches dominate dynamic branch working set and they tend to have short offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because conditional branches guide the control- flow inside functions, and software engineering principles favor small functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Consequently, short offsets dominate the branch offset distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In addition to CVP-1 traces, we analyze five more server ap- plications - Wordpress [53], Mediawiki [52], and Drupal [51] from Facebook’s HHVM OSS-performance benchmarks [8], Kafka [50] from Java DaCapo [14], and Finagle-HTTP [7] from Java Renaissance [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Further, these applications are compiled to x86 (CVP-1 and IPC-1 traces are compiled to 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Fraction of dynamic branches covered Number of bits required for branch target offsets Wordpress Mediawiki Kafka Finagle_http Drupal IPC-1 traces avg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 13: Target offset distribution in x86 compiled server applications and Arm64 IPC traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Arm64) which also enables us to assess the impact of ISA on target offset distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The results presented in Figure 13 show that the offset distribution in these applications is also very similar to that in IPC-1 traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The only difference is that x86 traces require slightly larger offsets (1 or 2-bits more) to achieve a similar dynamic branch coverage as the Arm64 (CVP-1 and IPC-1) traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' For example, 6-bit offsets cover about 54% branches in Arm64 traces, whereas x86 offsets need 8-bits to achieve 58% branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because x86 offsets specify the distance between branch PC and target in number of bytes because x86 instruction are variable size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In contrast, Arm64 offsets specify this distance in number of instructions because all instructions are 4-bytes, thus saving 2 offset bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As BTB-X needs to store slightly larger offsets for x86 than Arm64, we reassess its storage advantage over PDede and Conv-BTB for x86 architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' As each way in 8-way BTB-X needs to cover about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5% of branches, we size its ways to store offset of 0-, 5-, 6-, 7-, 9-, 12-, 20-, and 27-bits based on the offset distribution in x86 applications shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, each set needs 86-bits for offsets compared to 80-bit in Arm64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Consequently, BTB-X’s storage advantage is slightly lower for x86 than Arm64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, BTB-X still stores 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='18x more branches than Conv-BTB for x86 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x for Arm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Compared to PDede, BTB-X stores 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='21x more branches (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x for Arm64) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='9KB storage budget and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='31x more branches (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='34x for Arm64) at 58KB storage budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' (Section VI-B presents this analysis of Arm64 traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=') VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' RELATED WORK Mitigating BTB misses: BTB was first disclosed by Losq [36] and was further expanded by Lee et al [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Since BTB lies on the critical path for instruction delivery, there has been several proposals to increase its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Instead of accessing BTB with the PC of each individual instruction, Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' [54] proposed to access it with basic-block address and store not only the target but also the fall-through address in the BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In case the branch is predicted to be not taken, the fall- through address is used, after fetching the current basic-block, as the next PC for both instruction fetch as well as for the 11 next BTB access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The advantage of such a BTB organization over the conventional BTB organization is that it reduces BTB bandwidth and power requirements as a single access provides the next control flow divergence point, whereas the conventional organization requires as many accesses as the number of instructions until the next branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Whereas the initial proposal on basic-block-based BTB [54] stores full fall- through address, the later work [44] proposed to store the delta between two sequential basic-block addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Fagin [18] proposed to use the BTB storage more effectively by storing only the partial tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' To further amortize the tag storage cost, some designs proposed to share a BTB entry among multiple branches that reside in the same cache block [2], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Though these BTB designs aim to improve different aspects of BTB management, they all share a common trait, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=', they store full target addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Thus, the key idea of BTB-X can be applied to all of these BTB designs to reduce their target storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Prior work [10], [11], [23], [47]–[49] has also explored mechanisms to reduce the storage cost of branch targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Seznec [47], [48] proposed to break the target address into page number and offset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' and store a pointer to the page number, along with the page offset, in the BTB while the page number itself is stored in a separate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' It reduces the storage cost as a pointer to page number is smaller than the page number itself, and the page number for all the targets in a page is stored only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Hoogerbrugge [23] proposed to size some of the entries in a set for storing small target offsets, thus reducing BTB storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The state-of-the-art BTB design, PDede [49], combines these two ideas to address their individual limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Con- cretely, Sezenc’s design is sub-optimal for same-page branches as it unnecessarily stores (pointer to) their target page number even though it is same as the page number of their branch PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' In contrast, Hoogerbrugge’s design is sub-optimal for inter-page branches as it stores their full targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Inspired from Hoogerbrugge’s design, PDede sizes some entries in a set to store same-page branch targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' and similar to Seznec’s design, for inter-page branches, it stores pointers to page numbers instead of page numbers themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' PDede further reduces the inter-page target storage cost by dividing the page number into page- and region-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' However, as it is based on Seznec’s design, it also has to pay the addition latency cost of indi- rection between main-BTB and the page-/region-BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Micro BTB [22], proposes a flexible BTB entry structure where each entry can store either one branch, if its offset is large, or two branches if their offsets are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We show that all these designs are sub-optimal in exploiting the storage optimization opportunity presented by the uneven branch offset distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' BTB-X not only captures this opportunity but also avoids the BTB indirection of the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Apart from optimizing BTB organization, prior work [15], [16], [28], [31], [32] has also explored BTB prefilling/prefetch- ing to mitigate BTB misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The state-of-the-art in BTB prefetching is a profile guided software prefetcher, called Twig [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' It analyzes an application’s execution profile to identify critical BTB misses and then injects software prefetch instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' The prefetch instruction takes compressed branch PC and target as operands and its execution fills this informa- tion in BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These prefetching techniques are complementary to BTB organization and, thus, can be used along with BTB-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Mitigating L1-I misses: As L1-I misses continue to be a major performance limiter in server applications [12], [25], [46], prior work has proposed both hardware and software mechanisms to mitigate L1-I misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' On the hardware side, state-of-the-art temporal stream prefetchers [19], [20] record the L1-I miss/access history and replay it to discover prefetch candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' While such prefetchers are highly effective, their huge metadata storage cost renders them impractical despite recent attempts to address this weakness [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Fetch- directed prefetchers use in-core structures (BTB and branch direction predictor) to run ahead of the fetch unit to find prefetch candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' While the early work [43] focused on L1-I prefetching only, the state-of-the-art fetch-directed prefetchers [31], [32] also prefill into the BTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Several purely-software based approaches to instruction prefetching and improving the L1-I capacity has also been proposed [9], [13], [17], [29], [34], [37]–[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' These methods use data from application profiling to perform either compile- time, link-time or post-link time optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Since these methods are software-only they will benefit from the increased BTB capacity provided by the BTB-X organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' CONCLUSION The multi-megabyte instruction footprints of contemporary server applications cause frequent BTB and L1-I misses, which have become major performance limiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Because BTB capacity greatly affects front-end performance by dictating pipeline flush rate and the efficacy of fetch-directed instruction prefetching, commercial products allocate tens to hundreds of KBs of storage to BTBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We observe that the single largest contributor to the BTB storage cost is the cost of storing branch target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We further observe that BTB storage cost can be drastically reduced by storing target offsets instead of full or even compressed targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' This is because targets of most branches lie relatively close to the branches themselves and our analysis shows that more than 99% of offsets can be represented with at most half the bits required to store the full targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Based on these observations, we propose a storage- effective BTB organization, called BTB-X, that stores target offsets in place of target address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Furthermore, BTB-X, an 8-way set associative BTB, uses differently sized ways with each storing offsets of a different length, thus accounting for the uneven distribution of offset lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Overall, BTB- X is capable of storing about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='24x more branches than a conventional BTB and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='3x more branches than a state-of- the-art BTB organization within the same storage budget.' metadata={'source': 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Symposium on Microarchitecture, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' MICRO 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Washington, DC, USA: IEEE Computer Society Press, 1992, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' 129–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' APPENDIX A ARTIFACT APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Abstract We implement BTB-X in Champsim simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Our artifacts provide the following: 1) BTB-X implementation in Champ- sim, 2) Link to workload traces, 3) Scripts for generating configuration files, launching simulations, and collecting re- sults, and 4) Excel file for plotting the most important results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' We identify three key results for artifact evaluation: a) Branch Target Offset distribution (Figure 4), b) BTB MPKI reduction (Figure 9), and c) Performance improvement (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Meta-information Compilation: Tested with GCC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' It should also work with other recent GCC versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Code/Workloads: Download code/workloads from the provided link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Experiments: Modify the provided scripts (as described below) to run simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Metrics: IPC, BTB MPKI, Branch Target Offset distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Time needed to run experiments: Less than 30 minutes when running all traces in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Plotting graphs: Excel file, BTBX artifact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='xlsx, is provided to plot graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Access to artifacts Code: Download BTB-X implementation from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content=' Workloads: The workloads can be downloaded from https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='com/file/d/1qs8t8-YWc7lLoYbjbH d3lf1xdoYBznf/view?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfcweU/content/2301.03899v1.pdf'} +page_content='usp=sharing Place the workloads in 1𝜇𝑚), which can be easily +made by standard photo-lithography techniques, enabling large-area fabrication and scalable +manufacturing. Our demonstration provides a new technical route to solve the long-standing issue +of robust and portable thermal imaging with simultaneous spectral and polarimetric resolution. +By utilizing meta-optical designs and advanced computational imaging methods, our approach + +a +Fully-Passive +Quasi-Active +Blackbody Rad. (a.u.) +Peak wavelengths for +Trans. +room-temperature objects +& human body +T=300K +SWIR +0.5 +0.5 +Atm. +This Work: +Long-wave Infrared +Atmospheric Transparency Windows +0 +0 +0 +5 +10 +15 +Wavelength(um) +b +d +Mosaic Sensor +Filter Wheel +Interferometry +Not Feasible +Limited Spectral +Limited FOV +in LWIR +Resolution +Temperature Estimation +Material Estimation +This Work: +Linear +e +Polarizer +Object Detection +Low-SWaP, Large-FOv +Spectro-Polarimetric +LWIR Imaging +M3 +M2 +Spinning +LWIR Sensor +Control +Compressive +M1 +Sensing +Raw Signals +Spectro-Polarimetric +Meta-dataoffers a promising platform for next-generation high-contrast LWIR thermal imaging. +The architecture of our spinning-metasurface based spectro-polarimetric imaging system +is depicted in Fig. 1e. It comprises of a broadband linear polarizer, three anisotropic and +dispersive metasurfaces, and an LWIR imaging sensor. The polarizer is utilized to polarize the +incoming thermal radiation signals, and the metasurfaces are utilized to realize spectral filtering. +We design the metasurfaces with high anisotropy to produce distinct spectral responses for +orthogonal polarizations. Additionally, the metasurfaces’ dispersion rotates different wavelengths +of light to varying polarization orientations. By using the metasurfaces in tandem and axially +spinning the polarizer and metasurfaces to different angles, we obtain tunable transmission +spectra that sample the incident thermal radiation in its spectral and polarimetric channels. We +then reconstruct unknown spectra of imaging targets using compressive sensing and dictionary +learning algorithms. Dictionary learning generates a set of basis functions that represent the +unknown spectra in a sparse format [20]. Compressed sensing enables accurate reconstruction of +the sparse spectra from limited number of measurements [21]. Combining these two techniques +enables accurate and stable spectral reconstruction in the presence of noise and measurement +errors [22]. The four-dimensional spectro-polarimetric data generated by our system offers a +wealth of physical information about an imaging target, making it a valuable tool for physics- +driven machine vision [4,23], facilitating various tasks such as object detection and semantic +segmentation [24,25]. +2. +Design of Spinning Metasurfaces +To quantitatively describe the mechanism of the spinning-angle-controlled transmission spectra, +we represent the spectro-polarimetric response of the metasurfaces using Jones matrices. +Assuming that the transmission axis (x-axis) of the input linear polarizer is at 0 degrees, +the Jones matrix 𝐽𝑖 of a metasurface i with the x-axis at a spinning angle 𝜃𝑖 can be expressed as: +𝐽𝑖(𝜃𝑖, 𝜆) = 𝑅(−𝜃𝑖) · 𝐽𝑀𝑖 (𝜆) · 𝑅(𝜃𝑖) += +������ +𝑐𝑜𝑠(𝜃𝑖) +−𝑠𝑖𝑛(𝜃𝑖) +𝑠𝑖𝑛(𝜃𝑖) +𝑐𝑜𝑠(𝜃𝑖) +������ +������ +𝑡𝑖𝑝(𝜆) +0 +0 +𝑡𝑖𝑠(𝜆) +������ +������ +𝑐𝑜𝑠(𝜃𝑖) +𝑠𝑖𝑛(𝜃𝑖) +−𝑠𝑖𝑛(𝜃𝑖) +𝑐𝑜𝑠(𝜃𝑖) +������ +(1) +where R is the rotation matrix, 𝐽𝑀𝑖 contains the anisotropic transmission of the metasurface 𝑡𝑖𝑝 +and 𝑡𝑖𝑠 along the two principle axes p and s. Then, the Jones matrix of the three-metasurface +assembly is given by: +𝐽 (𝜃1, 𝜃2, 𝜃3, 𝜆) = 𝐽1 (𝜃1, 𝜆) · 𝐽2 (𝜃2, 𝜆) · 𝐽3 (𝜃3, 𝜆) · +(2) +Thus, the total transmission spectrum of the three spinning metasurfaces strongly depends +on the spinning-angle combinations Θ = (𝜃1, 𝜃2, 𝜃3) when the constituted metasurfaces are +strongly anisotropic and dispersive, i.e. 𝑡𝑖𝑝(𝜆) ≠ 𝑡𝑖𝑠(𝜆) (see Supplementary Materials for the +detailed analysis). We note that large differences between the spectral responses of the three +metasurfaces (M1, M2, M3) are also introduced to minimize the correlations between the +generated spectra, which can significantly improve the spectral reconstruction performance [26]. +We emphasize that our design generates a large set of distinct transmission spectra with only +three metasurfaces, while the total number of spectra in traditional mosaic array is limited to the +number of metasurfaces/filters used [12–18]. +Accordingly, we design the metasurfaces and experimentally achieve three key characteristics +for optimized spectro-polarimetric imaging performance: 1) Strong anisotropy and dispersion for +efficient wavelength demultiplexing; 2) High transmission and low self-emission for high signal +to noise ratio (SNR); 3) Small angular dependence for a large FOV. The unit cell of the three +designed metasurfaces are shown in Fig. 2 a-c. Strong dispersive anisotropy of the transmission + +Fig. 2. Design and characterization of the spinning metasurfaces. (a-c) Schematics of +the three different metasurface devices. (d) A scanning electron microscope (SEM) +image of a fabricated metasurface. Inset: an optical image of a 1-inch-diameter device +used for the imaging experiments, highlighting the large-area uniformity. (e) The +measured polarized transmission spectra (𝑡𝑖𝑝 and 𝑡𝑖𝑠) of the three metasurfaces (M1 +- M3) are depicted, displaying strong anisotropy and distinctive dispersion. (f) The +tunable transmission spectra controlled by the spinning angle combinations of the three +metasurfaces. Inset: an optical image of the motorized rotatory mount. The overall size +of the spinning metasurface module is smaller than 10 cm x 10 cm x 10 cm, making it +a promising platform for next-generation high-contrast LWIR thermal imaging. (g) The +normalized spatial transmittance of the module at two representative spinning angle +combinations Θ = (0◦, 0◦, 0◦) (top) and Θ = (90◦, 90◦, 90◦) (bottom). The red circles +correspond to a transmittance of 0.5 and the field of view of the imaging system is +estimated to be 25◦. +spectra can be observed in Fig. 2e. Additionally, we emphasize that large-area devices are +generally required for imaging applications to ensure sufficient numerical aperture. All the +metasurfaces designed here have feature sizes larger than 1 𝜇𝑚. Large-area devices (25.4mm +in diameter) with high structural quality and uniformity (Fig. 2d) can be rapidly fabricated by +standard photo-lithography techniques, enabling scalable manufacturing for practical applications. +This is in strong contrast to recent works on miniaturized spectrometers [27–35], where the +device footprint is on the micrometer scale and thus not suitable for imaging applications. +The tunable transmission spectra produced by our spinning metasurfaces are shown in Fig. 2f. +The distinct spectra are a result of the tuned spinning-angle combination Θ. We integrate +the three fabricated metasurfaces tandemly via compact rotatory mounts to independently +control the rotation of each metasurface (Inset of Fig. 2g). We also optimize the spinning- +angle combinations of the three spinning metasurfaces using genetic algorithms to generate +largely uncorrelated transmission spectra for optimal spectral reconstruction performance (see +Supplementary Materials for details). Additionally, we note that increasing the number of +metasurfaces can further improve the spectral resolution, but simultaneously reduces the SNR as +the peak transmissions of the LWIR devices are limited to around 0.6. However, our method has +the potential to scale up into the hyperspectral regime by adding more high-transmission LWIR +metasurfaces. + +d +b +c +a +d = 25.4 mm +ns +M1 +M2 +M3 +e +g +0.2 +X +0.6 +Trans. +0.4 +Spinning +- 25 +0.15 +0.2 +人 +M1 +Transmittance +0.8 +Transmittance +0.6 +(0.".0,.0) = 0 +0.6 +0.1 +0.4 +0.4 +0.2 +M2 +0.6 +M3 +0.05 +0.2 +0.4 +0 +0.2 +(.06..06--.06) = 0 +0 +X +8 +9 +10 +11 +12 +13 +14 +8 +10 +12 +14 +Wavelength (um) +Wavelength (μm)We also evaluate the FOV of our imaging module by integrating it with an LWIR thermal +camera and capturing images of a large area uniform blackbody. To determine the spatial +transmission efficiency, we normalize the signal counts of each pixel by the counts at the center +of the images. We also define the angular range with transmittance above 0.5 as the effective +FOV of a system. As seen in Figure 2g, our spinning metasurface module has an FOV of around +25 degrees, which is significantly larger than what can be achieved with interferometer-based +spectral imagers. +3. +Spectral Reconstruction +Fig. 3. Schematic of the spectral reconstruction process. The measured raw signal +(a) can be expressed by the pre-calibrated spectral response function (b) of the +imaging system multiplied by the spectrum of an imaging target (c). For the spectral +reconstrcution, the unknown spectrum 𝑃𝜆 is projected onto a sparse representation +basis 𝐷𝑘𝜆 using dictionary learning (c and d). This sparse representation 𝜙𝑘 is +then used for compressive sensing based reconstruction (e). The use of compressive +sensing and dictionary learning in the reconstruction process significantly improves the +reconstruction accuracy, making the spinning-metasurface-based spectro-polarimetric +imaging more robust against noise and measurement errors. +To extract the unknown spectro-polarimetric properties of various imaging targets, we use a +combination of dictionary learning and compressive sensing algorithms in the reconstruction +process. The tunable transmission spectra produced by our spinning metasurfaces (shown in Fig. +2g) are not narrowband, which means that the collected raw signals at different spinning-angle +combinations Θ do not directly reflect the spectral radiance at different wavelengths. Instead, +the collected signal 𝐼(Θ) at each pixel can be described as an integral of the spectral response +function 𝑅(Θ, 𝜆) multiplied by the ground truth spectrum 𝑃(𝜆) that we wish to obtain, i.e. +𝐼(Θ) = +∫ 𝜆max +𝜆min 𝑅(Θ, 𝜆)𝑃(𝜆)d𝜆. To solve for this equation, we discretize the spectral range of +interest and express it in a tensor form as shown in Eq. 3: +𝐼Θ = 𝑅Θ𝜆𝑃𝜆 +(3) + +b +Wavelength 入 +a +Dictionary Learning +Counts +Response +R +Counts +p +R +@1 +Sparse +Counts +R +Spinning Angle O +---- +Re1,入i +R01,入2 +R01,入 +e +10 +Spinning Angles D +R02,入1 +R02,入n +R02,入2 +102 +Ren,入1 +Wavelength 入 +Compressive +Measured +Unknown +Pre-calibrated Spectral Response Function +Raw Signal +Sensing +SpectrumWe emphasize that directly solving Eq. 3 does not produce accurate spectral reconstructions. +In theory, we can use measured signals 𝐼Θ and the pre-calibrated response function 𝑅𝜃𝜆 to +determine unknown spectra 𝑃𝜆 at each pixel of a scene. However, in practice, two limitations +impede the performance of spectral reconstruction: the problem becomes underdetermined when +there are many discretized wavelength bands, and measurement noise affects both 𝐼Θ and 𝑅Θ𝜆, +making the direct reconstruction method unstable and the results inaccurate. +To improve the accuracy and stability of spectral reconstruction, we use compressed sensing +and dictionary learning algorithms to solve Eq. 3. Specifically, we first use dictionary learning to +create a dictionary of basis functions 𝐷. These functions can represent any thermal radiation +spectrum in the space of spectra we are studying. We then project the unknown spectrum 𝑃𝜆 as a +linear combination of the basis functions in the dictionary (Fig. 3 b and c). We have, +𝑃𝜆 = 𝐷𝜆𝑘𝜙𝑘 +(4) +where 𝜙𝑘 is known as a sparse coding of the spectrum 𝑃𝜆. With this sparse representation, the +spectral reconstruction problem can be solved by first obtaining 𝜙recon: +𝜙recon = arg min +𝜙𝑘 ∥𝜙𝑘 ∥1 +s.t. ∥𝐼Θ − 𝑅Θ𝜆𝑃𝜆∥2 = +��𝐼𝑝 − 𝐴Θ𝑘𝜙𝑘 +�� +2 < 𝜖 +(5) +where 𝐴Θ𝑘 = 𝑅Θ𝜆𝐷𝜆𝑘, and 𝜖 is the residual error. Finally, the spectra 𝑃𝜆 at each pixel of a scene +is reconstructed by +𝑃recon = 𝐷𝜆𝑘 𝜙recon +(6) +Our reconstruction method significantly improves the reconstruction accuracy, making the +spinning-metasurface-based spectro-polarimetric imaging more robust against noise and measure- +ment errors. +4. +Spectro-Polarimetric Imaging +To evaluate the performance of our prototype imaging system, we conduct experiments using +a custom-designed ’PURDUE’ target made of letters constructed from titanium and a glass +substrate (Fig. 4a). Each letter has unique micro-grating structures (Inset of Fig. 4a) that generate +distinctive polarimetric signatures in the thermal radiation signal. The glass substrate also +features a characteristic emission peak around 11 𝜇𝑚. Note that we heat the image target +to 150◦𝐶 to generate high signal intensity. The reconstructed spectra of four representative +pixels are shown in Fig. 4 b-e. We compare them with the ground truth spectra measured by +a Fourier-transform infrared spectrometer, validating the effectiveness of our reconstruction +approach. The reconstructed spectral frames (Fig. 2 f) also exhibit high contrasts between +different wavelengths, demonstrating that the system can effectively reveal the LWIR spectral +properties of different targets. We note that the relatively low reconstruction accuracy at shorter +wavelengths (8 - 10 𝜇𝑚) results from the low transmission (low SNR) and the high correlation +(similarity) between the tuned spectra (Fig. 2f). +We also obtain the polarimetric information including degree the linear polarization (DOLP) +and the angle of linear polarization (AoLP) using the designed system. For polarimetric imaging, +we collectively rotate the spinning metasurfaces and the input polarizer, selecting four different +polarizations (0◦, 90◦, 45◦ and −45◦) while maintaining the same spectral transmission. We +use the first three Stokes parameters to quantify the polarimetric information associated with +each pixel, i.e. 𝑆0 = 𝐼0 + 𝐼90, 𝑆1 = 𝐼0 − 𝐼90, and 𝑆2 = 𝐼45 − 𝐼−45, where 𝐼0, 𝐼90, 𝐼45 and +𝐼−45 are the light intensity at polarization angles of 0◦, 90◦, 45◦ and −45◦, respectively. The +DoLP and AoLP are then calculated at each wavelength through DoLP = +√︃ +𝑆2 +1 + 𝑆2 +2/𝑆0 and + +Fig. 4. Spectro-polarimetric thermal imaging results. (a) An optical image of the +’PURDUE’ imaging target that is constructed from titanium letters on a glass substrate. +Inset: a zoomed-in optical image of the micro-structures in the letters, which generate +distinctive spectral and polarimetric signatures. (b-e) Reconstructed spectra of four +representative pixels (corresponding to the letter ’R’, ’U’, ’E’ and the glass substrate, +respectively) compared with the ground truth spectra measured by a Fourier-transform +infrared spectrometer. (f) Reconstructed spectral frames at 6 representative wavelengths. +The contrast between different frames demonstrates that the system can effectively +reveal the LWIR spectral properties of various materials and structures. (g-h) Degree- +of-linear-polarization and angle-of-linear-polarization frames. Distinctive polarimetric +signatures can be observed for each letter in the images. +AoLP = 𝑎𝑟𝑐𝑡𝑎𝑛(𝑆2/𝑆1). As shown in Fig. 4 h and i, we can clearly distinguish between different +letters based on their polarimetric signatures in the thermal radiation signal. The four-dimensional +spatial-spectro-polarimetric data-tesseract provides significantly more insight associated with an +object, making it a powerful tool for a wide range of imaging applications. +5. +Conclusion +Our results provide an innovative approach for spectro-polarimetric thermal imaging by combining +meta-optics and computational imaging. The low-SWaP (size, weight, and power) system opens +the door for physics-driven machine vision. The high-dimensional thermal image data can +significantly improve the performance of tasks such as depth estimation, object detection, and +semantic segmentation when only radiative heat signal is available. Furthermore, we foresee +that spectro-polarimetric thermal imaging can also be a powerful tool for scientific research, +allowing for non-destructive characterization in the infrared region to investigate a wide range of +novel physical phenomena, such as anisotropic thermal conduction [36] and directional radiative +heat transfer [37]. Overall, our work provides a key development in the rapidly growing field of +thermal imaging. +Funding. +The authors thank the Defense Advanced Research Projects Agency (DARPA) Nascent +Light-Matter Interactions (NLM) program for funding to pursue this research. +Disclosures. +The authors declare no conflicts of interest. +Data availability. +Data underlying the results presented in this paper are not publicly available at this +time but may be obtained from the authors upon reasonable request. +Supplemental document. +See Supplementary Materials for supporting content. + +d +a +c +e +Glass +Reconstruction +“E' +glass +Ground Truth +"U' +ince +BURDUE +0.5 +"R' +0.5 +0.5 +0.5 +Radia +Titanium +0 +0 +0 +0 +14 +10 +12 +14 +10 +12 +14 +10 +12 +14 +10 +12 +8 +8 +8 +8 +Wavelength (μm) +Wavelength (μm) +Wavelength (μm) +Wavelength (μm) +g +f +入= 8.3 μm +入 = 9.4 μm +PURDUE +DoLP +入 = 10.5 μm +0.06 +0.04 +1 +PURDUR +PURDUE +PURDUE +0.02 +0.8 +iance +0.6 +Radi +入 = 11.6 μm +入 = 13.8 μm +入 = 12.7 μm +0.4 +h +AoLP +ORDUE +50 +PURDOR +0.2 +PURDUR +PURDUR +0 +-506. +References +References +1. +A. 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Raman, “Broadband directional control of thermal emission,” Science 372, 393–397 +(2021). + diff --git a/itFJT4oBgHgl3EQfWyzG/content/tmp_files/load_file.txt b/itFJT4oBgHgl3EQfWyzG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..703bccdf7b919c5bfb30abd57ff3fddc30ed70a8 --- /dev/null +++ b/itFJT4oBgHgl3EQfWyzG/content/tmp_files/load_file.txt @@ -0,0 +1,619 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf,len=618 +page_content='Spinning Meta-Cam: Spectro-Polarimetric Long-wave Infrared Thermal Imaging based on Spinning Metasurfaces XUEJI WANG,1 ZIYI YANG,1 FANGLIN BAO,1,TYLER SENTZ,1 AND ZUBIN JACOB1,* 1Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 zjacob@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='edu Abstract: Spectro-polarimetric imaging in the long-wave infrared (LWIR) region is a powerful tool for capturing temperature, material composition, and surface morphology information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' However, current spectro-polarimetric LWIR imagers are often bulky and severely limited in spectral resolution and field of view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' In this work, we present a new paradigm for spectro-polarimetric demultiplexing by combining large-area meta-optical devices and advanced computational imaging algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We use the intrinsic dispersion and polarization modulation of anisotropic spinning metasurfaces to achieve simultaneous spectral and polarimetric resolution without the need for bulky filter wheels or interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Our spinning-metasurface-based spectro-polarimetric module is robust, compact (< 10 x 10 x 10 cm) and has a wide field of view (25◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Our approach represents a significant advancement in the field of thermal imaging, allowing for high-quality, information-rich thermal image data for a wide range of applications such as astronomical exploration, medical diagnosis, and agricultural monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' © 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Introduction Advances in machine vision technology have spurred a need for high-resolution, informative images across a range of applications scenarios [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Infrared thermography is a powerful tool for capturing temperature, material composition, and surface morphology information about objects, even in situations with limited external lighting [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The long-wave infrared (LWIR) spectral region is particularly useful for infrared thermography as most room-temperature objects emit thermal radiation at these wavelengths, according to Planck’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Additionally, the LWIR atmospheric transmission window reduces the effect of environmental turbulence on the thermal radiation signal in this region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' As a result, LWIR thermal imaging has become a crucial technology frontier in various applications such as astronomical exploration [6, 7], medical diagnosis [8], and agricultural monitoring [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Furthermore, demultiplexing the LWIR thermal radiation into its spectral and polarimetric components has been heralded as the next-generation solution for applications such as methane sensing [10] and thermal facial recognition [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' However, the commonly used mosaic filter approach for spectral demultiplexing [12–16] is not practical for LWIR thermal imaging, due to the limited number of pixels in LWIR focal plane arrays (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' State-of-the-art LWIR spectral imagers instead rely on infrared bandpass filters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1c) or interferometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1d), but these methods have limitations such as bulky filter wheels with limited spectral resolution, or unrobust interferometers with a limited field of view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Recently, metasurface-based spectral and polarimetric imaging has demonstrated huge potential in the visible region of spectrum [17–19], but using infrared metasurfaces in LWIR thermal imaging is still an open technological frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We emphasize that directly integrating metasurfaces on a thermal sensor significantly changes the heat transport properties of the sensor and make this approach incompatible with the widely arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='11519v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='optics] 27 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Long-wave infrared (LWIR) spectro-polarimetric thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (a) The room-temperature blackbody radiation (shown in red) and the atmospheric transmission spectrum (shown as a shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The LWIR spectral region is crucial for thermal imaging due to its peaked room-temperature thermal radiation spectrum and the atmospheric transparency window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (b-d) Conventional methods for spectral imaging, such as using a mosaic sensor (b), a filter wheel (c), or interferometry (d), either pose limitations or are infeasible for LWIR thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (e) In this study, we propose a new approach for spectro-polarimetric thermal imaging, achieved by combining large-area spinning metasurfaces and compressive sensing reconstruction algorithms adopted microbolometer technology in LWIR thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' To address these limitations, in this work, we put forth a new paradigm for spectro-polarimetric thermal imaging using large-area meta-optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The spectral demultiplexing is achieved through the intrinsic dispersion and polarization modulation in anisotropic metasurfaces, and spectral reconstruction is realized using compressive sensing and dictionary learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Our designs employ simple 2D structures with large feature sizes (> 1𝜇𝑚), which can be easily made by standard photo-lithography techniques, enabling large-area fabrication and scalable manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Our demonstration provides a new technical route to solve the long-standing issue of robust and portable thermal imaging with simultaneous spectral and polarimetric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' By utilizing meta-optical designs and advanced computational imaging methods, our approach a Fully-Passive Quasi-Active Blackbody Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=') Peak wavelengths for Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' room-temperature objects & human body T=300K SWIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 Atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' This Work: Long-wave Infrared Atmospheric Transparency Windows 0 0 0 5 10 15 Wavelength(um) b d Mosaic Sensor Filter Wheel Interferometry Not Feasible Limited Spectral Limited FOV in LWIR Resolution Temperature Estimation Material Estimation This Work: Linear e Polarizer Object Detection Low-SWaP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Large-FOv Spectro-Polarimetric LWIR Imaging M3 M2 Spinning LWIR Sensor Control Compressive M1 Sensing Raw Signals Spectro-Polarimetric Meta-dataoffers a promising platform for next-generation high-contrast LWIR thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The architecture of our spinning-metasurface based spectro-polarimetric imaging system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' It comprises of a broadband linear polarizer, three anisotropic and dispersive metasurfaces, and an LWIR imaging sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The polarizer is utilized to polarize the incoming thermal radiation signals, and the metasurfaces are utilized to realize spectral filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We design the metasurfaces with high anisotropy to produce distinct spectral responses for orthogonal polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Additionally, the metasurfaces’ dispersion rotates different wavelengths of light to varying polarization orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' By using the metasurfaces in tandem and axially spinning the polarizer and metasurfaces to different angles, we obtain tunable transmission spectra that sample the incident thermal radiation in its spectral and polarimetric channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We then reconstruct unknown spectra of imaging targets using compressive sensing and dictionary learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Dictionary learning generates a set of basis functions that represent the unknown spectra in a sparse format [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Compressed sensing enables accurate reconstruction of the sparse spectra from limited number of measurements [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Combining these two techniques enables accurate and stable spectral reconstruction in the presence of noise and measurement errors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The four-dimensional spectro-polarimetric data generated by our system offers a wealth of physical information about an imaging target, making it a valuable tool for physics- driven machine vision [4,23], facilitating various tasks such as object detection and semantic segmentation [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Design of Spinning Metasurfaces To quantitatively describe the mechanism of the spinning-angle-controlled transmission spectra, we represent the spectro-polarimetric response of the metasurfaces using Jones matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Assuming that the transmission axis (x-axis) of the input linear polarizer is at 0 degrees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' the Jones matrix 𝐽𝑖 of a metasurface i with the x-axis at a spinning angle 𝜃𝑖 can be expressed as: 𝐽𝑖(𝜃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 𝜆) = 𝑅(−𝜃𝑖) · 𝐽𝑀𝑖 (𝜆) · 𝑅(𝜃𝑖) = ������ 𝑐𝑜𝑠(𝜃𝑖) −𝑠𝑖𝑛(𝜃𝑖) 𝑠𝑖𝑛(𝜃𝑖) 𝑐𝑜𝑠(𝜃𝑖) ������ ������ 𝑡𝑖𝑝(𝜆) 0 0 𝑡𝑖𝑠(𝜆) ������ ������ 𝑐𝑜𝑠(𝜃𝑖) 𝑠𝑖𝑛(𝜃𝑖) −𝑠𝑖𝑛(𝜃𝑖) 𝑐𝑜𝑠(𝜃𝑖) ������ (1) where R is the rotation matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 𝐽𝑀𝑖 contains the anisotropic transmission of the metasurface 𝑡𝑖𝑝 and 𝑡𝑖𝑠 along the two principle axes p and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Then, the Jones matrix of the three-metasurface assembly is given by: 𝐽 (𝜃1, 𝜃2, 𝜃3, 𝜆) = 𝐽1 (𝜃1, 𝜆) · 𝐽2 (𝜃2, 𝜆) · 𝐽3 (𝜃3, 𝜆) · (2) Thus, the total transmission spectrum of the three spinning metasurfaces strongly depends on the spinning-angle combinations Θ = (𝜃1, 𝜃2, 𝜃3) when the constituted metasurfaces are strongly anisotropic and dispersive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 𝑡𝑖𝑝(𝜆) ≠ 𝑡𝑖𝑠(𝜆) (see Supplementary Materials for the detailed analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We note that large differences between the spectral responses of the three metasurfaces (M1, M2, M3) are also introduced to minimize the correlations between the generated spectra, which can significantly improve the spectral reconstruction performance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We emphasize that our design generates a large set of distinct transmission spectra with only three metasurfaces, while the total number of spectra in traditional mosaic array is limited to the number of metasurfaces/filters used [12–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Accordingly, we design the metasurfaces and experimentally achieve three key characteristics for optimized spectro-polarimetric imaging performance: 1) Strong anisotropy and dispersion for efficient wavelength demultiplexing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2) High transmission and low self-emission for high signal to noise ratio (SNR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3) Small angular dependence for a large FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The unit cell of the three designed metasurfaces are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2 a-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Strong dispersive anisotropy of the transmission Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Design and characterization of the spinning metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (a-c) Schematics of the three different metasurface devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (d) A scanning electron microscope (SEM) image of a fabricated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Inset: an optical image of a 1-inch-diameter device used for the imaging experiments, highlighting the large-area uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (e) The measured polarized transmission spectra (𝑡𝑖𝑝 and 𝑡𝑖𝑠) of the three metasurfaces (M1 M3) are depicted, displaying strong anisotropy and distinctive dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (f) The tunable transmission spectra controlled by the spinning angle combinations of the three metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Inset: an optical image of the motorized rotatory mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The overall size of the spinning metasurface module is smaller than 10 cm x 10 cm x 10 cm, making it a promising platform for next-generation high-contrast LWIR thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (g) The normalized spatial transmittance of the module at two representative spinning angle combinations Θ = (0◦, 0◦, 0◦) (top) and Θ = (90◦, 90◦, 90◦) (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The red circles correspond to a transmittance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 and the field of view of the imaging system is estimated to be 25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' spectra can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Additionally, we emphasize that large-area devices are generally required for imaging applications to ensure sufficient numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' All the metasurfaces designed here have feature sizes larger than 1 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Large-area devices (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4mm in diameter) with high structural quality and uniformity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2d) can be rapidly fabricated by standard photo-lithography techniques, enabling scalable manufacturing for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' This is in strong contrast to recent works on miniaturized spectrometers [27–35], where the device footprint is on the micrometer scale and thus not suitable for imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The tunable transmission spectra produced by our spinning metasurfaces are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The distinct spectra are a result of the tuned spinning-angle combination Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We integrate the three fabricated metasurfaces tandemly via compact rotatory mounts to independently control the rotation of each metasurface (Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We also optimize the spinning- angle combinations of the three spinning metasurfaces using genetic algorithms to generate largely uncorrelated transmission spectra for optimal spectral reconstruction performance (see Supplementary Materials for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Additionally, we note that increasing the number of metasurfaces can further improve the spectral resolution, but simultaneously reduces the SNR as the peak transmissions of the LWIR devices are limited to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' However, our method has the potential to scale up into the hyperspectral regime by adding more high-transmission LWIR metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' d b c a d = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 mm ns M1 M2 M3 e g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 Spinning 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 人 M1 Transmittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='8 Transmittance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='0) = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 M3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='.06--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='06) = 0 0 X 8 9 10 11 12 13 14 8 10 12 14 Wavelength (um) Wavelength (μm)We also evaluate the FOV of our imaging module by integrating it with an LWIR thermal camera and capturing images of a large area uniform blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' To determine the spatial transmission efficiency, we normalize the signal counts of each pixel by the counts at the center of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We also define the angular range with transmittance above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 as the effective FOV of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' As seen in Figure 2g, our spinning metasurface module has an FOV of around 25 degrees, which is significantly larger than what can be achieved with interferometer-based spectral imagers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Spectral Reconstruction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Schematic of the spectral reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The measured raw signal (a) can be expressed by the pre-calibrated spectral response function (b) of the imaging system multiplied by the spectrum of an imaging target (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' For the spectral reconstrcution, the unknown spectrum 𝑃𝜆 is projected onto a sparse representation basis 𝐷𝑘𝜆 using dictionary learning (c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' This sparse representation 𝜙𝑘 is then used for compressive sensing based reconstruction (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The use of compressive sensing and dictionary learning in the reconstruction process significantly improves the reconstruction accuracy, making the spinning-metasurface-based spectro-polarimetric imaging more robust against noise and measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' To extract the unknown spectro-polarimetric properties of various imaging targets, we use a combination of dictionary learning and compressive sensing algorithms in the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The tunable transmission spectra produced by our spinning metasurfaces (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2g) are not narrowband, which means that the collected raw signals at different spinning-angle combinations Θ do not directly reflect the spectral radiance at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Instead, the collected signal 𝐼(Θ) at each pixel can be described as an integral of the spectral response function 𝑅(Θ, 𝜆) multiplied by the ground truth spectrum 𝑃(𝜆) that we wish to obtain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 𝐼(Θ) = ∫ 𝜆max 𝜆min 𝑅(Θ, 𝜆)𝑃(𝜆)d𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' To solve for this equation, we discretize the spectral range of interest and express it in a tensor form as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3: 𝐼Θ = 𝑅Θ𝜆𝑃𝜆 (3) b Wavelength 入 a Dictionary Learning Counts Response R Counts p R @1 Sparse Counts R Spinning Angle O ---- Re1,入i R01,入2 R01,入 e 10 Spinning Angles D R02,入1 R02,入n R02,入2 102 Ren,入1 Wavelength 入 Compressive Measured Unknown Pre-calibrated Spectral Response Function Raw Signal Sensing SpectrumWe emphasize that directly solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3 does not produce accurate spectral reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' In theory, we can use measured signals 𝐼Θ and the pre-calibrated response function 𝑅𝜃𝜆 to determine unknown spectra 𝑃𝜆 at each pixel of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' However, in practice, two limitations impede the performance of spectral reconstruction: the problem becomes underdetermined when there are many discretized wavelength bands, and measurement noise affects both 𝐼Θ and 𝑅Θ𝜆, making the direct reconstruction method unstable and the results inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' To improve the accuracy and stability of spectral reconstruction, we use compressed sensing and dictionary learning algorithms to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Specifically, we first use dictionary learning to create a dictionary of basis functions 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' These functions can represent any thermal radiation spectrum in the space of spectra we are studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We then project the unknown spectrum 𝑃𝜆 as a linear combination of the basis functions in the dictionary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 3 b and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We have, 𝑃𝜆 = 𝐷𝜆𝑘𝜙𝑘 (4) where 𝜙𝑘 is known as a sparse coding of the spectrum 𝑃𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' With this sparse representation, the spectral reconstruction problem can be solved by first obtaining 𝜙recon: 𝜙recon = arg min 𝜙𝑘 ∥𝜙𝑘 ∥1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' ∥𝐼Θ − 𝑅Θ𝜆𝑃𝜆∥2 = ��𝐼𝑝 − 𝐴Θ𝑘𝜙𝑘 �� 2 < 𝜖 (5) where 𝐴Θ𝑘 = 𝑅Θ𝜆𝐷𝜆𝑘, and 𝜖 is the residual error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Finally, the spectra 𝑃𝜆 at each pixel of a scene is reconstructed by 𝑃recon = 𝐷𝜆𝑘 𝜙recon (6) Our reconstruction method significantly improves the reconstruction accuracy, making the spinning-metasurface-based spectro-polarimetric imaging more robust against noise and measure- ment errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Spectro-Polarimetric Imaging To evaluate the performance of our prototype imaging system, we conduct experiments using a custom-designed ’PURDUE’ target made of letters constructed from titanium and a glass substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Each letter has unique micro-grating structures (Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4a) that generate distinctive polarimetric signatures in the thermal radiation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The glass substrate also features a characteristic emission peak around 11 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Note that we heat the image target to 150◦𝐶 to generate high signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The reconstructed spectra of four representative pixels are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4 b-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We compare them with the ground truth spectra measured by a Fourier-transform infrared spectrometer, validating the effectiveness of our reconstruction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The reconstructed spectral frames (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2 f) also exhibit high contrasts between different wavelengths, demonstrating that the system can effectively reveal the LWIR spectral properties of different targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We note that the relatively low reconstruction accuracy at shorter wavelengths (8 - 10 𝜇𝑚) results from the low transmission (low SNR) and the high correlation (similarity) between the tuned spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We also obtain the polarimetric information including degree the linear polarization (DOLP) and the angle of linear polarization (AoLP) using the designed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' For polarimetric imaging, we collectively rotate the spinning metasurfaces and the input polarizer, selecting four different polarizations (0◦, 90◦, 45◦ and −45◦) while maintaining the same spectral transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' We use the first three Stokes parameters to quantify the polarimetric information associated with each pixel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 𝑆0 = 𝐼0 + 𝐼90, 𝑆1 = 𝐼0 − 𝐼90, and 𝑆2 = 𝐼45 − 𝐼−45, where 𝐼0, 𝐼90, 𝐼45 and 𝐼−45 are the light intensity at polarization angles of 0◦, 90◦, 45◦ and −45◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The DoLP and AoLP are then calculated at each wavelength through DoLP = √︃ 𝑆2 1 + 𝑆2 2/𝑆0 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Spectro-polarimetric thermal imaging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (a) An optical image of the ’PURDUE’ imaging target that is constructed from titanium letters on a glass substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Inset: a zoomed-in optical image of the micro-structures in the letters, which generate distinctive spectral and polarimetric signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (b-e) Reconstructed spectra of four representative pixels (corresponding to the letter ’R’, ’U’, ’E’ and the glass substrate, respectively) compared with the ground truth spectra measured by a Fourier-transform infrared spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (f) Reconstructed spectral frames at 6 representative wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The contrast between different frames demonstrates that the system can effectively reveal the LWIR spectral properties of various materials and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' (g-h) Degree- of-linear-polarization and angle-of-linear-polarization frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Distinctive polarimetric signatures can be observed for each letter in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' AoLP = 𝑎𝑟𝑐𝑡𝑎𝑛(𝑆2/𝑆1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 4 h and i, we can clearly distinguish between different letters based on their polarimetric signatures in the thermal radiation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The four-dimensional spatial-spectro-polarimetric data-tesseract provides significantly more insight associated with an object, making it a powerful tool for a wide range of imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Conclusion Our results provide an innovative approach for spectro-polarimetric thermal imaging by combining meta-optics and computational imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The low-SWaP (size, weight, and power) system opens the door for physics-driven machine vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The high-dimensional thermal image data can significantly improve the performance of tasks such as depth estimation, object detection, and semantic segmentation when only radiative heat signal is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Furthermore, we foresee that spectro-polarimetric thermal imaging can also be a powerful tool for scientific research, allowing for non-destructive characterization in the infrared region to investigate a wide range of novel physical phenomena, such as anisotropic thermal conduction [36] and directional radiative heat transfer [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Overall, our work provides a key development in the rapidly growing field of thermal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The authors thank the Defense Advanced Research Projects Agency (DARPA) Nascent Light-Matter Interactions (NLM) program for funding to pursue this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' Supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' See Supplementary Materials for supporting content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' d a c e Glass Reconstruction “E\' glass Ground Truth "U\' ince BURDUE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 "R\' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 Radia Titanium 0 0 0 0 14 10 12 14 10 12 14 10 12 14 10 12 8 8 8 8 Wavelength (μm) Wavelength (μm) Wavelength (μm) Wavelength (μm) g f 入= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='3 μm 入 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 μm PURDUE DoLP 入 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='5 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='04 1 PURDUR PURDUE PURDUE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='8 iance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 Radi 入 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='6 μm 入 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='8 μm 入 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='7 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='4 h AoLP ORDUE 50 PURDOR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content='2 PURDUR PURDUR 0 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' References References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFJT4oBgHgl3EQfWyzG/content/2301.11519v1.pdf'} +page_content=' M.' metadata={'source': 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HUNG AND ALEXANDER MORET´O +Abstract. We study the codegree isomorphism problem for finite simple groups. In par- +ticular, we show that such a group is determined by the codegrees (counting multiplicity) +of its irreducible characters. The proof is uniform for all simple groups and only depends +on the classification by means of Artin-Tits’ simple order theorem. +1. Introduction +The group ring/algebra isomorphism problem has a long history, dating back to the work +of G. Higman [Hig40] in the forties on integral group rings of abelian groups and its study +has lead to important discoveries in group theory, ring theory and representation theory (see +the recent survey [Mar22] and references therein). Its most general form is: for a ring/field +F and finite groups G and H, does a group ring/algebra isomorphism FG – FH imply a +shared property of G and H or even a group isomorphism G – H? The problem includes a +number of more prominent cases: the Integral Isomorphism Problem [Hig40, San85, Sco86, +Her01], the Complex-group-algebras Isomorphism Problem [Bra63, Isa86, BNOT15, SIS20], +and the Modular Isomorphism Problem [Des56, Pas74, May14, Mar22]. +The isomorphism problem for complex group algebras (when F “ C), by the well- +known Wedderburn theorem, is equivalent to the one for character degrees: which finite +group is determined by its multiset of character degrees (i.e., the set of character degrees +counting multiplicities)? This has been confirmed in the affirmative for (quasi)simple groups +[BNOT15]. +The version without multiplicity (and therefore stronger) for finite simple +groups is a conjecture proposed by B. Huppert in the late nineties [Hup00], which has +been also extensively studied over the past two decades [Hup06, NT15, BTZ17]. +The +results to date on these character-degree isomorphism problems (both the group-algebras +problem and Huppert’s conjecture) were achieved essentially on a case-by-case basis using +the classification of finite simple groups. +This paper is concerned with the codegree isomorphism problem for finite simple groups. +For a character χ of a finite group G, the codegree of χ is codpχq :“ |G : kerpχq|{χp1q. +2010 Mathematics Subject Classification. Primary 20C15, 20C30, 20C33, 20D06. +Key words and phrases. Character codegrees, isomorphism problem, finite simple groups, Huppert’s +conjecture. +Part of this work was done while the first author was visiting the Vietnam Institute for Advanced Study +in Mathematics (VIASM), and he thanks the VIASM for its hospitality and financial support. +The research of the second author is supported by Ministerio de Ciencia e Innovaci´on (Grant +PID2019-103854GB-I00 funded by MCIN/AEI/ 10.13039/501100011033) and Generalitat Valenciana +CIAICO/2021/163. +1 + +2 +N. N. HUNG AND A. MORET ´O +This notion was first introduced and studied (in a slightly different form) by D. Chillag +and M. Herzog [CH89] and D. Chillag, A. Mann, and O. Manz [CMM91]. It was later +developed into the current form used nowadays by G. Qian, Y. Wang, and H. Wei [QWW07] +and has been proved to have remarkable connections with the structure of finite groups +[QWW07, Isa11, DL16, Mor21, Qia21, CN22]. +Let codpGq denote the set of all the codegrees of irreducible characters of G. Recently, +there has been great interest in the codegree analogue of Huppert’s conjecture (Prob- +lem 20.79 of the Kourovka Notebook [Kou20]), to which we will refer to as Huppert’s +codegree conjecture: +(HCC) +Let S be a finite nonabelian simple group and G a finite group. +Then codpGq “ codpSq if and only if G – S. +The approach so far to this problem is more or less similar to Huppert’s original method, +and therefore, unfortunately, is still case-by-case [BAK21, Aha22, GKL+22, GZY22, LY23]. +Let codpGq “ tc1 ă c2 ă ... ă cku and mGpciq be the number of irreducible characters of +G with codegree ci. The multiset +CpGq :“ tpci, mGpciqq : 1 ď i ď ku +is called the group pseudo-algebra of G, which can be viewed as the codegree counterpart +of the aforementioned complex group algebra CG. A natural weaker version of (HCC) asks +whether G and S must be isomorphic if CpGq “ CpSq. Our first principal result solves this +codegree-with-multiplicity isomorphism problem. +Theorem A. Let S be a finite simple group and G a finite group. Then +CpGq “ CpSq if and only if G – S. +The main novelty of this paper is a more uniform approach to these codegree problems +with as little case-by-case analysis as possible. Our proof of Theorem A, somewhat surpris- +ingly, only relies on the classification via the so-called simple order theorem (also known +as the Artin-Tits theorem [Art55b, KLST90]), which states that two non-isomorphic finite +simple groups have the same order if and only if they are either PSL4p2q and PSL3p4q or +Ω2n`1pqq and PSp2npqq for some n ě 3 and odd q. This is perhaps the first time that a +result of this type is proved uniformly for all simple groups. +The following are two key ingredients in the proof of Theorem A. We find it remarkable +that they admit strikingly elementary proofs. The first provides a characterization of perfect +groups in terms of codegrees. (Recall that a group is perfect if it coincides with its derived +subgroup.) +Theorem B. A finite nontrivial group G is perfect if and only if G has no prime character +codegrees. +The second ingredient we are referring to is an order-divisibility property involving char- +acter codegrees of finite simple groups. +Theorem C. Suppose that S is a finite simple group and G a finite group such that +codpSq Ď codpGq. +Then |S| divides |G|. +In particular, if G and S are simple groups +and codpGq “ codpSq then |G| “ |S|. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +3 +Our next main result is far stronger than the second statement in Theorem C, but does +require the classification. +Theorem D. Let S and H be finite simple groups such that codpSq Ď codpHq. Then +S – H. +Theorem D in fact is the first step in proving Huppert’s codegree conjecture (HCC). Let +G be any finite group and H a simple group such that codpGq “ codpHq, and N be a +maximal normal subgroup of G so that S :“ G{N is simple. In order to prove G – H, one +would first need to establish S – H, under the assumption codpSq Ď codpGq “ codpHq. +This is precisely what we do in Theorem D (see Theorem 8.3). +Remark 1. Using Theorem D, we will prove in a subsequent paper [HMT] that (HCC) +holds for all sporadic groups, alternating groups, groups of Lie type of low rank, and, for +the first time in the degree/codegree isomorphism problem, groups of Lie type of arbitrary +rank over a field of prime order. Furthermore, perhaps unexpectedly, we reduce (HCC) to +a problem on p-groups. +Remark 2. The proof of Theorem D is fairly complicated and combines several tech- +niques. In particular, it essentially utilizes, besides Theorem C, some deep results on the +representation theory of finite simple groups, including the classification of prime-power- +degree representations [MZ01, BBOO01], lower/upper bounds for the largest degree of +irreducible representations [VK85, LMT13], and the existence of p-defect zero characters +[Mic86, Wil88, GO96]. Along the way we prove an effective and explicit upper bound for +the largest character degree bpSq of an exceptional group S of Lie type (Theorem 5.3). (See +Lemma 5.4 for previous related work on symmetric and alternating groups [VK85] and +classical groups [LMT13].) +Remark 3. We need bounds for the largest character degree bpSq in order to control the +behavior of fpSq :“ |S|{bpSq – the smallest nontrivial codegree of S. While the relevance +of the smallest (or low-degree in general) characters of (quasi/almost)simple groups is well- +known in group representation theory (see [MM21] for the latest results), the smallest +codegree had not been studied much before. This invariant arises naturally in the proof +of Theorem D (see Lemma 5.1) and measures the relative growth of bpSq compared to |S|. +Our proof would be much simpler if one can show that f is divisibly increasing among +nonabelian simple groups, by which we mean that, if S and H are nonabelian simple of +different orders such that |S| divides |H|, then fpSq ă fpHq. +We indeed confirm this +phenomenon in many cases, particularly when one of the two groups involved is alternating +(see Propositions 7.1, 7.3, and 8.1). +The layout of the paper is as follows. +In Section 2, we prove some results on prime +character codegrees, including Theorem B, and provide a short proof of a theorem of Riese +and Schmid (see Theorem 2.7) on prime-power codegrees. Section 3 is devoted to the proof +of Theorem C and its consequences. Using the results in the preceding sections, we prove +Theorem A in Section 4. Results on bounding the largest character degree are presented +in Section 5. Finally, the proof of Theorem D is carried out in Sections 6, 7, and 8. + +4 +N. N. HUNG AND A. MORET ´O +2. Prime-power codegrees +In this section we prove some results on prime-power character codegrees. These results +show that, in contrast to character degrees, there are significant restrictions on the structure +of groups with faithful irreducible characters of prime/prime-power codegree. +We mainly follow the notation from [Isa76] for character theory and [Atl1, Car85] for +finite simple groups. Throughout, for a positive integer n and a prime p, we write np to +denote the maximal p-power divisor of n and np1 :“ n{np to denote the maximal divisor not +divisible by p of n. Let N IJ G and θ P IrrpNq. We write IrrpG|θq for the set of irreducible +constituents of θG and IrrpG|Nq for the set of irreducible characters of G whose kernels do +not contain N. If G is a group, πpGq is the set of primes that divide |G|. If n is an integer, +πpnq is the set of primes that divide n, and if S is a set of integers, then πpSq is the set of +primes that divide some member of S. As usual, cdpGq :“ tχp1q : χ P IrrpGqu is the set +of all irreducible character degrees of G. Other notation will be recalled or defined when +necessary. +We begin by collecting some known facts on character codegrees that we will use without +explicit mention. +Lemma 2.1. Let G be a finite group and χ P IrrpGq. The following hold: +(i) If χ is not the principal character, then codpχq ą χp1q. +(ii) If N IJ G and N ď kerpχq, then the codegree of χ as a character of G coincides with +the codegree of χ viewed as a character of G{N. +(iii) If N IJIJ G and θ P IrrpNq lies under χ, then codpθq divides codpχq. +(iv) πpGq “ πpcodpGqq. +(v) If G is abelian, then codpGq “ opGq, where opGq is the set of orders of the elements +of G. +Proof. Part (i) is [DL16, Lem. 2.1]. Parts (ii) and (iii) are contained in [QWW07, Lem. 2.1], +and part (iv) is [QWW07, Lem. 2.4]. Now, we prove part (v). The inclusion opGq Ď codpGq +follows from [DL16, Lem. 2.2]. Conversely, if d P codpGq there exists χ P IrrpGq such that +d “ |G : kerpχq| (note that since G is abelian, χ is linear). Since G{ kerpχq is cyclic, we +conclude that G has elements of order d. +□ +2.1. Prime codegrees: characterizing perfect groups. The goal in this subsection is +to provide a characterization of perfect groups in terms of the absence of prime codegrees. +Theorem 2.2. Let G be a finite group. Suppose that there exists χ P IrrpGq faithful such +that codpχq “ p is a prime number. Then G is either the cyclic group of order p or a +Frobenius group with Frobenius kernel of order p. +Proof. We argue by induction on |G|. +Let N IJ G be minimal such that there exists +θ P IrrpNq lying under χ with codpθq “ p. Then +|G| +χp1q “ p “ codpθq “ rN : kerpθqs +θp1q +, +and we deduce that +p “ codpχq ą χp1q “ rG : Ns| kerpθq|θp1q. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +5 +In particular, p does not divide any of the three factors in the right hand side. +Suppose first that N ă G. By the inductive hypothesis, N{ kerpθq is cyclic of order +p or a Frobenius group with Frobenius kernel K{ kerpθq of order p. In the latter case, if +λ P IrrpK{ kerpθqq lies under θ then codpλq “ p. This contradicts the choice of N. (Note +that K is normal in G because K is characteristic in N.) +Hence, we may assume that N{ kerpθq is cyclic of order p. By Clifford’s theorem the +faithful character χN is a sum of G-conjugates of θ. Let T be a complete set of represen- +tatives in G for these conjugates. By [Isa76, Lem. 2.21], the intersection of kerpθgq, where +g runs over T, is trivial. We conclude that N embeds into the direct product +ź +gPT +N{ kerpθgq. +Each of the direct factors has order p, and so N is an elementary abelian p-group. Since +p does not divide | kerpθq|, we conclude that N is cyclic of order p. As θ is linear and +χp1q “ |G|{p, we now have χ “ θG. It follows that G is a Frobenius group with kernel N, +as desired. +Now, we consider the case N “ G. Let M be a maximal normal subgroup of G. Since +N “ G the codegree of any irreducible character of N lying under χ is 1. This means +that χM “ χp1q1M. But χ is faithful, so we deduce that M “ 1 and G is simple. If G is +abelian, then it is the cyclic group of order p. If G is not abelian, then |G|{p “ χp1q ă +a +|G| +and it follows that |G| ă p2. By Sylow’s theorems, it follows that G has a normal Sylow +p-subgroup. This contradiction completes the proof. +□ +The following consequence of Theorem 2.2 is Theorem B. +Corollary 2.3. A finite nontrivial group G is perfect if and only if G does not have any +prime character codegree. +Proof. By Lemma 2.2, if G has an irreducible character χ of prime codegree then G{ kerpχq +is solvable. In particular, G is not perfect. Conversely, if G is not perfect, then the abelian +group G{G1 has some irreducible character of prime codegree. +□ +2.2. Prime power codegrees: the Riese-Schmid theorem. Chillag and Herzog proved +in [CH89, Thm. 1] that a simple group does not possess nontrivial irreducible characters of +prime power codegree. The proof relied on a case by case analysis of the simple groups, using +the fact that, most of the times, they have p-blocks of defect zero. This was generalized +by Riese and Schmid in [RS98, Cor. +3] to quasisimple groups, using also block theory +and the classification. We offer a short proof of this result that only depends on an easy +consequence of the classification, which is due to W. Kimmerle, R. Lyons, R. Sandling, and +D. N. Teague [KLST90, Thm. 3.6]: +Lemma 2.4. For every finite simple group S and prime p, |S| ă p|S|p1q2. +The following is a restatement of [RS98, Lem. 1] in the language of codegrees. +Lemma 2.5. Let p be a prime. Let G be a finite group and χ P IrrpGq faithful. Then +codpχq is a power of p if and only if χ is induced from a Sylow p-subgroup of G. + +6 +N. N. HUNG AND A. MORET ´O +Theorem 2.6. A quasisimple group G does not possess nonprincipal characters of prime +power codegree. +Proof. Suppose that there exists 1G ‰ χ P IrrpGq of p-power codegree. Let K :“ kerpχq +and note that K ď ZpGq. By Lemma 2.5, χ is induced from a Sylow p-subgroup of G{K. +Therefore, +χp1q ě |G{K|p1. +By Lemma 2.4, we know that +|G{ZpGq| ă p|G{ZpGq|p1q2 ď p|G{K|p1q2. +Hence, by [Isa76, Cor. 2.30], +χp1q ď |G : ZpGq|1{2 ă |G{K|p1, +which violates the inequality above. +□ +The next result is a restatement in terms of character codegrees of Theorem B of [RS98]. +The proof in [RS98] uses Brauer’s first and third main theorems. Recall that if a group G +has trivial p1-core Op1pGq, then it is defined to be p-constrained if the p-core OppGq contains +its centralizer. +Theorem 2.7 (Riese-Schmid). Let G be a finite group and let p be a prime. Suppose that +χ P IrrpGq is faithful of p-power codegree. Then Op1pGq “ 1 and G is p-constrained. +Proof. By Theorem 2.6, we know that G is not simple. Let N be a minimal normal subgroup +of G and let θ P IrrpNq lying under χ. Since θ has p-power codegree (by Lemma 2.1(iii)) +and codpθq ą 1 (note that since χ is faithful, θ ‰ 1N), we deduce that N is either an +elementary abelian p-subgroup or a direct product of nonabelian simple groups of order +divisible by p. In particular, Op1pGq “ 1. +We claim that N is an elementary abelian p-group. Suppose that N “ S1 ˆ ¨ ¨ ¨ ˆ St, +with Si – S for some nonabelian simple group S of order divisible by p. We wish to reach a +contradiction. Since θ ‰ 1N, there exists a nonprincipal ψ P IrrpSiq lying under θ for some +i. Note that codpψq is a power of p and this contradicts Theorem 2.6. The claim follows. +Write P :“ OppGq and C :“ CGpPq. Note that C X P “ ZpPq and Op1pCq “ 1 (because +Op1pGq “ 1). We want to see that C ď P. Assume not. Take K subnormal in G such that +ZpPq ď K ď C and K{ZpPq is simple. Since Op1pCq “ 1 and G does not have nonabelian +minimal normal subgroups, we conclude that K1 is quasisimple. +Now, take γ P IrrpK1q lying under χ. Again, we have that γ is not principal and codpγq +is a p-power. This contradicts Theorem 2.6. +□ +We end this section with a variation of Theorem 2.6. +Theorem 2.8. Let G be a finite group. Suppose that p is a prime and χ P IrrpGq is faithful +of p-power codegree. Then codpχq exceeds the p-part of the product of the orders of the +nonabelian composition factors in a composition series of G. In particular, if K{L is a +non-abelian chief factor of G, then codpχq ą |K{L|p. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +7 +Proof. Let n be the product of the orders of the non-abelian composition factors in a +composition series of G. +Using again that χ is induced from a Sylow p-subgroup and +[KLST90], we have +codpχq ą χp1q ě |G|p1 ě np1 ą np, +as wanted. +□ +3. An order-divisibility result for codegrees +The next result is the first part of Theorem C, which will be crucial in the proofs of our +main theorems. +Theorem 3.1. Suppose that S is a finite simple group and G a finite group such that +codpSq Ď codpGq. Then |S| divides |G|. +Proof. Let d1, ..., dk be all the degrees of nontrivial irreducible characters of S, and let mi +(1 ď i ď k) be the number of those characters of degree di. By the assumption, for each i, +there exists χi P IrrpGq such that +|S| +di +“ rG : kerpχiqs +χip1q +“ +|G| +χip1q| kerpχiq|. +It follows that +kÿ +i“1 +mid2 +i +|S|2 “ +kÿ +i“1 +miχip1q2| kerpχiq|2 +|G|2 +, +and thus +řk +i“1 miχip1q2| kerpχiq|2 +|G|2 +“ +řk +i“1 mid2 +i +|S|2 +“ |S| ´ 1 +|S|2 +. +Therefore |S|2 divides |G|2p|S| ´ 1q, and the theorem follows. +□ +We record some consequences of Theorem 3.1 that will be needed in subsequent sections. +Corollary 3.2. Suppose that S and H are finite simple group such that codpSq “ codpHq. +Then |S| “ |H|. +Proof. This directly follows from Theorem 3.1. +□ +Remark 4. It is a basic fact that the group algebra of a finite group determines the order +of the group, just note that |G| “ ř +χPIrrpGq χp1q2, but it is not known whether the group +pseudo-algebra determines the order of the group (see [Mor22, Question 3.2]). However, we +do not think that the analogue of Corollary 3.2 for character degrees can be proved without +the classification. +Lemma 3.3. Suppose that S and H are finite nonabelian simple groups such that codpSq Ď +codpHq. Let x :“ |H|{|S|. Then x P N and dx P cdpHq for every 1 ‰ d P cdpSq. +Proof. We know that x P N by Theorem 3.1. For each 1 ‰ d P cdpSq, we have |S|{d P +codpHq, and thus there exists some χ P IrrpHq such that |S|{d “ |H|{χp1q, implying that +χp1q “ dx, as claimed. +□ + +8 +N. N. HUNG AND A. MORET ´O +Lemma 3.4. Let S and H be finite simple groups of Lie type. Suppose that the defining +characteristic of H is p and codpSq Ď codpHq. Then |S|p1 “ |H|p1 and there exists χ P +IrrpSq such that χp1q “ |S|p. +Proof. We first observe that |S| is divisible by p because otherwise every codegree of S is not +divisible by p but the only nontrivial codegree of H not divisible by p is |H|p1 “ |H|{StHp1q. +By [Mic86, Wil88], S has an irreducible character, say χ, of p-defect 0, so that codpχq is +coprime to p. Therefore we have +|S|{χp1q “ |H|p1. +It follows from Theorem 3.1 that χp1q|H|p1 “ |S| divides |H|, implying that |S|p1 “ |H|p1 +and χp1q “ |S|p. +□ +Remark that when p ě 5, the above result holds for all S. This is because, in such +case, irreducible p-defect zero characters still exist in alternating groups (by [GO96]) and +in sporadic simple groups [Atl1]. +Lemma 3.5. Let S be a finite simple group of Lie type and H a finite nonabelian simple +group. Suppose that codpSq Ď codpHq and there are primes p ‰ r such that H has a unique +character degree divisible by each |H|p and |H|r. Then |S| “ |H| and cdpSq Ď cdpHq. +Proof. Repeating the arguments in the proof of Lemma 3.4, we have |S|p1 “ |H|p1 and +|S|r1 “ |H|r1, implying that |S| “ |H| and cdpSq Ď cdpHq. +□ +4. Group pseudo-algebras of simple groups: Theorem A +In this section we prove Theorem A, using the results in the preceding sections. +Theorem 3.1 is useful in proving results concerning codegrees of finite simple groups. +One of them is the next theorem, whose proof makes use of the simple order theorem. +Recall that the simple order theorem asserts that two non-isomorphic finite simple groups +have the same order if and only if they are either PSL4p2q and PSL3p4q (of order 20, 160) +or Ω2n`1pqq and PSp2npqq (odd-dimensional orthogonal and symplectic groups, of order +p1{2qqn2 śn +i“1pq2i ´ 1q) for some n ě 3 and odd q (see [KLST90, Thm. 5.1] for instance). +It was proved by E. Artin [Art55a, Art55b] in the fifties for known families of simple +groups at the time, and completed by J. Tits for the remaining families discovered later +on (see [KLST90]). Artin’s method is to consider certain invariants associated to (orders +of) simple groups that can be computed explicitly and are able to distinguish the groups +easily. Therefore, the simple order theorem currently relies on the classification of finite +simple groups. +Theorem 4.1. Suppose that S and H are finite simple groups such that codpSq “ codpHq. +Then S – H. +Proof. The statement is trivial when both of S and H are abelian. If one of the two groups +is nonabelian, then, by Corollary 2.3, so is the other. So assume that both S and H are +nonabelian. By Theorem 3.2, we have |S| “ |H| and hence it follows that cdpSq “ cdpHq. +Assume to the contrary that S is not isomorphic to H. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +9 +By the simple order theorem, we have +tS, Hu “ tPSL4p2q, PSL3p4qu +or +tS, Hu “ tΩ2n`1pqq, PSp2npqqu +for some odd prime power q “ pℓ and n ě 3. The former case is eliminated using [Atl1], so +we just need to show that cdpΩ2n`1pqqq ‰ cdpPSp2npqqq for indicated n and q. +By Lusztig’s classification of ordinary characters of finite groups of Lie type (see [DM91, +Chapter 13]), irreducible characters of G :“ Sp2npqq are parameterized by pairs ppsq, ψq +where psq is the conjugacy class of a semisimple element s P G˚ :“ SO2n`1pqq and ψ is a +unipotent character of the centralizer C :“ CG˚psq. Moreover, the degree of the character +χppsq,ψq associated to ppsq, ψq is +χppsq,ψqp1q “ rG˚ : Csp1ψp1q. +Let α P t˘1u such that 4 | pqm ´ αq and consider a semisimple element s P G˚ with +spectrum t1, ´1, ..., ´1u such that C – GOα +2n (see [Ng10, Lem. 2.2]). Such s will then +belong to Ω2n`1pqq “ rG˚, G˚s, implying that the semisimple character χppsq,1Cq associated +to the pair ppsq, 1Cq is trivial on ZpSp2npqqq, by [NT13, Lem. 4.4]. We therefore have an +irreducible character of PSp2npqq of degree +χpsqp1q “ p|SO2n`1pqq|{|GOα +2n|qp1 “ pqn ` αq{2. +To see that cdpPSp2npqqq ‰ cdpΩ2n`1pqqq (for odd q and n ě 3), it is enough to show that +pqn`αq{2 is not character degree of Ω2n`1pqq. By [TZ96, Thm. 6.1], under our assumptions +on n and q and the additional condition pn, qq ‰ p3, 3q, the minimal (nontrivial) irreducible +character of Spin2n`1pqq has degree +dpSpin2n`1pqqq “ +" +pq2n ´ 1q{pq2 ´ 1q +if q ě 5 +pqn ´ 1qpqn ´ qq{2pq ` 1q +if q “ 3, +(For the definition of classical groups of various isogeny types, including the odd-dimensional +spin groups, we refer the reader to [Car85, p. 40].) Note that Spin2n`1pqq is a central +extension of Ω2n`1pqq and so every character degree of Ω2n`1pqq is one of Spin2n`1pqq. It +is now easy to check that dpSpin2n`1pqqq ą pqn ` αq{2 for n ě 3. For the remaining case +pn, qq “ p3, 3q, we note that dpSpin2n`1pqqq “ 27, which is still greater than pqn ` αq{2 “ +13. +□ +Certainly, one has to do much more work to relax the hypothesis in Theorem 4.1 to +codpSq Ď codpHq; that is, to obtain Theorem D. +Lemma 4.2. Let n ě 3 and q be an odd prime power. Then cdpΩ2n`1pqqq Ę cdpPSp2npqqq +and cdpPSp2npqqq Ę cdpΩ2n`1pqqq. +Proof. We have seen in the proof of Theorem 4.1 that PSp2npqq possesses an irreducible +character of degree pqn ` αq{2, where α P t˘1u such that 4 | pqn ´ αq, and furthermore +pqn ` αq{2 R cdpΩ2n`1pqqq. Therefore, it suffices to show cdpΩ2n`1pqqq Ę cdpPSp2npqqq. +We claim that both +pq2n ´ 1q{pq2 ´ 1q and qpq2n ´ 1q{pq2 ´ 1q + +10 +N. N. HUNG AND A. MORET ´O +are elements of cdpΩ2n`1pqqq for q odd. +Let G :“ Spin2n`1pqq, the universal cover of +Ω2n`1pqq. The dual group G˚ of G (in the sense of [DM91, Def. 13.10]) is the projective +conformal symplectic group PCSp2npqq, which is the quotient of rG “ CSp2npqq by its +center Zp rGq » Cq´1. +Consider a semisimple element s P rG with spectrum Specpsq “ +t´1, ´1, 1, ..., 1u and +C rGpsq – pSp2pqq ˆ Sp2n´2pqqq ¨ Cq´1 +(see [Ng10, Lem. 2.4]). Let s˚ be the image of s under the natural homomorphism from +rG to G˚. It is easy to see that, by the choice of s, C rGpsq is the complete inverse image of +CG˚ps˚q under this homomorphism, and thus CG˚ps˚q “ C rGpsq{Zp rGq and +rG˚ : CG˚ps˚qsp1 “ r rG : C rGpsqsp1 “ +|Sp2npqq|p1 +|Sp2pqq|p1|Sp2n´2pqq|p1 “ q2n ´ 1 +q2 ´ 1 , +where p is the defining characteristic of G. +Consider the canonical homomorphism f : Sp2pqq ˆ Sp2n´2pqq ãÑ C rGpsq Ñ CG˚ps˚q. +Using [DM91, Prop. 13.20], we know that unipotent characters of Sp2pqq ˆ Sp2n´2pqq are +of the form θ ˝ f where θ runs over the unipotent characters of CG˚ps˚q. In particular, as +Sp2pqq – SL2pqq has unipotent characters of degrees 1 and q, CG˚ps˚q has two unipotent +characters of degree 1 and q as well. By the conclusion of the previous paragraph, the +Lusztig series EpG, ps˚qq of G associated to the conjugacy class ps˚q of G˚ contains two +characters of degrees pq2n ´ 1q{pq2 ´ 1q and qpq2n ´ 1q{pq2 ´ 1q. +Note that s˚ P PSp2npqq “ rG˚, G˚s and |ZpGq| “ |G˚{pG˚q1| “ 2, and therefore every +character in the Lusztig series EpG, ps˚qq restricts trivially to ZpGq (see [NT13, Lem. 4.4]), +and so can be viewed as a character of G{ZpGq “ Ω2n`1pqq. The claim is completely proved. +Suppose first that q ą 3 and assume to the contrary that cdpΩ2n`1pqqq Ď cdpPSp2npqqq. +Then we have pq2n ´ 1q{pq2 ´ 1q P cdpPSp2npqqq, so that pq2n ´ 1q{pq2 ´ 1q P cdpSp2npqqq. +Let χ P IrrpSp2npqqq such that χp1q “ pq2n ´ 1q{pq2 ´ 1q. Now, as q ą 3, we have pq2n ´ +1q{pq2 ´1q ă pq2n ´1q{2pq `1q, and therefore, by the classification of irreducible characters +of Sp2npqq of degrees up to pq2n ´ 1q2pq ` 1q ([TZ96, Thm. 5.2]), we deduce that χ must +be one of the Weyl characters of degree pqn ˘ 1q{2 or the minimal unipotent one of degree +pqn ´ 1qpqn ´ qq{2pq ` 1q. A simple check reveals that none of these degrees matches the +degree of χ. +Now we suppose q “ 3. (In such case, pq2n ´ 1q{pq2 ´ 1q is indeed a character degree +of PSp2npqq.) +As the case of Ω7p3q and PSp6p3q can be checked directly, we suppose +furthermore that n ě 4. We then have qpq2n ´ 1q{pq2 ´ 1q ă pq2n ´ 1qpqn´1 ´ qq{2pq2 ´ 1q. +Examining the degrees up to pq2n´1qpqn´1´qq{2pq2´1q of irreducible characters of Sp2npqq +available in [Ng10, Cor. 4.2], we observe that none of them is equal to qpq2n ´ 1q{pq2 ´ 1q. +We have shown that qpq2n ´ 1q{pq2 ´ 1q R cdpSp2npqqq, and therefore, by the above claim, +cdpΩ2n`1pqqq Ę cdpPSp2npqqq, as desired. +□ +We can now prove our first main Theorem A, which in fact follows from the following +slightly stronger result. If G and H are groups we say that CpGq Ď CpHq if codpGq Ď +codpHq and mGpcq ď mHpcq for every c P codpGq. +Theorem 4.3. Let H be a finite simple group and G a nontrivial finite group such that +CpGq Ď CpHq. Then G – H. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +11 +Proof. Suppose first that H is abelian of prime order p. Then codpGq Ď codpHq “ t1, pu. +Therefore, by Lemma 2.4 of [DL16], G is an elementary abelian p-group and since kpGq ď p, +we conclude that G is cyclic of order p, as wanted. +So we may assume that H is nonabelian. By Corollary 2.3 and the assumption CpGq Ď +CpHq, we have that G is perfect. Let N IJ G such that S :“ G{N is nonabelian simple. +Now codpSq Ď codpGq Ď codpHq and therefore, by Theorem 3.1, we have |S| divides |H|. +Note that CpSq Ď CpGq Ď CpHq. Therefore there exists a subset I Ď IrrpHqzt1Hu and +a bijection f : IrrpSqzt1Su Ñ I such that +|S| +χp1q “ +|H| +fpχqp1q +for every χ P IrrpSqzt1Su. It follows that +ÿ +χPIrrpSqzt1Su +χp1q2 +|S|2 “ +ÿ +ψPI +ψp1q2 +|H|2 , +and thus +|S| ´ 1 +|S|2 +ď |H| ´ 1 +|H|2 +. +As the function px ´ 1q{x2 is decreasing on r2, 8q, we deduce that |S| ě |H|. +The conclusions of the last two paragraphs show that |S| “ |H|. +If S – H then +CpG{Nq “ CpSq “ CpHq Ě CpGq and so G{N and G have the same number of con- +jugacy classes, which is possible only when N “ 1, and we are done. +So assume by contradiction that S fl H. Using again the simple order theorem, we +have tS, Hu “ tPSL4p2q, PSL3p4qu or tS, Hu “ tΩ2n`1pqq, PSp2npqqu for some n ě +3 and odd q. +For the former pair, using the character tables of both PSL4p2q and +PSL3p4q available in [Atl1], one observes that 7 P cdpPSL4p2qqzcdpPSL3p4qq and 63 P +cdpPSL3p4qqzcdpPSL4p2qq, implying that none of codpPSL3p4qq and codpPSL2p4qq con- +tains the other, and this violates the fact that codpSq Ď codpHq. +The latter pair was +already handled in Lemma 4.2. +□ +5. The largest character degree of finite simple groups +Let bpGq denote the largest degree of an irreducible character of a finite group G. Recall +from the Introduction that if S is a simple group, then fpSq :“ |S|{bpSq is the smallest +nontrivial character codegree of S. The following elementary fact explains the relevance of +fpSq, and therefore bpSq. +Lemma 5.1. Let S and H be finite simple groups such that codpSq Ď codpHq. +Then +fpSq ě fpHq. +Proof. The hypothesis implies that fpSq P codpSq Ď codpHq. Since fpHq is the smallest +nontrivial member of codpHq, it follows that fpSq ě fpHq. +□ +Under the hypothesis of Lemma 5.1, we showed in Theorem C that |S| divides |H|. We +will see in later sections that, in many cases, the two conditions fpSq ě fpHq and |S| +divides |H| are enough to force S – H, as stated in Theorem D. + +12 +N. N. HUNG AND A. MORET ´O +Browsing through character tables of small-order simple groups in [Atl1], one notices +that if |H| is a multiple of |S| and |H| ą |S|, then bpHq ą bpSq. However, it seems that the +largest character degree grows slower than the order – that is, fpHq ą fpSq. This is not so +easy to prove generally, but we do confirm it in several cases, particularly when either S or +H is an alternating group (see Sections 7 and 8). +We shall need effective (both lower and upper) bounds for the largest degree of an irre- +ducible character of simple groups. For symmetric groups, asymptotic and explicit bounds +were obtained by A. M. Vershik and S. V. Kerov in [VK85] which can be used to derive the +corresponding bounds for alternating groups. For a group S of Lie type in characteristic +p, an obvious (and in fact very tight!) lower bound for bpSq is the degree StSp1q “ |S|p +of the Steinberg character StS. When S is of classical type, explicit upper bounds have +been worked out by M. Larsen, G. Malle, and P. H. Tiep in [LMT13]. Unfortunately, upper +bounds for exceptional groups achieved in [LMT13] are only asymptotic and its proof does +not allow one to obtain an explicit bound. We obtain Theorem 5.3 below that we believe +will be useful in other applications. +Lemma 5.2. Let G be a simple algebraic group over the algebraic closure of a finite field +of order q in characteristic p, F : G Ñ G a Steinberg endomorphism, and G :“ GF be the +corresponding finite group of Lie type. Let r be the rank of G. Then +pq ´ 1qr ¨ |G|p ď |G|p1 ď qr ¨ |G|p. +Proof. Note that finite groups GF of the same isogeny type have the same order, so we may +work with G being of simply-connected type. The inequalities are then straightforward to +verify using the order formulas for finite groups of Lie type available in [Atl1, p. xvi]. +□ +Theorem 5.3. Let S be a simple exceptional group of Lie type defined over a field of order +q in characteristic p. Then the following hold: +(i) bpSq ă 256|S|p. +(ii) If q ą 2, then bpSq ă 26|S|p. +Proof. The Tits group can be verified easily, so we assume that S ‰ 2F4p2q1. We then may +find a simple algebraic group G of adjoint type and a Steinberg endomorphism F : G Ñ G +such that S “ rG, Gs where G :“ GF (see [MT11, Prop. 24.21]). Clearly it suffices to show +that bpGq ă 256StGp1q. +Let pG˚, F ˚q be the dual pair of pG, Fq, so G˚ will be the corresponding simple algebraic +group of simply connected type, and set G˚ :“ pG˚qF ˚. As mentioned before, Lusztig’s +classification on complex characters of finite groups of Lie type implies that the set of +irreducible complex characters of G is partitioned into Lusztig series EpG, psqq associated +to various conjugacy classes psq of semisimple elements of G˚. +Furthermore, there is a +bijection χ ÞÑ ψ from EpG, psqq to EpCG˚psq, p1qq such that +(1) +χp1q “ +|G|p1 +|CG˚psq|p1 ψp1q. +The detailed structure of centralizers of semisimple elements in a finite exceptional groups +of Lie type was determined by Carter [Car78], Deriziotis [Der77], and Deriziotis-Liebeck +[DL85]. A well-known result of Steinberg states that the centralizer CG˚psq of a semisimple + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +13 +element s is a connected reductive subgroup of maximal rank in G˚. +Such connected +subgroup has a decomposition CG˚psq “ ST, where S is a semisimple subgroup, T is a +central torus, S X T is finite, and |pCG˚psqqF ˚| “ |S˚||T˚| (see [DL85, p. 48]). When s is +in G˚, the centralizer CG˚psq is F ˚-stable and CG˚psq “ pCG˚psqqF ˚; and so +|CG˚psq| “ |S||T| +where S :“ SF ˚ and T :“ TF ˚. Let r be the semisimple rank of G˚ and q (that will be a +power of p) the absolute value of all eigenvalues of F on the character group of an F-stable +maximal torus of G. Possible values for |S| and |T| are available in [Der77, DL85]. In +particular, we have +|S| “ +ź +i +|Lripqaiq| +and +|T| “ +ź +j +Φjpqq, +where Lripqaiqs are finite groups of Lie type (of simply-connected type) of rank ri defined +over a field of order qai and Φjpqqs are cyclotomic polynomials (and also polynomials of the +forms q2 ˘ +? +2q ` 1, q2 ˘ +? +3q ` 1, or q4 ˘ +? +2q3 ` q2 ˘ +? +2q ` 1 for Suzuki and Ree groups) +evaluated at q. As CG˚psq has maximal rank, we furthermore have +(2) +ÿ +i +airi ` +ÿ +j +degpΦjq “ r. +Now formula (1) implies that the typical degree of an irreducible character of G is of the +form +χp1q “ +|G|p1 +ś +i |Lripqaiq|p1 ś +j Φjpqqψp1q, +where ψ P EpCG˚psq, p1qq, a unipotent character of CG˚psq. By [LMT13, Thm. 3.1], for +any finite group of Lie type GF , where G is a simple algebraic group in characteristic p and +F a Steinberg endomorphism on G, the degree Stp1q “ |GF |p of the Steinberg character +St of GF is strictly larger than the degree of any other unipotent character. Therefore, +the degrees of unipotent characters of CG˚psq, which are in fact the same as those of the +semisimple group S, are bounded by ś +j |Lripqaiq|p. It follows that +bpGq ď +|G|p1 +ś +i |Lripqaiq|p1 ś +j Φjpqq +ź +j +|Lripqaiq|p, +By Lemma 5.2, +|Lripqaiq|p +|Lripqaiq|p1 ď +1 +pqai ´ 1qri ď +1 +pq ´ 1qairi . +Also, it is easy to see that +Φjpqq ě pq ´ 1qdeg Φj. +We therefore deduce that +bpGq ď +|G|p1 +pq ´ 1q +ř +i airi`ř +j degpΦjq “ +|G|p1 +pq ´ 1qr . + +14 +N. N. HUNG AND A. MORET ´O +On the other hand, we have |G|p1 ď |G|pqr by again Lemma 5.2, and it follows that +bpGq +|G|p +ď +qr +pq ´ 1qr . +As the rank r is at most 8 for exceptional groups, the desired inequalities follow. +□ +Bounds for bpSq of alternating groups and classical groups are collected in the following. +Lemma 5.4. Let S be a finite simple group, n a positive integer, and q a prime power. +(i) For S “ An with n ě 5, +1 +2e´1.28255?n? +n! ď bpSq ď e´0.11565?n? +n!. +In particular, +bpSq ą 1 +2e´1.28255?np2πnq1{4pn{eqn{2 +(ii) For S “ An with n ě 5, +bpAn`1q ě +2pn ` 1q +?8n ` 1 ` 3bpAnq. +(iii) For S “ PSLnpqq with n ě 2, +bpSq ă 13p1 ` logqpn ` 1qq2.54StSp1q. +(iv) For S “ PSUnpqq with n ě 3, +bpSq ă 2p2 ` logqpn ` 1q1.27StSp1q. +(v) For S P tΩ2n`1pqq, PSp2npqq, PΩ˘ +2npqqu with n ě 2 and q odd, +bpSq ă 38p1 ` logqp2n ` 1qq1.27StSp1q. +(vi) For S P tΩ2n`1pqq, PSp2npqq, PΩ˘ +2npqqu with n ě 2 and q even, +bpSq ă 8p1 ` logqp2n ` 1qq1.27StSp1q. +Proof. Part (i) follows from [VK85, Thm. 1] and Parts (iii)-(vi) follow from [LMT13, Thm. +5.1, 5.2, and 5.3]. Let us prove Part (ii). +Let χ P IrrpSnq such that χp1q “ bpSnq. Let λ be the partition of n corresponding to +χ and Yλ be the Young diagram associated to λ. By the well-known branching rule, the +induction χSn`1 of χ from Sn to Sn`1 is the sum of irreducible characters corresponding to +the partitions of n ` 1 whose associated Young diagrams are obtained from Yλ by adding a +suitable node. The number of those suitable nodes is at most p?8n ` 1`1q{2 (see [HHN16, +p. 1950]) and at most one of the resulting Young diagrams is symmetric. We deduce that +pn ` 1qbpAnq ď pn ` 1qχp1q “ χSn`1p1q ď +?8n ` 1 ` 3 +2 +bpAn`1q, +and the result follows. +□ + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +15 +6. Theorem D: Groups of Lie type +In this section we prove Theorem D when the groups involved are of Lie type. +In the following, for simplicity, we say that two groups have the same defining charac- +teristic if there is a common characteristic over which the groups can be defined. +Proposition 6.1. Let S and H be finite simple groups of Lie type. Suppose that codpSq Ď +codpHq. Then S and H have the same defining characteristic. +Proof. Suppose that the defining characteristic of H is p. By Lemma 3.4, |S|p1 “ |H|p1 and +there is χ P IrrpSq such that χp1q “ |S|p. By Lemma 3.3, it follows that +(3) +d ¨ |H|p +χp1q P cdpHq for every d P cdpSq. +Certainly if χ is the Steinberg character of S then we are done. So we assume otherwise +and aim to find a contradiction or end up with a case where H can be defined in another +characteristic not equal to p. By the classification of prime-power-degree representations +of quasi-simple groups [MZ01, Thm. 1.1], we arrive at the following possibilities of S and +χp1q. +(i) S “ PSL2pqq, χp1q P tq ˘ 1u or q is odd and χp1q P tpq ˘ 1q{2u. Observe that χp1q +cannot be pq ˘ 1q{2 because otherwise, by taking d “ 2χp1q, we would have 2|H|p P cdpHq, +which is impossible. +So χp1q “ q ` α “ px for some α P t˘1u and x P N. +Suppose +first that q “ 2t for some t ě 2. Then 2t ` α “ px. By Mihailescu’s theorem [Mih04] +(previously known as Catalan’s conjecture), either x “ 1 so that 2t ` α is a (Mersenne +or Fermat) prime or α “ 1 and t “ 3. In the latter case, p “ 3 and |H|31 “ |S|31 “ 56, +forcing H to be 2G2p3q1, which turns out to be isomorphic to S “ PSL2p8q, as desired. So +it remains to consider the former case: q ` α “ 2t ` α “ p is the defining characteristic +of H. Now |H|p1 “ |S|p1 “ qpq ´ αq. Let pa be the order of the underlying field of H. +It is clear from the order formulas of simple groups of Lie type (see [Atl1, p. xvi]) that +|H|p ă |H|p1{ppa ´ 1q ď |H|p1{pp ´ 1q. We therefore deduce that +|H|p ă qpq ´ αq +q ` α ´ 1. +Thus we must have α “ ´1 and |H|p “ p. +Now H is a simple group of Lie type in +characteristic p such that |H| “ ppp ` 1qpp ` 2q. This is impossible as for such a group H, +one can check from the order formula that |H|p1 ă p|H|pq2. +Next we suppose q ě 5 is odd. Then p “ 2 and q ` α “ |S|2. Again by Mihailescu’s +theorem, either q is a prime or α “ ´1 and q “ 9. The case q “ 9 is eliminated in the same +way as before. So assume that |H|21 “ qpq ´ αq{2 and q is a prime. Note that when H is +not of type A1, every prime divisor of |H|p1 is smaller than +a +|H|p1. Therefore our group H +must be PSL2pq1q for some 2-power q1, implying that qpq ´ αq{2 “ q2 +1 ´ 1. This, however, +returns no relevant solutions. +(ii) For the remaining possibilities of S and χ, the character χ has a decent small degree +and we are able to produce an irreducible character of S whose degree is a proper multiple +of χp1q. Condition (3) then implies that a proper multiple of |H|p is a character degree + +16 +N. N. HUNG AND A. MORET ´O +of H, which is impossible. This required character turns out to be chosen as a unipotent +character in most cases. We refer the reader to [Car85, §13.8] for the description of unipotent +characters of finite classical groups. +The next possibility of S and χp1q is S “ PSLnpqq, q ą 2, n is an odd prime, pn, q´1q “ 1, +and χp1q “ pqn ´ 1q{pq ´ 1q. If n “ 3 then SL3pqq has an irreducible character of degree +q3 ´1 (see [SF73]), which is a proper multiple of χp1q “ q2 `q`1. For n ě 5, the unipotent +character parameterized by the partition p2, n ´ 2q with degree +χp2,n´2qp1q “ pqn ´ 1qpqn´1 ´ q2q +pq ´ 1qpq2 ´ 1q +fulfills our requirement. +Another possibility is S “ PSUnpqq, n is an odd prime, pn, q ` 1q “ 1, and χp1q “ +pqn ` 1q{pq ` 1q. This case is handled similarly as in the previous one. Here we note that, +when n ě 5 is odd, the unipotent character parameterized by the partition p2, n ´ 2q has +degree +χp2,n´2qp1q “ pqn ` 1qpqn´1 ´ q2q +pq ` 1qpq2 ´ 1q +. +The next case is S “ PSp2npqq, n ě 2, q “ ℓt with ℓ an odd prime, tn is a 2-power, and +χp1q “ pqn ` 1q{2. Now the unipotent character parameterized by the symbol +`0 1 +n +˘ +has +required degree +χp0 +1 +n qp1q “ pqn ` 1qpqn ` qq +2pq ` 1q +. +The last possibility involving a family of groups is S “ PSp2np3q, n ě 3 is a prime, and +χp1q “ p3n ´ 1q{2. Then S has a unipotent character with degree +χp0 +1 +n +´ +qp1q “ p3n ´ 1qp3n ´ 3q +8 +. +(iii) pS, χp1qq P tpSp6p2q, 7q, pSp6p2q, 27q, p2F4p2q1, 27q, pG2p2q1, 7q, pG2p2q1, 27q, pG2p3q, 64qu. +First assume that S “ G2p2q1 and χp1q “ 27. Then p “ 3 and |H|31 “ |S|31 “ 224. It +is easy to check that there is no such simple group S of Lie type in characteristic 3 with +27 | |S|3. In all other cases one can find a character ψ P IrrpSq such that χp1q | ψp1q but +χp1q ă ψp1q. Therefore ψp1q|H|p{χp1q is a proper multiple of |H|p and thus cannot be a +character degree of H, violating condition (3). +□ +As seen in Lemma 3.4 and Proposition 6.1, we face the situation where two simple groups +S and H of Lie type have the same characteristics p and |S|p1 “ |H|p1. It is surprising to +us that this turns out to happen only when |S| “ |H| (see Proposition 6.3 below), and +therefore they are among the coincidences appeared in the simple order theorem. +We slightly modify two of the invariants in Artin’s proof of the simple order theorem (for +classical groups) to prove our results. +Definition 6.2. Let S be a finite group of Lie type in characteristic p. Let ω “ ωpSq and +ψ “ ψpSq respectively denote the largest and the second largest orders of p modulo a prime +divisor of |S|p1. We will refer to ωpSq and ψpSq as the Artin invariants of S. + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +17 +In fact, when p is the dominant prime of |S|, these ωpSq and ψpSq coincide with Artin’s +invariants defined in [Art55b]. We remark that there are only a few cases involving Mersenne +and Fermat primes where p is not dominant in |S|, and they are listed in [KLST90, Thm. +3.3]. +Assume for now that S is not one of G2p2q1, 2G2p3q1, and 2F4p2q1. (Note that S1 “ +G2p2q1 – PSU3p9q and S2 “ 2G2p3q1 – PSL2p8q and, even though we do not allow S1 (or +S2) to be viewed as a Lie type group over a field of order 2 (or 3), we do allow it to be +viewed as one over the field of order 9 (or 8).) Let q “ pt be the order of the underlying field +of S. It is well-known that the order |S| then has the standard cyclotomic factorization in +terms of q as +|S| “ 1 +dqk ź +i +Φipqqei, +where d, k, eis depend on S and can be found in [Atl1, Table 6] and [KLST90, Tables C1, +C2, and C3] for instance, and Φipqq is the value of the ith cyclotomic polynomial evaluated +at q. Replacing q by pn and factorizing each Φipxtq further into cyclotomic polynomials of +x, one has +|S| “ 1 +dpkt ź +i +Φippqfi +for certain positive integers fis depending on S. +Using Zsigmondy’s theorem, it is not difficult to see that, except for some ‘small’ cases, +the invariants ωpSq and ψpSq are precisely the largest and second largest, respectively, +index i such that Φippq appears in the cyclotomic factorization of |S| (see [KLST90, Lem. +4.6]). We refer the reader to [KLST90, Tables A1, A2 and A3] for the list of exceptions +and the values of their Artin’s invariants, including the groups G2p2q1, 2G2p3q1, and 2F4p2q1 +we excluded earlier. We reproduce in Table 1 the values of ωpSq and ψpSq for the generic +case only. +Proposition 6.3. Let p be a prime. Suppose that S and H are two non-isomorphic simple +groups of Lie type in characteristic p and |S|p1 “ |H|p1. Then tS, Hu “ tPSL4p2q, PSL3p4qu +or tS, Hu “ tΩ2n`1pqq, PSp2npqqu for some n ě 3 and odd q. In particular, |S| “ |H|. +Proof. By the assumptions, we have ωpSq “ ωpHq and ψpSq “ ψpHq. First we consider +the case where both S and H are generic so that their invariants ω and ψ are available +in Table 1. Comparing those values, we can find all the collections of groups with equal +values of ω and ψ. We list these collections in Table 2 (each row in the table is one such +collection). Now one simply compare the p1-parts of orders of groups in each collection. It +turns out that the only pair with the same p1-parts of orders is tΩ2n`1pqq, PSp2npqqu with +some n ě 3 and odd q. +Assume now that at least one of the two groups, say S, is non-generic. That is, S is +among the exceptions listed in the second column of Table 1. The values of the invariants +ω and ψ of these groups are available in [KLST90, Tables A2 and A3]. The analysis is +basically the same as in the generic case, but more tedious. We first find all the possible +groups H with ωpSq “ ωpHq and ψpSq “ ψpHq, and then compare |S|p1 and |H|p1, where +p is the defining characteristic of S and H. + +18 +N. N. HUNG AND A. MORET ´O +Table 1. ωpSq and ψpSq for simple groups of Lie type: generic case. +S +(q “ pt) +Conditions +(p a Mersenne prime) +ωpSq +ψpSq +PSLnpqq, n ě 2 +pn, qq ‰ p2, 26q, p3, 22q, p3, 23q, +p4, 22q, p6, 2q, p7, 2q, p2, p2q, p3, pq +nt +pn ´ 1qt +PSU4pqq +6t +4t +PSUnpqq, n ě 3 odd +pn, qq ‰ p3, 23q, p5, 2q, p3, pq +2nt +2pn ´ 2qt +PSUnpqq, n ě 6 even +pn, qq ‰ p6, 2q +2pn ` 1qt +2pn ´ 1qt +Ω2n`1pqq, n ě 2 +pn, qq ‰ p2, 28q, p3, 2q, p4, 2q, p2, pq +2nt +2pn ´ 1qt +PSp2npqq, n ě 3 +2nt +2pn ´ 1qt +PΩ` +2npqq, n ě 4 +pn, qq ‰ p4, 2q, p5, 2q +2pn ´ 1qt +2pn ´ 2qt +PΩ´ +2npqq, n ě 4 +pn, qq ‰ p4, 2q +2nt +2pn ´ 1qt +2B2p2tq, t ě 3 odd +t ” 3pmod6q +4t +4t{3 +2B2p2tq, t ě 3 odd +t ” ˘1pmod6q +4t +t +G2pqq, q ě 3 +q ‰ 4 +6t +3t +2G2p3tq, t ě 3 odd +6t +2t +3D4pqq +q ‰ 2 +12t +6t +F4pqq +12t +8t +2F4p2tq, t ě 3 odd +12t +6t +E6pqq +12t +9t +2E6pqq +18t +12t +E7pqq +18t +14t +E8pqq +30t +24t +Let us demonstrate the case S “ PSL3p4q as an example. Then ωpSq “ 4 and ψpSq “ 3. +But there are only two other simple groups of Lie type with the same values of ω and ψ, +namely PSL4p2q and PΩ` +8 p2q. However, |PΩ` +8 p2q|21 ‰ |PSL3p4q|21 “ |PSL4p2q|21, and so +we come up with another possible pair for tS, Hu, namely tPSL4p2q, PSL3p4qu, as stated +in the theorem. +□ +The next theorem improves Theorem 4.1 when the relevant groups are of Lie type. +Theorem 6.4. Let S and H be finite simple groups of Lie type such that codpSq Ď codpHq. +Then S – H. +Proof. By Lemma 3.4 and Propositions 6.1 and 6.3, we have that S and H fall into one +of two pairs of groups concluded in Proposition 6.3. The result now follows by Lemma +4.2. +□ +7. Theorem D: The mixed case of alternating groups and groups of Lie type +In this section, we prove Theorem D in the mixed situation where the set of codegrees of +an alternating group S is contained in that of a simple group H of Lie type, or vice versa. +In the following proposition we remark that the condition on m is necessary, due to the +coincidences of isomorphic simple groups: A5 – PSL2p4q – PSL2p5q, A6 – PSL2p9q, and + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +19 +Table 2. Simple groups of Lie type with the same values of ω and ψ: generic case. +PSLnpp2sq, Ω2n`1ppsq, PSp2nppsq, PΩ` +2pn`1qppsq, PΩ´ +2nppsq +PSL3pp2sq, PSU4ppsq, Ω7ppsq, PSp6ppsq, PΩ` +8 ppsq +PSL2pp6sq, Ω5pp3sq, G2pp2sq, 3D4ppsq +PSL2pp3sq, G2ppsq +PSL3pp4sq, PSU4pp2sq, Ω7pp2sq, PSp6pp2sq, PΩ` +8 pp2sq, F4ppsq +PSL2p26sq, Ω5p23sq, G2p22sq, 3D4p2sq, 2F4p2sq, s ě 3 odd +PSL4pp3sq, E6ppsq +PSL4pp6sq, Ω9pp3sq, PΩ` +10pp3sq, PΩ´ +8 pp3sq, E6pp2sq +PSL3pp6sq, PSU4pp3sq, Ω7pp3sq, PSp6pp3sq, PΩ` +8 pp3sq, 2E6ppsq +PSL3pp12sq, Ω7pp6sq, PSp6pp6sq, PΩ` +8 pp6sq, F4pp3sq, 2E6pp2sq +PSL5pp6sq, Ω11pp3sq, PSp10pp3sq, PΩ` +12pp3sq, PΩ´ +10pp3sq, E8ppsq +PSUnpqq, PSUn´1pqq, n ě 7 odd +PSU3p22sq, 2B2p23sq, s odd +PSU3p3sq, 2G2p3sq, s ě 3 odd +PSU9ppsq, PSU8ppsq, E7ppsq +Ω2n`1pqq, PSp2npqq, n ě 3, q odd +A8 – PSL4p2q. We also recall that fpXq :“ |X|{bpXq, where bpXq is the largest character +degree of X. +Proposition 7.1. Suppose m “ 7 or m ě 9. Let H be a simple group of Lie type. If |Am| +divides |H|, then fpHq ą fpAmq. As a consequence, codpAmq Ę codpHq. +Proof. Let p be the defining characteristic and q, a power of p, the order of the underlying +field of H. Consider first the case H being of exceptional type. Using Lemma 5.3, we have +fpHq ą +|H| +256|H|p +“ +1 +256|H|p1. +As |Am| divides |H|, it follows that fpHq ą p1{256q|Am|p1. Therefore, to prove the theorem, +it suffices to show +bpAmq ě 256|Am|p. +Let us assume for now that that m ě 10. In particular, the dominant prime in |Am| “ m!{2 +is 2. We therefore just need to show bpAmq ě 256|Am|2. As |Am|2 ď 2m´2, for this we want +to show +bpAmq ě 64 ¨ 2m. +Note that bpA19q “ 64, 664, 600 ą 64 ¨ 219 (see [McK86] for the degree of the largest irre- +ducible characters and associated partitions of symmetric groups of degree up to 75, from +which one can deduce the exact value or a good bound for the one of corresponding alter- +nating groups). Now one just inducts on m with the help of Lemma 5.4(ii) to achieve the +desired bound for m ě 19. +Suppose that m ď 18 and recall that we are still dealing with exceptional groups. When +q “ 2, the proposition can be verified directly, so assume that q ě 3. +In such case, +bpHq ă 26|H|p by Lemma 5.3, and whence the above estimate can be refined so that we + +20 +N. N. HUNG AND A. MORET ´O +only need to prove bpAmq ě 26|Am|p, which turns out to be true for all 18 ě n ě 13. For the +remaining values m ď 12, the arguments go as follows. First we are done if fpAmq ď +a +|H|, +as fpHq ą +a +|H|, so we may assume that |H| ă fpAmq2. For each m ď 12, there are indeed +no possibilities for H satisfying |H| ă fpAmq2 and |Am| divides |H|. +Following the same idea as in the case of exceptional groups, but using Lemma 5.4 +instead, we can show that in fact fpHq ą fpAmq for every H of classical type and m ě 19. +Let us present the details for only the linear groups. +Consider H “ PSLnpqq for some n ě 2 and q a prime power. By Lemma 5.4(i), we have +fpAmq “ +m! +2bpAmq ď e1.28255?mpm!q1{2. +Thus, if |H| ě e2.5651?mm! then fpHq ą +a +|H| ě e1.28255?mpm!q1{2 ě fpAmq and we would +be done. We therefore can assume that |H| ă e2.5651?mm!, which in particular implies +that n ă m. +Using Lemma 5.4(iii), we see that, as before, it is enough to show that +bpAmq ě 13p1 ` logqpn ` 1qq2.54|Am|p. Since m ě n ` 1 and |Am|p ď 2m´2, for this it is +sufficient to show that +bpAmq ě 13p1 ` log2 mq2.542m´2. +This last inequality is indeed true for m “ 20, and therefore is true for all m ě 20, +by induction and Lemma 5.4(ii). Checking directly, we see that the inequality bpAmq ě +13p1 ` logqpn ` 1qq2.54|Am|2 is still valid for n “ 19. +As for the exceptional types, we are left to consider the small cases m ď 18. Again we +are done if fpAmq ď +a +|H|, so we may assume that |H| ă fpAmq2. For each m, we search +for relevant H satisfying |H| ă fpAmq2 and |Am| divides |H| and find that, for such an H, +the inequality fpHq ą fpAmq always holds true. +□ +We shall need the following result on 2-defect zero and 3-defect zero characters of alter- +nating groups, which easily follows from earlier work of F. Garvan, D. Kim and D. Stanton +[GKS90] on the so-called p-core partitions. They are partitions having no hook lengths di- +visible by p. Using Garvan-Kim-Stanton’s result, A. Granville and K. Ono [GO96] proved +the existence of p-defect zero characters with p ě 5 in symmetric and alternating groups. +Lemma 7.2. Let m be a positive integer. +(i) Am has a 2-defect zero irreducible character if and only if m “ 2k2 ` k or m “ +2k2 ` k ` 2 for some k P N. +(ii) Am has a 3-defect zero irreducible character if and only if there is a prime ℓ ” 2pmod +3q such that the the exact power of ℓ dividing 3m ` 1 is odd. +Proof. See the discussion in [GO96, pp. 333-334]. +□ +Proposition 7.3. Let S be a simple group of Lie type and 8 ‰ m ě 7 an integer. Then +codpSq Ę codpAmq. In fact, if |S| divides |Am| and m ě 44, then fpSq ă fpAmq. +Proof. Assume by contradiction that codpSq Ď codpAmq. By Lemma 5.1, we then have +fpSq ě fpAmq. +Suppose that the defining characteristic of S is p. Observe that fpSq ď |S|{StSp1q “ +|S|p1. Furthermore, |S|p1 ă |S|2 +p (see [CHMN15, Proof of Thm. 12]) and |S|p ď |Am|p by + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +21 +Theorem 3.1. Therefore we have fpSq ă p|Am|pq2. Assume for a moment that m ě 10 so +that |Am|p ď |Am|2 ď 2m´2. We now have fpSq ă 22m´4. On the other hand, it is clear +that fpAmq ą +a +m!{2. Therefore, we would be done if m! ě 24m´7. By the well-known +estimate m! ą +? +2πmpm{eqm, this is certainly true when m ě 44. So we may now suppose +that m ď 43. +As mentioned above, every simple group of Lie type, and therefore S in particular, has +a 2-defect zero irreducible character, which means that S has an odd codegree and so does +Am as codpSq Ď codpAmq. It follows that m “ 2k2 `k or m “ 2k2 `k`2 for some k P N, by +Lemma 7.2(i). This forces m to be one of 10, 12, 21, 23, 36, or 38. By the same reason, Am +has a codegree not divisible by 3 and so Lemma 7.2(ii) further narrows down the choices +for m: m P t10, 12, 21, 36u. In fact, when m “ 21 or 36, we still have fpAmq ą |Am|2 +2, and +since |Am|2 +2 ą fpSq, it follows that fpAmq ą fpSq, which is a contradiction. +Suppose m “ 10. The inequality fpAmq ă p|Am|pq2 then forces p “ 2 or 3. If p “ 2 +then |S|21 “ |A10|{χp1q, where χ P IrrpA10q is one of the two 2-defect zero irreducible +characters of equal degree 384, implying |S|21 “ 10!{p2 ¨ 384q “ 4725. It is easy to see +from [Atl1] that there is no such group of Lie type in characteristic 2. +If p “ 3 then +|S|31 “ 10!{p2 ¨ 567q “ 3200 since A10 has a unique 3-defect zero character of degree 567, +which again leads to a contradiction as there is no such group in characteristic 3. The case +m “ 12 is treated similarly and we skip the details. +□ +8. Theorem D: Alternating and sporadic groups +Proposition 8.1. Let m ă n be positive integers. Then fpAmq ă fpAnq. Consequently, +codpAmq Ę codpAm`1q. +Proof. It suffices to show that bpAm`1q ă pm ` 1qbpAmq. Let χ P IrrpAm`1q such that +χp1q “ bpAm`1q. As shown in [HHN16, p. 1956], such χ must be the restriction of an irre- +ducible character, say ψ, of Sm`1 whose associated partition, say λ, is not self-conjugate. +In particular, χp1q “ ψp1q. As in the proof on Lemma 5.4(ii), let Yλ be the Young diagram +associated to λ. The restriction ψSm of ψ to Sn is the sum of irreducible characters corre- +sponding to the partitions of n whose associated Young diagrams are obtained from Yλ by +removing a suitable node. The number of those suitable nodes is at most p?8m ` 9 ´ 1q{2, +so +bpAm`1q “ ψp1q ď +?8m ` 9 ´ 1 +2 +bpSmq. +Since bpSmq ă 2bpAmq as already mentioned above, it follows that +bpAm`1q ă p +? +8m ` 9 ´ 1qbpAmq, +which implies our desired inequality bpAm`1q ă pm ` 1qbpAmq for m ě 5. The result is +easily checked for smaller m. +□ +Proposition 8.2. Theorem D is true when either S or H is a sporadic simple group. +Proof. The case where both S and H are sporadic simple groups can be verified by using +the available data in [Atl1]. +Suppose that S is a sporadic group and H “ Am for some m ě 5. Let pS be the largest +prime divisor of |S|. By Theorem 3.1, we have |S| divides |Am|, so pS ď m. By Lemma + +22 +N. N. HUNG AND A. MORET ´O +5.1, we have fpSq ě fpAmq ą +a +m!{2. It follows that fpSq ě +a +pS!{2. Again using [Atl1], +it can be checked that this can never happen. +Next we assume that S is a sporadic group and H is a simple group of Lie type in +characteristic p. Suppose first that S has an irreducible character, say χ, of p-defect zero. +Then, as argued in the proof of Lemma 3.4, we have |S|p1 “ |H|p1 and χp1q “ |S|p. In +particular, χp1q is a prime power, and therefore, [MZ01, Thm. 1.1] yields +pS, p, χp1qq P tpM11{M12, 11, 11q, pM11, 2, 16q, pM24{Co2{Co3, 23, 23qu +(We note that M12 has another irreducible character of prime power degree, namely 16, +but the character is not of 2-defect zero and thus does not fit our situation.) However, for +each of these possibilities, there is no simple group of Lie type H in characteristic p such +that |H|p1 “ |S|p1. Next, we suppose that S has no characters of p-defect zero. By [GO96, +Cor. 2], +p P t2, 3u and S P tM12, M22, M24, J2, HS, Suz, Ru, Co1, Co3, BMu. +Now we just apply Lemma 5.4 and argue similarly as in the proof of Proposition 7.1, with +S in place of Am, to arrive at fpHq ą fpSq, and thus it follows from Lemma 5.1 that +codpSq Ę codpHq. +Now we consider the case where S “ Am for some m ě 5 and H a sporadic simple group. +Using Theorem 3.1, we have m!{2 divides |H| and so m is at most pH ´ 1, where pH is the +smallest prime not dividing |H|. This constraint is enough to ensure that fpAmq ă fpHq, +and thus codpAmq Ę codpHq by Lemma 5.1. +Finally we consider the case where S is a simple group of Lie type and H a sporadic +simple group. As in the proof of Proposition 7.3 , we have fpHq ď p|H|pq2, where p is the +defining characteristic of S. The only possible p satisfying such condition is p “ 2. Now +|S|21 is an odd codegree of S, and hence of H, and so |S|21 “ |H|{χp1q for some 2-defect zero +character χ P IrrpHq. There are in fact only 16 sporadic simple groups having a 2-defect +zero irreducible character. For such a group and such a character, there are no S satisfying +the indicated condition. This concludes the proof. +□ +Theorem D follows from Theorem 6.4 and Propositions 7.1, 7.3, 8.1, and 8.2. +For future work on Huppert’s codegree conjecture (HCC), we record the following im- +mediate consequence of Theorem D. +Theorem 8.3. Let S be a finite nonabelian simple group. +Let G be a minimal coun- +terexample to (HCC) with respect to S – that is, G is minimal subject to the conditions +codpGq “ codpSq and G fl S. Then G has a unique minimal normal subgroup N and +G{N – S. +Proof. Let N be a maximal normal subgroup of G. Since codpG{Nq Ď codpGq “ codpSq +and G{N is simple, it follows from Theorem D that G{N – S. +Furthermore, by the +minimality of G as a counterexample, we have that N is a minimal normal subgroup of G. +(If G has a normal subgroup M such that M ă N, then codpG{Nq Ď codpG{Mq Ď codpGq, +forcing codpG{Mq “ codpSq.) Also, N is the unique minimal normal subgroup of G since, +otherwise, G “ S ˆ S, which violates the assumption codpGq “ codpSq. +□ + +THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS +23 +We conclude the paper with a couple of remarks. First, the group pseudo-algebra CpGq +seems to better distinguish finite groups than the usual complex group algebra CG. For +instance, while any two abelian groups A and B of the same order have the same complex +group algebra CA “ CB, it was shown in [Mor22] that A – B if and only if CpAq “ CpBq. +It has even been speculated that a finite group G and an abelian group A are isomorphic +if and only if CpGq “ CpAq. 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Algebra 113 (1988), +511-522. 3, 8 +Department of Mathematics, The University of Akron, Akron, OH 44325, USA +Email address: hungnguyen@uakron.edu +Departamento de Matem´aticas, Universidad de Valencia, 46100 Burjassot, Valencia, Spain +Email address: alexander.moreto@uv.es + diff --git a/l9AyT4oBgHgl3EQfk_jh/content/tmp_files/load_file.txt b/l9AyT4oBgHgl3EQfk_jh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed4f86eba56a281bf8dd038a53bd08ed21d3e06b --- /dev/null +++ b/l9AyT4oBgHgl3EQfk_jh/content/tmp_files/load_file.txt @@ -0,0 +1,1411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf,len=1410 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='00446v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='GR] 1 Jan 2023 THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS NGUYEN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND ALEXANDER MORET´O Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We study the codegree isomorphism problem for finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In par- ticular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits’ simple order theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Introduction The group ring/algebra isomorphism problem has a long history, dating back to the work of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Higman [Hig40] in the forties on integral group rings of abelian groups and its study has lead to important discoveries in group theory, ring theory and representation theory (see the recent survey [Mar22] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Its most general form is: for a ring/field F and finite groups G and H, does a group ring/algebra isomorphism FG – FH imply a shared property of G and H or even a group isomorphism G – H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The problem includes a number of more prominent cases: the Integral Isomorphism Problem [Hig40, San85, Sco86, Her01], the Complex-group-algebras Isomorphism Problem [Bra63, Isa86, BNOT15, SIS20], and the Modular Isomorphism Problem [Des56, Pas74, May14, Mar22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The isomorphism problem for complex group algebras (when F “ C), by the well- known Wedderburn theorem, is equivalent to the one for character degrees: which finite group is determined by its multiset of character degrees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=', the set of character degrees counting multiplicities)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This has been confirmed in the affirmative for (quasi)simple groups [BNOT15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The version without multiplicity (and therefore stronger) for finite simple groups is a conjecture proposed by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Huppert in the late nineties [Hup00], which has been also extensively studied over the past two decades [Hup06, NT15, BTZ17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The results to date on these character-degree isomorphism problems (both the group-algebras problem and Huppert’s conjecture) were achieved essentially on a case-by-case basis using the classification of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This paper is concerned with the codegree isomorphism problem for finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For a character χ of a finite group G, the codegree of χ is codpχq :“ |G : kerpχq|{χp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Primary 20C15, 20C30, 20C33, 20D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Character codegrees, isomorphism problem, finite simple groups, Huppert’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Part of this work was done while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM), and he thanks the VIASM for its hospitality and financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The research of the second author is supported by Ministerio de Ciencia e Innovaci´on (Grant PID2019-103854GB-I00 funded by MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='13039/501100011033) and Generalitat Valenciana CIAICO/2021/163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O This notion was first introduced and studied (in a slightly different form) by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Chillag and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Herzog [CH89] and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Chillag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Mann, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Manz [CMM91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It was later developed into the current form used nowadays by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Qian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Wei [QWW07] and has been proved to have remarkable connections with the structure of finite groups [QWW07, Isa11, DL16, Mor21, Qia21, CN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let codpGq denote the set of all the codegrees of irreducible characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Recently, there has been great interest in the codegree analogue of Huppert’s conjecture (Prob- lem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='79 of the Kourovka Notebook [Kou20]), to which we will refer to as Huppert’s codegree conjecture: (HCC) Let S be a finite nonabelian simple group and G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then codpGq “ codpSq if and only if G – S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The approach so far to this problem is more or less similar to Huppert’s original method, and therefore, unfortunately, is still case-by-case [BAK21, Aha22, GKL+22, GZY22, LY23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let codpGq “ tc1 ă c2 ă .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' ă cku and mGpciq be the number of irreducible characters of G with codegree ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The multiset CpGq :“ tpci, mGpciqq : 1 ď i ď ku is called the group pseudo-algebra of G, which can be viewed as the codegree counterpart of the aforementioned complex group algebra CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A natural weaker version of (HCC) asks whether G and S must be isomorphic if CpGq “ CpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Our first principal result solves this codegree-with-multiplicity isomorphism problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a finite simple group and G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then CpGq “ CpSq if and only if G – S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The main novelty of this paper is a more uniform approach to these codegree problems with as little case-by-case analysis as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Our proof of Theorem A, somewhat surpris- ingly, only relies on the classification via the so-called simple order theorem (also known as the Artin-Tits theorem [Art55b, KLST90]), which states that two non-isomorphic finite simple groups have the same order if and only if they are either PSL4p2q and PSL3p4q or Ω2n`1pqq and PSp2npqq for some n ě 3 and odd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This is perhaps the first time that a result of this type is proved uniformly for all simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The following are two key ingredients in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We find it remarkable that they admit strikingly elementary proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The first provides a characterization of perfect groups in terms of codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (Recall that a group is perfect if it coincides with its derived subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A finite nontrivial group G is perfect if and only if G has no prime character codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The second ingredient we are referring to is an order-divisibility property involving char- acter codegrees of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S is a finite simple group and G a finite group such that codpSq Ď codpGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |S| divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, if G and S are simple groups and codpGq “ codpSq then |G| “ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 3 Our next main result is far stronger than the second statement in Theorem C, but does require the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S and H be finite simple groups such that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then S – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D in fact is the first step in proving Huppert’s codegree conjecture (HCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be any finite group and H a simple group such that codpGq “ codpHq, and N be a maximal normal subgroup of G so that S :“ G{N is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In order to prove G – H, one would first need to establish S – H, under the assumption codpSq Ď codpGq “ codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This is precisely what we do in Theorem D (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Theorem D, we will prove in a subsequent paper [HMT] that (HCC) holds for all sporadic groups, alternating groups, groups of Lie type of low rank, and, for the first time in the degree/codegree isomorphism problem, groups of Lie type of arbitrary rank over a field of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Furthermore, perhaps unexpectedly, we reduce (HCC) to a problem on p-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The proof of Theorem D is fairly complicated and combines several tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, it essentially utilizes, besides Theorem C, some deep results on the representation theory of finite simple groups, including the classification of prime-power- degree representations [MZ01, BBOO01], lower/upper bounds for the largest degree of irreducible representations [VK85, LMT13], and the existence of p-defect zero characters [Mic86, Wil88, GO96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Along the way we prove an effective and explicit upper bound for the largest character degree bpSq of an exceptional group S of Lie type (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (See Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 for previous related work on symmetric and alternating groups [VK85] and classical groups [LMT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We need bounds for the largest character degree bpSq in order to control the behavior of fpSq :“ |S|{bpSq – the smallest nontrivial codegree of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' While the relevance of the smallest (or low-degree in general) characters of (quasi/almost)simple groups is well- known in group representation theory (see [MM21] for the latest results), the smallest codegree had not been studied much before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This invariant arises naturally in the proof of Theorem D (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1) and measures the relative growth of bpSq compared to |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Our proof would be much simpler if one can show that f is divisibly increasing among nonabelian simple groups, by which we mean that, if S and H are nonabelian simple of different orders such that |S| divides |H|, then fpSq ă fpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We indeed confirm this phenomenon in many cases, particularly when one of the two groups involved is alternating (see Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The layout of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In Section 2, we prove some results on prime character codegrees, including Theorem B, and provide a short proof of a theorem of Riese and Schmid (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='7) on prime-power codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Section 3 is devoted to the proof of Theorem C and its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using the results in the preceding sections, we prove Theorem A in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Results on bounding the largest character degree are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Finally, the proof of Theorem D is carried out in Sections 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Prime-power codegrees In this section we prove some results on prime-power character codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' These results show that, in contrast to character degrees, there are significant restrictions on the structure of groups with faithful irreducible characters of prime/prime-power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We mainly follow the notation from [Isa76] for character theory and [Atl1, Car85] for finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Throughout, for a positive integer n and a prime p, we write np to denote the maximal p-power divisor of n and np1 :“ n{np to denote the maximal divisor not divisible by p of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let N IJ G and θ P IrrpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We write IrrpG|θq for the set of irreducible constituents of θG and IrrpG|Nq for the set of irreducible characters of G whose kernels do not contain N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If G is a group, πpGq is the set of primes that divide |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If n is an integer, πpnq is the set of primes that divide n, and if S is a set of integers, then πpSq is the set of primes that divide some member of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As usual, cdpGq :“ tχp1q : χ P IrrpGqu is the set of all irreducible character degrees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Other notation will be recalled or defined when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We begin by collecting some known facts on character codegrees that we will use without explicit mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a finite group and χ P IrrpGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The following hold: (i) If χ is not the principal character, then codpχq ą χp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (ii) If N IJ G and N ď kerpχq, then the codegree of χ as a character of G coincides with the codegree of χ viewed as a character of G{N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (iii) If N IJIJ G and θ P IrrpNq lies under χ, then codpθq divides codpχq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (iv) πpGq “ πpcodpGqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (v) If G is abelian, then codpGq “ opGq, where opGq is the set of orders of the elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Part (i) is [DL16, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Parts (ii) and (iii) are contained in [QWW07, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1], and part (iv) is [QWW07, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now, we prove part (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The inclusion opGq Ď codpGq follows from [DL16, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Conversely, if d P codpGq there exists χ P IrrpGq such that d “ |G : kerpχq| (note that since G is abelian, χ is linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since G{ kerpχq is cyclic, we conclude that G has elements of order d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Prime codegrees: characterizing perfect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The goal in this subsection is to provide a characterization of perfect groups in terms of the absence of prime codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that there exists χ P IrrpGq faithful such that codpχq “ p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then G is either the cyclic group of order p or a Frobenius group with Frobenius kernel of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We argue by induction on |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let N IJ G be minimal such that there exists θ P IrrpNq lying under χ with codpθq “ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |G| χp1q “ p “ codpθq “ rN : kerpθqs θp1q , and we deduce that p “ codpχq ą χp1q “ rG : Ns| kerpθq|θp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 5 In particular, p does not divide any of the three factors in the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose first that N ă G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the inductive hypothesis, N{ kerpθq is cyclic of order p or a Frobenius group with Frobenius kernel K{ kerpθq of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In the latter case, if λ P IrrpK{ kerpθqq lies under θ then codpλq “ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This contradicts the choice of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (Note that K is normal in G because K is characteristic in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Hence, we may assume that N{ kerpθq is cyclic of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Clifford’s theorem the faithful character χN is a sum of G-conjugates of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let T be a complete set of represen- tatives in G for these conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By [Isa76, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='21], the intersection of kerpθgq, where g runs over T, is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We conclude that N embeds into the direct product ź gPT N{ kerpθgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Each of the direct factors has order p, and so N is an elementary abelian p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since p does not divide | kerpθq|, we conclude that N is cyclic of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As θ is linear and χp1q “ |G|{p, we now have χ “ θG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that G is a Frobenius group with kernel N, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now, we consider the case N “ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let M be a maximal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since N “ G the codegree of any irreducible character of N lying under χ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This means that χM “ χp1q1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' But χ is faithful, so we deduce that M “ 1 and G is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If G is abelian, then it is the cyclic group of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If G is not abelian, then |G|{p “ χp1q ă a |G| and it follows that |G| ă p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Sylow’s theorems, it follows that G has a normal Sylow p-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ The following consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2 is Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A finite nontrivial group G is perfect if and only if G does not have any prime character codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2, if G has an irreducible character χ of prime codegree then G{ kerpχq is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, G is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Conversely, if G is not perfect, then the abelian group G{G1 has some irreducible character of prime codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Prime power codegrees: the Riese-Schmid theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Chillag and Herzog proved in [CH89, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1] that a simple group does not possess nontrivial irreducible characters of prime power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The proof relied on a case by case analysis of the simple groups, using the fact that, most of the times, they have p-blocks of defect zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This was generalized by Riese and Schmid in [RS98, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3] to quasisimple groups, using also block theory and the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We offer a short proof of this result that only depends on an easy consequence of the classification, which is due to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Kimmerle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lyons, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Sandling, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Teague [KLST90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For every finite simple group S and prime p, |S| ă p|S|p1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The following is a restatement of [RS98, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1] in the language of codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let p be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a finite group and χ P IrrpGq faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then codpχq is a power of p if and only if χ is induced from a Sylow p-subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A quasisimple group G does not possess nonprincipal characters of prime power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that there exists 1G ‰ χ P IrrpGq of p-power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let K :“ kerpχq and note that K ď ZpGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='5, χ is induced from a Sylow p-subgroup of G{K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, χp1q ě |G{K|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4, we know that |G{ZpGq| ă p|G{ZpGq|p1q2 ď p|G{K|p1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Hence, by [Isa76, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='30], χp1q ď |G : ZpGq|1{2 ă |G{K|p1, which violates the inequality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ The next result is a restatement in terms of character codegrees of Theorem B of [RS98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The proof in [RS98] uses Brauer’s first and third main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Recall that if a group G has trivial p1-core Op1pGq, then it is defined to be p-constrained if the p-core OppGq contains its centralizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='7 (Riese-Schmid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a finite group and let p be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that χ P IrrpGq is faithful of p-power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then Op1pGq “ 1 and G is p-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6, we know that G is not simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let N be a minimal normal subgroup of G and let θ P IrrpNq lying under χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since θ has p-power codegree (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1(iii)) and codpθq ą 1 (note that since χ is faithful, θ ‰ 1N), we deduce that N is either an elementary abelian p-subgroup or a direct product of nonabelian simple groups of order divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, Op1pGq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We claim that N is an elementary abelian p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that N “ S1 ˆ ¨ ¨ ¨ ˆ St, with Si – S for some nonabelian simple group S of order divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We wish to reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since θ ‰ 1N, there exists a nonprincipal ψ P IrrpSiq lying under θ for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that codpψq is a power of p and this contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Write P :“ OppGq and C :“ CGpPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that C X P “ ZpPq and Op1pCq “ 1 (because Op1pGq “ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We want to see that C ď P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Take K subnormal in G such that ZpPq ď K ď C and K{ZpPq is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since Op1pCq “ 1 and G does not have nonabelian minimal normal subgroups, we conclude that K1 is quasisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now, take γ P IrrpK1q lying under χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Again, we have that γ is not principal and codpγq is a p-power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ We end this section with a variation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that p is a prime and χ P IrrpGq is faithful of p-power codegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then codpχq exceeds the p-part of the product of the orders of the nonabelian composition factors in a composition series of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, if K{L is a non-abelian chief factor of G, then codpχq ą |K{L|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let n be the product of the orders of the non-abelian composition factors in a composition series of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using again that χ is induced from a Sylow p-subgroup and [KLST90], we have codpχq ą χp1q ě |G|p1 ě np1 ą np, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' An order-divisibility result for codegrees The next result is the first part of Theorem C, which will be crucial in the proofs of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S is a finite simple group and G a finite group such that codpSq Ď codpGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |S| divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=', dk be all the degrees of nontrivial irreducible characters of S, and let mi (1 ď i ď k) be the number of those characters of degree di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the assumption, for each i, there exists χi P IrrpGq such that |S| di “ rG : kerpχiqs χip1q “ |G| χip1q| kerpχiq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that kÿ i“1 mid2 i |S|2 “ kÿ i“1 miχip1q2| kerpχiq|2 |G|2 , and thus řk i“1 miχip1q2| kerpχiq|2 |G|2 “ řk i“1 mid2 i |S|2 “ |S| ´ 1 |S|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore |S|2 divides |G|2p|S| ´ 1q, and the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ We record some consequences of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 that will be needed in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S and H are finite simple group such that codpSq “ codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |S| “ |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This directly follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is a basic fact that the group algebra of a finite group determines the order of the group, just note that |G| “ ř χPIrrpGq χp1q2, but it is not known whether the group pseudo-algebra determines the order of the group (see [Mor22, Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' However, we do not think that the analogue of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2 for character degrees can be proved without the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S and H are finite nonabelian simple groups such that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let x :“ |H|{|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then x P N and dx P cdpHq for every 1 ‰ d P cdpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We know that x P N by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For each 1 ‰ d P cdpSq, we have |S|{d P codpHq, and thus there exists some χ P IrrpHq such that |S|{d “ |H|{χp1q, implying that χp1q “ dx, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S and H be finite simple groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that the defining characteristic of H is p and codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |S|p1 “ |H|p1 and there exists χ P IrrpSq such that χp1q “ |S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We first observe that |S| is divisible by p because otherwise every codegree of S is not divisible by p but the only nontrivial codegree of H not divisible by p is |H|p1 “ |H|{StHp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By [Mic86, Wil88], S has an irreducible character, say χ, of p-defect 0, so that codpχq is coprime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore we have |S|{χp1q “ |H|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 that χp1q|H|p1 “ |S| divides |H|, implying that |S|p1 “ |H|p1 and χp1q “ |S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Remark that when p ě 5, the above result holds for all S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This is because, in such case, irreducible p-defect zero characters still exist in alternating groups (by [GO96]) and in sporadic simple groups [Atl1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a finite simple group of Lie type and H a finite nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that codpSq Ď codpHq and there are primes p ‰ r such that H has a unique character degree divisible by each |H|p and |H|r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then |S| “ |H| and cdpSq Ď cdpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Repeating the arguments in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4, we have |S|p1 “ |H|p1 and |S|r1 “ |H|r1, implying that |S| “ |H| and cdpSq Ď cdpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Group pseudo-algebras of simple groups: Theorem A In this section we prove Theorem A, using the results in the preceding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 is useful in proving results concerning codegrees of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' One of them is the next theorem, whose proof makes use of the simple order theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Recall that the simple order theorem asserts that two non-isomorphic finite simple groups have the same order if and only if they are either PSL4p2q and PSL3p4q (of order 20, 160) or Ω2n`1pqq and PSp2npqq (odd-dimensional orthogonal and symplectic groups, of order p1{2qqn2 śn i“1pq2i ´ 1q) for some n ě 3 and odd q (see [KLST90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1] for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It was proved by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Artin [Art55a, Art55b] in the fifties for known families of simple groups at the time, and completed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Tits for the remaining families discovered later on (see [KLST90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Artin’s method is to consider certain invariants associated to (orders of) simple groups that can be computed explicitly and are able to distinguish the groups easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, the simple order theorem currently relies on the classification of finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S and H are finite simple groups such that codpSq “ codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then S – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The statement is trivial when both of S and H are abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If one of the two groups is nonabelian, then, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, so is the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So assume that both S and H are nonabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2, we have |S| “ |H| and hence it follows that cdpSq “ cdpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume to the contrary that S is not isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 9 By the simple order theorem, we have tS, Hu “ tPSL4p2q, PSL3p4qu or tS, Hu “ tΩ2n`1pqq, PSp2npqqu for some odd prime power q “ pℓ and n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The former case is eliminated using [Atl1], so we just need to show that cdpΩ2n`1pqqq ‰ cdpPSp2npqqq for indicated n and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lusztig’s classification of ordinary characters of finite groups of Lie type (see [DM91, Chapter 13]), irreducible characters of G :“ Sp2npqq are parameterized by pairs ppsq, ψq where psq is the conjugacy class of a semisimple element s P G˚ :“ SO2n`1pqq and ψ is a unipotent character of the centralizer C :“ CG˚psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Moreover, the degree of the character χppsq,ψq associated to ppsq, ψq is χppsq,ψqp1q “ rG˚ : Csp1ψp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let α P t˘1u such that 4 | pqm ´ αq and consider a semisimple element s P G˚ with spectrum t1, ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=', ´1u such that C – GOα 2n (see [Ng10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Such s will then belong to Ω2n`1pqq “ rG˚, G˚s, implying that the semisimple character χppsq,1Cq associated to the pair ppsq, 1Cq is trivial on ZpSp2npqqq, by [NT13, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We therefore have an irreducible character of PSp2npqq of degree χpsqp1q “ p|SO2n`1pqq|{|GOα 2n|qp1 “ pqn ` αq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' To see that cdpPSp2npqqq ‰ cdpΩ2n`1pqqq (for odd q and n ě 3), it is enough to show that pqn`αq{2 is not character degree of Ω2n`1pqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By [TZ96, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1], under our assumptions on n and q and the additional condition pn, qq ‰ p3, 3q, the minimal (nontrivial) irreducible character of Spin2n`1pqq has degree dpSpin2n`1pqqq “ " pq2n ´ 1q{pq2 ´ 1q if q ě 5 pqn ´ 1qpqn ´ qq{2pq ` 1q if q “ 3, (For the definition of classical groups of various isogeny types, including the odd-dimensional spin groups, we refer the reader to [Car85, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Note that Spin2n`1pqq is a central extension of Ω2n`1pqq and so every character degree of Ω2n`1pqq is one of Spin2n`1pqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is now easy to check that dpSpin2n`1pqqq ą pqn ` αq{2 for n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For the remaining case pn, qq “ p3, 3q, we note that dpSpin2n`1pqqq “ 27, which is still greater than pqn ` αq{2 “ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Certainly, one has to do much more work to relax the hypothesis in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 to codpSq Ď codpHq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' that is, to obtain Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let n ě 3 and q be an odd prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then cdpΩ2n`1pqqq Ę cdpPSp2npqqq and cdpPSp2npqqq Ę cdpΩ2n`1pqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We have seen in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 that PSp2npqq possesses an irreducible character of degree pqn ` αq{2, where α P t˘1u such that 4 | pqn ´ αq, and furthermore pqn ` αq{2 R cdpΩ2n`1pqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, it suffices to show cdpΩ2n`1pqqq Ę cdpPSp2npqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We claim that both pq2n ´ 1q{pq2 ´ 1q and qpq2n ´ 1q{pq2 ´ 1q 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O are elements of cdpΩ2n`1pqqq for q odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G :“ Spin2n`1pqq, the universal cover of Ω2n`1pqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The dual group G˚ of G (in the sense of [DM91, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='10]) is the projective conformal symplectic group PCSp2npqq, which is the quotient of rG “ CSp2npqq by its center Zp rGq » Cq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Consider a semisimple element s P rG with spectrum Specpsq “ t´1, ´1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=', 1u and C rGpsq – pSp2pqq ˆ Sp2n´2pqqq ¨ Cq´1 (see [Ng10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let s˚ be the image of s under the natural homomorphism from rG to G˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is easy to see that, by the choice of s, C rGpsq is the complete inverse image of CG˚ps˚q under this homomorphism, and thus CG˚ps˚q “ C rGpsq{Zp rGq and rG˚ : CG˚ps˚qsp1 “ r rG : C rGpsqsp1 “ |Sp2npqq|p1 |Sp2pqq|p1|Sp2n´2pqq|p1 “ q2n ´ 1 q2 ´ 1 , where p is the defining characteristic of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Consider the canonical homomorphism f : Sp2pqq ˆ Sp2n´2pqq ãÑ C rGpsq Ñ CG˚ps˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using [DM91, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='20], we know that unipotent characters of Sp2pqq ˆ Sp2n´2pqq are of the form θ ˝ f where θ runs over the unipotent characters of CG˚ps˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, as Sp2pqq – SL2pqq has unipotent characters of degrees 1 and q, CG˚ps˚q has two unipotent characters of degree 1 and q as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the conclusion of the previous paragraph, the Lusztig series EpG, ps˚qq of G associated to the conjugacy class ps˚q of G˚ contains two characters of degrees pq2n ´ 1q{pq2 ´ 1q and qpq2n ´ 1q{pq2 ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that s˚ P PSp2npqq “ rG˚, G˚s and |ZpGq| “ |G˚{pG˚q1| “ 2, and therefore every character in the Lusztig series EpG, ps˚qq restricts trivially to ZpGq (see [NT13, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4]), and so can be viewed as a character of G{ZpGq “ Ω2n`1pqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The claim is completely proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose first that q ą 3 and assume to the contrary that cdpΩ2n`1pqqq Ď cdpPSp2npqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then we have pq2n ´ 1q{pq2 ´ 1q P cdpPSp2npqqq, so that pq2n ´ 1q{pq2 ´ 1q P cdpSp2npqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let χ P IrrpSp2npqqq such that χp1q “ pq2n ´ 1q{pq2 ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now, as q ą 3, we have pq2n ´ 1q{pq2 ´1q ă pq2n ´1q{2pq `1q, and therefore, by the classification of irreducible characters of Sp2npqq of degrees up to pq2n ´ 1q2pq ` 1q ([TZ96, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2]), we deduce that χ must be one of the Weyl characters of degree pqn ˘ 1q{2 or the minimal unipotent one of degree pqn ´ 1qpqn ´ qq{2pq ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A simple check reveals that none of these degrees matches the degree of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now we suppose q “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (In such case, pq2n ´ 1q{pq2 ´ 1q is indeed a character degree of PSp2npqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') As the case of Ω7p3q and PSp6p3q can be checked directly, we suppose furthermore that n ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We then have qpq2n ´ 1q{pq2 ´ 1q ă pq2n ´ 1qpqn´1 ´ qq{2pq2 ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Examining the degrees up to pq2n´1qpqn´1´qq{2pq2´1q of irreducible characters of Sp2npqq available in [Ng10, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2], we observe that none of them is equal to qpq2n ´ 1q{pq2 ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We have shown that qpq2n ´ 1q{pq2 ´ 1q R cdpSp2npqqq, and therefore, by the above claim, cdpΩ2n`1pqqq Ę cdpPSp2npqqq, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ We can now prove our first main Theorem A, which in fact follows from the following slightly stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If G and H are groups we say that CpGq Ď CpHq if codpGq Ď codpHq and mGpcq ď mHpcq for every c P codpGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let H be a finite simple group and G a nontrivial finite group such that CpGq Ď CpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then G – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose first that H is abelian of prime order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then codpGq Ď codpHq “ t1, pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 of [DL16], G is an elementary abelian p-group and since kpGq ď p, we conclude that G is cyclic of order p, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So we may assume that H is nonabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3 and the assumption CpGq Ď CpHq, we have that G is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let N IJ G such that S :“ G{N is nonabelian simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now codpSq Ď codpGq Ď codpHq and therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we have |S| divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that CpSq Ď CpGq Ď CpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore there exists a subset I Ď IrrpHqzt1Hu and a bijection f : IrrpSqzt1Su Ñ I such that |S| χp1q “ |H| fpχqp1q for every χ P IrrpSqzt1Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that ÿ χPIrrpSqzt1Su χp1q2 |S|2 “ ÿ ψPI ψp1q2 |H|2 , and thus |S| ´ 1 |S|2 ď |H| ´ 1 |H|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As the function px ´ 1q{x2 is decreasing on r2, 8q, we deduce that |S| ě |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The conclusions of the last two paragraphs show that |S| “ |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If S – H then CpG{Nq “ CpSq “ CpHq Ě CpGq and so G{N and G have the same number of con- jugacy classes, which is possible only when N “ 1, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So assume by contradiction that S fl H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using again the simple order theorem, we have tS, Hu “ tPSL4p2q, PSL3p4qu or tS, Hu “ tΩ2n`1pqq, PSp2npqqu for some n ě 3 and odd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For the former pair, using the character tables of both PSL4p2q and PSL3p4q available in [Atl1], one observes that 7 P cdpPSL4p2qqzcdpPSL3p4qq and 63 P cdpPSL3p4qqzcdpPSL4p2qq, implying that none of codpPSL3p4qq and codpPSL2p4qq con- tains the other, and this violates the fact that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The latter pair was already handled in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The largest character degree of finite simple groups Let bpGq denote the largest degree of an irreducible character of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Recall from the Introduction that if S is a simple group, then fpSq :“ |S|{bpSq is the smallest nontrivial character codegree of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The following elementary fact explains the relevance of fpSq, and therefore bpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S and H be finite simple groups such that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then fpSq ě fpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The hypothesis implies that fpSq P codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since fpHq is the smallest nontrivial member of codpHq, it follows that fpSq ě fpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Under the hypothesis of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we showed in Theorem C that |S| divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We will see in later sections that, in many cases, the two conditions fpSq ě fpHq and |S| divides |H| are enough to force S – H, as stated in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O Browsing through character tables of small-order simple groups in [Atl1], one notices that if |H| is a multiple of |S| and |H| ą |S|, then bpHq ą bpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' However, it seems that the largest character degree grows slower than the order – that is, fpHq ą fpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This is not so easy to prove generally, but we do confirm it in several cases, particularly when either S or H is an alternating group (see Sections 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We shall need effective (both lower and upper) bounds for the largest degree of an irre- ducible character of simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For symmetric groups, asymptotic and explicit bounds were obtained by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Vershik and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Kerov in [VK85] which can be used to derive the corresponding bounds for alternating groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For a group S of Lie type in characteristic p, an obvious (and in fact very tight!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') lower bound for bpSq is the degree StSp1q “ |S|p of the Steinberg character StS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' When S is of classical type, explicit upper bounds have been worked out by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Larsen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Malle, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Tiep in [LMT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Unfortunately, upper bounds for exceptional groups achieved in [LMT13] are only asymptotic and its proof does not allow one to obtain an explicit bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We obtain Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3 below that we believe will be useful in other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a simple algebraic group over the algebraic closure of a finite field of order q in characteristic p, F : G Ñ G a Steinberg endomorphism, and G :“ GF be the corresponding finite group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let r be the rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then pq ´ 1qr ¨ |G|p ď |G|p1 ď qr ¨ |G|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that finite groups GF of the same isogeny type have the same order, so we may work with G being of simply-connected type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The inequalities are then straightforward to verify using the order formulas for finite groups of Lie type available in [Atl1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' xvi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a simple exceptional group of Lie type defined over a field of order q in characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then the following hold: (i) bpSq ă 256|S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (ii) If q ą 2, then bpSq ă 26|S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The Tits group can be verified easily, so we assume that S ‰ 2F4p2q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We then may find a simple algebraic group G of adjoint type and a Steinberg endomorphism F : G Ñ G such that S “ rG, Gs where G :“ GF (see [MT11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Clearly it suffices to show that bpGq ă 256StGp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let pG˚, F ˚q be the dual pair of pG, Fq, so G˚ will be the corresponding simple algebraic group of simply connected type, and set G˚ :“ pG˚qF ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As mentioned before, Lusztig’s classification on complex characters of finite groups of Lie type implies that the set of irreducible complex characters of G is partitioned into Lusztig series EpG, psqq associated to various conjugacy classes psq of semisimple elements of G˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Furthermore, there is a bijection χ ÞÑ ψ from EpG, psqq to EpCG˚psq, p1qq such that (1) χp1q “ |G|p1 |CG˚psq|p1 ψp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The detailed structure of centralizers of semisimple elements in a finite exceptional groups of Lie type was determined by Carter [Car78], Deriziotis [Der77], and Deriziotis-Liebeck [DL85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' A well-known result of Steinberg states that the centralizer CG˚psq of a semisimple THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 13 element s is a connected reductive subgroup of maximal rank in G˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Such connected subgroup has a decomposition CG˚psq “ ST, where S is a semisimple subgroup, T is a central torus, S X T is finite, and |pCG˚psqqF ˚| “ |S˚||T˚| (see [DL85, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' When s is in G˚, the centralizer CG˚psq is F ˚-stable and CG˚psq “ pCG˚psqqF ˚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' and so |CG˚psq| “ |S||T| where S :“ SF ˚ and T :“ TF ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let r be the semisimple rank of G˚ and q (that will be a power of p) the absolute value of all eigenvalues of F on the character group of an F-stable maximal torus of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Possible values for |S| and |T| are available in [Der77, DL85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, we have |S| “ ź i |Lripqaiq| and |T| “ ź j Φjpqq, where Lripqaiqs are finite groups of Lie type (of simply-connected type) of rank ri defined over a field of order qai and Φjpqqs are cyclotomic polynomials (and also polynomials of the forms q2 ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q ` 1, q2 ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3q ` 1, or q4 ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q3 ` q2 ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q ` 1 for Suzuki and Ree groups) evaluated at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As CG˚psq has maximal rank, we furthermore have (2) ÿ i airi ` ÿ j degpΦjq “ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now formula (1) implies that the typical degree of an irreducible character of G is of the form χp1q “ |G|p1 ś i |Lripqaiq|p1 ś j Φjpqqψp1q, where ψ P EpCG˚psq, p1qq, a unipotent character of CG˚psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By [LMT13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1], for any finite group of Lie type GF , where G is a simple algebraic group in characteristic p and F a Steinberg endomorphism on G, the degree Stp1q “ |GF |p of the Steinberg character St of GF is strictly larger than the degree of any other unipotent character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, the degrees of unipotent characters of CG˚psq, which are in fact the same as those of the semisimple group S, are bounded by ś j |Lripqaiq|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that bpGq ď |G|p1 ś i |Lripqaiq|p1 ś j Φjpqq ź j |Lripqaiq|p, By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2, |Lripqaiq|p |Lripqaiq|p1 ď 1 pqai ´ 1qri ď 1 pq ´ 1qairi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Also, it is easy to see that Φjpqq ě pq ´ 1qdeg Φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We therefore deduce that bpGq ď |G|p1 pq ´ 1q ř i airi`ř j degpΦjq “ |G|p1 pq ´ 1qr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 14 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O On the other hand, we have |G|p1 ď |G|pqr by again Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2, and it follows that bpGq |G|p ď qr pq ´ 1qr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As the rank r is at most 8 for exceptional groups, the desired inequalities follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Bounds for bpSq of alternating groups and classical groups are collected in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a finite simple group, n a positive integer, and q a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (i) For S “ An with n ě 5, 1 2e´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='28255?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' ď bpSq ď e´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='11565?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='. In particular, bpSq ą 1 2e´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='28255?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='np2πnq1{4pn{eqn{2 (ii) For S “ An with n ě 5, bpAn`1q ě 2pn ` 1q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8n ` 1 ` 3bpAnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (iii) For S “ PSLnpqq with n ě 2, bpSq ă 13p1 ` logqpn ` 1qq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='54StSp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (iv) For S “ PSUnpqq with n ě 3, bpSq ă 2p2 ` logqpn ` 1q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='27StSp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (v) For S P tΩ2n`1pqq, PSp2npqq, PΩ˘ 2npqqu with n ě 2 and q odd, bpSq ă 38p1 ` logqp2n ` 1qq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='27StSp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (vi) For S P tΩ2n`1pqq, PSp2npqq, PΩ˘ 2npqqu with n ě 2 and q even, bpSq ă 8p1 ` logqp2n ` 1qq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='27StSp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Part (i) follows from [VK85, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1] and Parts (iii)-(vi) follow from [LMT13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let us prove Part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let χ P IrrpSnq such that χp1q “ bpSnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let λ be the partition of n corresponding to χ and Yλ be the Young diagram associated to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the well-known branching rule, the induction χSn`1 of χ from Sn to Sn`1 is the sum of irreducible characters corresponding to the partitions of n ` 1 whose associated Young diagrams are obtained from Yλ by adding a suitable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The number of those suitable nodes is at most p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8n ` 1`1q{2 (see [HHN16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1950]) and at most one of the resulting Young diagrams is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We deduce that pn ` 1qbpAnq ď pn ` 1qχp1q “ χSn`1p1q ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8n ` 1 ` 3 2 bpAn`1q, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D: Groups of Lie type In this section we prove Theorem D when the groups involved are of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In the following, for simplicity, we say that two groups have the same defining charac- teristic if there is a common characteristic over which the groups can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S and H be finite simple groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then S and H have the same defining characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that the defining characteristic of H is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4, |S|p1 “ |H|p1 and there is χ P IrrpSq such that χp1q “ |S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, it follows that (3) d ¨ |H|p χp1q P cdpHq for every d P cdpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Certainly if χ is the Steinberg character of S then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So we assume otherwise and aim to find a contradiction or end up with a case where H can be defined in another characteristic not equal to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the classification of prime-power-degree representations of quasi-simple groups [MZ01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1], we arrive at the following possibilities of S and χp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (i) S “ PSL2pqq, χp1q P tq ˘ 1u or q is odd and χp1q P tpq ˘ 1q{2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Observe that χp1q cannot be pq ˘ 1q{2 because otherwise, by taking d “ 2χp1q, we would have 2|H|p P cdpHq, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So χp1q “ q ` α “ px for some α P t˘1u and x P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose first that q “ 2t for some t ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then 2t ` α “ px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Mihailescu’s theorem [Mih04] (previously known as Catalan’s conjecture), either x “ 1 so that 2t ` α is a (Mersenne or Fermat) prime or α “ 1 and t “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In the latter case, p “ 3 and |H|31 “ |S|31 “ 56, forcing H to be 2G2p3q1, which turns out to be isomorphic to S “ PSL2p8q, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So it remains to consider the former case: q ` α “ 2t ` α “ p is the defining characteristic of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now |H|p1 “ |S|p1 “ qpq ´ αq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let pa be the order of the underlying field of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is clear from the order formulas of simple groups of Lie type (see [Atl1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' xvi]) that |H|p ă |H|p1{ppa ´ 1q ď |H|p1{pp ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We therefore deduce that |H|p ă qpq ´ αq q ` α ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Thus we must have α “ ´1 and |H|p “ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now H is a simple group of Lie type in characteristic p such that |H| “ ppp ` 1qpp ` 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This is impossible as for such a group H, one can check from the order formula that |H|p1 ă p|H|pq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Next we suppose q ě 5 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then p “ 2 and q ` α “ |S|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Again by Mihailescu’s theorem, either q is a prime or α “ ´1 and q “ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The case q “ 9 is eliminated in the same way as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So assume that |H|21 “ qpq ´ αq{2 and q is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that when H is not of type A1, every prime divisor of |H|p1 is smaller than a |H|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore our group H must be PSL2pq1q for some 2-power q1, implying that qpq ´ αq{2 “ q2 1 ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This, however, returns no relevant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (ii) For the remaining possibilities of S and χ, the character χ has a decent small degree and we are able to produce an irreducible character of S whose degree is a proper multiple of χp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Condition (3) then implies that a proper multiple of |H|p is a character degree 16 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O of H, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This required character turns out to be chosen as a unipotent character in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We refer the reader to [Car85, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8] for the description of unipotent characters of finite classical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The next possibility of S and χp1q is S “ PSLnpqq, q ą 2, n is an odd prime, pn, q´1q “ 1, and χp1q “ pqn ´ 1q{pq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If n “ 3 then SL3pqq has an irreducible character of degree q3 ´1 (see [SF73]), which is a proper multiple of χp1q “ q2 `q`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For n ě 5, the unipotent character parameterized by the partition p2, n ´ 2q with degree χp2,n´2qp1q “ pqn ´ 1qpqn´1 ´ q2q pq ´ 1qpq2 ´ 1q fulfills our requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Another possibility is S “ PSUnpqq, n is an odd prime, pn, q ` 1q “ 1, and χp1q “ pqn ` 1q{pq ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This case is handled similarly as in the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Here we note that, when n ě 5 is odd, the unipotent character parameterized by the partition p2, n ´ 2q has degree χp2,n´2qp1q “ pqn ` 1qpqn´1 ´ q2q pq ` 1qpq2 ´ 1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The next case is S “ PSp2npqq, n ě 2, q “ ℓt with ℓ an odd prime, tn is a 2-power, and χp1q “ pqn ` 1q{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now the unipotent character parameterized by the symbol `0 1 n ˘ has required degree χp0 1 n qp1q “ pqn ` 1qpqn ` qq 2pq ` 1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The last possibility involving a family of groups is S “ PSp2np3q, n ě 3 is a prime, and χp1q “ p3n ´ 1q{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then S has a unipotent character with degree χp0 1 n ´ qp1q “ p3n ´ 1qp3n ´ 3q 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (iii) pS, χp1qq P tpSp6p2q, 7q, pSp6p2q, 27q, p2F4p2q1, 27q, pG2p2q1, 7q, pG2p2q1, 27q, pG2p3q, 64qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' First assume that S “ G2p2q1 and χp1q “ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then p “ 3 and |H|31 “ |S|31 “ 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is easy to check that there is no such simple group S of Lie type in characteristic 3 with 27 | |S|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In all other cases one can find a character ψ P IrrpSq such that χp1q | ψp1q but χp1q ă ψp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore ψp1q|H|p{χp1q is a proper multiple of |H|p and thus cannot be a character degree of H, violating condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ As seen in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we face the situation where two simple groups S and H of Lie type have the same characteristics p and |S|p1 “ |H|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is surprising to us that this turns out to happen only when |S| “ |H| (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3 below), and therefore they are among the coincidences appeared in the simple order theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We slightly modify two of the invariants in Artin’s proof of the simple order theorem (for classical groups) to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a finite group of Lie type in characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let ω “ ωpSq and ψ “ ψpSq respectively denote the largest and the second largest orders of p modulo a prime divisor of |S|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We will refer to ωpSq and ψpSq as the Artin invariants of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 17 In fact, when p is the dominant prime of |S|, these ωpSq and ψpSq coincide with Artin’s invariants defined in [Art55b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We remark that there are only a few cases involving Mersenne and Fermat primes where p is not dominant in |S|, and they are listed in [KLST90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume for now that S is not one of G2p2q1, 2G2p3q1, and 2F4p2q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (Note that S1 “ G2p2q1 – PSU3p9q and S2 “ 2G2p3q1 – PSL2p8q and, even though we do not allow S1 (or S2) to be viewed as a Lie type group over a field of order 2 (or 3), we do allow it to be viewed as one over the field of order 9 (or 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Let q “ pt be the order of the underlying field of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is well-known that the order |S| then has the standard cyclotomic factorization in terms of q as |S| “ 1 dqk ź i Φipqqei, where d, k, eis depend on S and can be found in [Atl1, Table 6] and [KLST90, Tables C1, C2, and C3] for instance, and Φipqq is the value of the ith cyclotomic polynomial evaluated at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Replacing q by pn and factorizing each Φipxtq further into cyclotomic polynomials of x, one has |S| “ 1 dpkt ź i Φippqfi for certain positive integers fis depending on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Zsigmondy’s theorem, it is not difficult to see that, except for some ‘small’ cases, the invariants ωpSq and ψpSq are precisely the largest and second largest, respectively, index i such that Φippq appears in the cyclotomic factorization of |S| (see [KLST90, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We refer the reader to [KLST90, Tables A1, A2 and A3] for the list of exceptions and the values of their Artin’s invariants, including the groups G2p2q1, 2G2p3q1, and 2F4p2q1 we excluded earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We reproduce in Table 1 the values of ωpSq and ψpSq for the generic case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let p be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S and H are two non-isomorphic simple groups of Lie type in characteristic p and |S|p1 “ |H|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then tS, Hu “ tPSL4p2q, PSL3p4qu or tS, Hu “ tΩ2n`1pqq, PSp2npqqu for some n ě 3 and odd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, |S| “ |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the assumptions, we have ωpSq “ ωpHq and ψpSq “ ψpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' First we consider the case where both S and H are generic so that their invariants ω and ψ are available in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Comparing those values, we can find all the collections of groups with equal values of ω and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We list these collections in Table 2 (each row in the table is one such collection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now one simply compare the p1-parts of orders of groups in each collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It turns out that the only pair with the same p1-parts of orders is tΩ2n`1pqq, PSp2npqqu with some n ě 3 and odd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume now that at least one of the two groups, say S, is non-generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' That is, S is among the exceptions listed in the second column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The values of the invariants ω and ψ of these groups are available in [KLST90, Tables A2 and A3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The analysis is basically the same as in the generic case, but more tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We first find all the possible groups H with ωpSq “ ωpHq and ψpSq “ ψpHq, and then compare |S|p1 and |H|p1, where p is the defining characteristic of S and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 18 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' ωpSq and ψpSq for simple groups of Lie type: generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' S (q “ pt) Conditions (p a Mersenne prime) ωpSq ψpSq PSLnpqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 2 pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 26q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 22q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 23q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 22q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' pq nt pn ´ 1qt PSU4pqq 6t 4t PSUnpqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 3 odd pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 23q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' pq 2nt 2pn ´ 2qt PSUnpqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 6 even pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q 2pn ` 1qt 2pn ´ 1qt Ω2n`1pqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 2 pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 28q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' pq 2nt 2pn ´ 1qt PSp2npqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 3 2nt 2pn ´ 1qt PΩ` 2npqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 4 pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' p5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q 2pn ´ 1qt 2pn ´ 2qt PΩ´ 2npqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 4 pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' qq ‰ p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2q 2nt 2pn ´ 1qt 2B2p2tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' t ě 3 odd t ” 3pmod6q 4t 4t{3 2B2p2tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' t ě 3 odd t ” ˘1pmod6q 4t t G2pqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' q ě 3 q ‰ 4 6t 3t 2G2p3tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' t ě 3 odd 6t 2t 3D4pqq q ‰ 2 12t 6t F4pqq 12t 8t 2F4p2tq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' t ě 3 odd 12t 6t E6pqq 12t 9t 2E6pqq 18t 12t E7pqq 18t 14t E8pqq 30t 24t Let us demonstrate the case S “ PSL3p4q as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then ωpSq “ 4 and ψpSq “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' But there are only two other simple groups of Lie type with the same values of ω and ψ, namely PSL4p2q and PΩ` 8 p2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' However, |PΩ` 8 p2q|21 ‰ |PSL3p4q|21 “ |PSL4p2q|21, and so we come up with another possible pair for tS, Hu, namely tPSL4p2q, PSL3p4qu, as stated in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ The next theorem improves Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 when the relevant groups are of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S and H be finite simple groups of Lie type such that codpSq Ď codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then S – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 and Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, we have that S and H fall into one of two pairs of groups concluded in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The result now follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D: The mixed case of alternating groups and groups of Lie type In this section, we prove Theorem D in the mixed situation where the set of codegrees of an alternating group S is contained in that of a simple group H of Lie type, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In the following proposition we remark that the condition on m is necessary, due to the coincidences of isomorphic simple groups: A5 – PSL2p4q – PSL2p5q, A6 – PSL2p9q, and THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 19 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Simple groups of Lie type with the same values of ω and ψ: generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSLnpp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω2n`1ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp2nppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 2pn`1qppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ´ 2nppsq PSL3pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSU4ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω7ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp6ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 8 ppsq PSL2pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω5pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' G2pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3D4ppsq PSL2pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' G2ppsq PSL3pp4sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSU4pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω7pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp6pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 8 pp2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' F4ppsq PSL2p26sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω5p23sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' G2p22sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3D4p2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2F4p2sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' s ě 3 odd PSL4pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' E6ppsq PSL4pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω9pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 10pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ´ 8 pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' E6pp2sq PSL3pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSU4pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω7pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp6pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 8 pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2E6ppsq PSL3pp12sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω7pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp6pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 8 pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' F4pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2E6pp2sq PSL5pp6sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ω11pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp10pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ` 12pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PΩ´ 10pp3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' E8ppsq PSUnpqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSUn´1pqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 7 odd PSU3p22sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2B2p23sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' s odd PSU3p3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2G2p3sq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' s ě 3 odd PSU9ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSU8ppsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' E7ppsq Ω2n`1pqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' PSp2npqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' n ě 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' q odd A8 – PSL4p2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We also recall that fpXq :“ |X|{bpXq, where bpXq is the largest character degree of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose m “ 7 or m ě 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let H be a simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If |Am| divides |H|, then fpHq ą fpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As a consequence, codpAmq Ę codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let p be the defining characteristic and q, a power of p, the order of the underlying field of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Consider first the case H being of exceptional type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, we have fpHq ą |H| 256|H|p “ 1 256|H|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As |Am| divides |H|, it follows that fpHq ą p1{256q|Am|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, to prove the theorem, it suffices to show bpAmq ě 256|Am|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let us assume for now that that m ě 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, the dominant prime in |Am| “ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' {2 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We therefore just need to show bpAmq ě 256|Am|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As |Am|2 ď 2m´2, for this we want to show bpAmq ě 64 ¨ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Note that bpA19q “ 64, 664, 600 ą 64 ¨ 219 (see [McK86] for the degree of the largest irre- ducible characters and associated partitions of symmetric groups of degree up to 75, from which one can deduce the exact value or a good bound for the one of corresponding alter- nating groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now one just inducts on m with the help of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4(ii) to achieve the desired bound for m ě 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that m ď 18 and recall that we are still dealing with exceptional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' When q “ 2, the proposition can be verified directly, so assume that q ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In such case, bpHq ă 26|H|p by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, and whence the above estimate can be refined so that we 20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O only need to prove bpAmq ě 26|Am|p, which turns out to be true for all 18 ě n ě 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For the remaining values m ď 12, the arguments go as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' First we are done if fpAmq ď a |H|, as fpHq ą a |H|, so we may assume that |H| ă fpAmq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For each m ď 12, there are indeed no possibilities for H satisfying |H| ă fpAmq2 and |Am| divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Following the same idea as in the case of exceptional groups, but using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 instead, we can show that in fact fpHq ą fpAmq for every H of classical type and m ě 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let us present the details for only the linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Consider H “ PSLnpqq for some n ě 2 and q a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4(i), we have fpAmq “ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2bpAmq ď e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='28255?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='mpm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='q1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Thus, if |H| ě e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='5651?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' then fpHq ą a |H| ě e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='28255?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='mpm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='q1{2 ě fpAmq and we would be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We therefore can assume that |H| ă e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='5651?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=', which in particular implies that n ă m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4(iii), we see that, as before, it is enough to show that bpAmq ě 13p1 ` logqpn ` 1qq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='54|Am|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since m ě n ` 1 and |Am|p ď 2m´2, for this it is sufficient to show that bpAmq ě 13p1 ` log2 mq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='542m´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This last inequality is indeed true for m “ 20, and therefore is true for all m ě 20, by induction and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Checking directly, we see that the inequality bpAmq ě 13p1 ` logqpn ` 1qq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='54|Am|2 is still valid for n “ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As for the exceptional types, we are left to consider the small cases m ď 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Again we are done if fpAmq ď a |H|, so we may assume that |H| ă fpAmq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For each m, we search for relevant H satisfying |H| ă fpAmq2 and |Am| divides |H| and find that, for such an H, the inequality fpHq ą fpAmq always holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ We shall need the following result on 2-defect zero and 3-defect zero characters of alter- nating groups, which easily follows from earlier work of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Garvan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Stanton [GKS90] on the so-called p-core partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' They are partitions having no hook lengths di- visible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Garvan-Kim-Stanton’s result, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Granville and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ono [GO96] proved the existence of p-defect zero characters with p ě 5 in symmetric and alternating groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let m be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (i) Am has a 2-defect zero irreducible character if and only if m “ 2k2 ` k or m “ 2k2 ` k ` 2 for some k P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (ii) Am has a 3-defect zero irreducible character if and only if there is a prime ℓ ” 2pmod 3q such that the the exact power of ℓ dividing 3m ` 1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' See the discussion in [GO96, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 333-334].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a simple group of Lie type and 8 ‰ m ě 7 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then codpSq Ę codpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In fact, if |S| divides |Am| and m ě 44, then fpSq ă fpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume by contradiction that codpSq Ď codpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we then have fpSq ě fpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that the defining characteristic of S is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Observe that fpSq ď |S|{StSp1q “ |S|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Furthermore, |S|p1 ă |S|2 p (see [CHMN15, Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 12]) and |S|p ď |Am|p by THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 21 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore we have fpSq ă p|Am|pq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Assume for a moment that m ě 10 so that |Am|p ď |Am|2 ď 2m´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' We now have fpSq ă 22m´4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' On the other hand, it is clear that fpAmq ą a m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Therefore, we would be done if m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' ě 24m´7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the well-known estimate m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2πmpm{eqm, this is certainly true when m ě 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' So we may now suppose that m ď 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As mentioned above, every simple group of Lie type, and therefore S in particular, has a 2-defect zero irreducible character, which means that S has an odd codegree and so does Am as codpSq Ď codpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that m “ 2k2 `k or m “ 2k2 `k`2 for some k P N, by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This forces m to be one of 10, 12, 21, 23, 36, or 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By the same reason, Am has a codegree not divisible by 3 and so Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2(ii) further narrows down the choices for m: m P t10, 12, 21, 36u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In fact, when m “ 21 or 36, we still have fpAmq ą |Am|2 2, and since |Am|2 2 ą fpSq, it follows that fpAmq ą fpSq, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose m “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The inequality fpAmq ă p|Am|pq2 then forces p “ 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If p “ 2 then |S|21 “ |A10|{χp1q, where χ P IrrpA10q is one of the two 2-defect zero irreducible characters of equal degree 384, implying |S|21 “ 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' {p2 ¨ 384q “ 4725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It is easy to see from [Atl1] that there is no such group of Lie type in characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' If p “ 3 then |S|31 “ 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' {p2 ¨ 567q “ 3200 since A10 has a unique 3-defect zero character of degree 567, which again leads to a contradiction as there is no such group in characteristic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The case m “ 12 is treated similarly and we skip the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D: Alternating and sporadic groups Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let m ă n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then fpAmq ă fpAnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Consequently, codpAmq Ę codpAm`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It suffices to show that bpAm`1q ă pm ` 1qbpAmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let χ P IrrpAm`1q such that χp1q “ bpAm`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As shown in [HHN16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1956], such χ must be the restriction of an irre- ducible character, say ψ, of Sm`1 whose associated partition, say λ, is not self-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, χp1q “ ψp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As in the proof on Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4(ii), let Yλ be the Young diagram associated to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The restriction ψSm of ψ to Sn is the sum of irreducible characters corre- sponding to the partitions of n whose associated Young diagrams are obtained from Yλ by removing a suitable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The number of those suitable nodes is at most p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8m ` 9 ´ 1q{2, so bpAm`1q “ ψp1q ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='8m ` 9 ´ 1 2 bpSmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since bpSmq ă 2bpAmq as already mentioned above, it follows that bpAm`1q ă p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 8m ` 9 ´ 1qbpAmq, which implies our desired inequality bpAm`1q ă pm ` 1qbpAmq for m ě 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The result is easily checked for smaller m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem D is true when either S or H is a sporadic simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The case where both S and H are sporadic simple groups can be verified by using the available data in [Atl1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose that S is a sporadic group and H “ Am for some m ě 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let pS be the largest prime divisor of |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we have |S| divides |Am|, so pS ď m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By Lemma 22 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' HUNG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' MORET ´O 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we have fpSq ě fpAmq ą a m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It follows that fpSq ě a pS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Again using [Atl1], it can be checked that this can never happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Next we assume that S is a sporadic group and H is a simple group of Lie type in characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Suppose first that S has an irreducible character, say χ, of p-defect zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then, as argued in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4, we have |S|p1 “ |H|p1 and χp1q “ |S|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' In particular, χp1q is a prime power, and therefore, [MZ01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1] yields pS, p, χp1qq P tpM11{M12, 11, 11q, pM11, 2, 16q, pM24{Co2{Co3, 23, 23qu (We note that M12 has another irreducible character of prime power degree, namely 16, but the character is not of 2-defect zero and thus does not fit our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') However, for each of these possibilities, there is no simple group of Lie type H in characteristic p such that |H|p1 “ |S|p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Next, we suppose that S has no characters of p-defect zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' By [GO96, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2], p P t2, 3u and S P tM12, M22, M24, J2, HS, Suz, Ru, Co1, Co3, BMu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now we just apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 and argue similarly as in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, with S in place of Am, to arrive at fpHq ą fpSq, and thus it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1 that codpSq Ę codpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now we consider the case where S “ Am for some m ě 5 and H a sporadic simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, we have m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' {2 divides |H| and so m is at most pH ´ 1, where pH is the smallest prime not dividing |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This constraint is enough to ensure that fpAmq ă fpHq, and thus codpAmq Ę codpHq by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Finally we consider the case where S is a simple group of Lie type and H a sporadic simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' As in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3 , we have fpHq ď p|H|pq2, where p is the defining characteristic of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' The only possible p satisfying such condition is p “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Now |S|21 is an odd codegree of S, and hence of H, and so |S|21 “ |H|{χp1q for some 2-defect zero character χ P IrrpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' There are in fact only 16 sporadic simple groups having a 2-defect zero irreducible character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For such a group and such a character, there are no S satisfying the indicated condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ Theorem D follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='4 and Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='1, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For future work on Huppert’s codegree conjecture (HCC), we record the following im- mediate consequence of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let S be a finite nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let G be a minimal coun- terexample to (HCC) with respect to S – that is, G is minimal subject to the conditions codpGq “ codpSq and G fl S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Then G has a unique minimal normal subgroup N and G{N – S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Let N be a maximal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Since codpG{Nq Ď codpGq “ codpSq and G{N is simple, it follows from Theorem D that G{N – S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Furthermore, by the minimality of G as a counterexample, we have that N is a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' (If G has a normal subgroup M such that M ă N, then codpG{Nq Ď codpG{Mq Ď codpGq, forcing codpG{Mq “ codpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=') Also, N is the unique minimal normal subgroup of G since, otherwise, G “ S ˆ S, which violates the assumption codpGq “ codpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' □ THE CODEGREE ISOMORPHISM PROBLEM FOR FINITE SIMPLE GROUPS 23 We conclude the paper with a couple of remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' First, the group pseudo-algebra CpGq seems to better distinguish finite groups than the usual complex group algebra CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' For instance, while any two abelian groups A and B of the same order have the same complex group algebra CA “ CB, it was shown in [Mor22] that A – B if and only if CpAq “ CpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' It has even been speculated that a finite group G and an abelian group A are isomorphic if and only if CpGq “ CpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' This, if true, would indicate that abelian groups have very distinctive character codegrees (counting multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Theorems A shows that simple groups indeed have very distinctive codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Our results are likely to remain true for quasi and/or almost simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' However, at the time of this writing, we do not see yet a uniform proof for these larger families of groups as the one presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' References [Aha22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Ahanjideh, Nondivisibility among irreducible character co-degrees, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 105 (2022), 68-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 2 [Art55a] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Artin, The orders of the linear groups, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 8 (1955) 355-365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 8 [Art55b] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Artin, The orders of the classical simple groups, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Pure Appl.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Willems, Blocks of defect zero in finite simple groups of Lie type, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' Algebra 113 (1988), 511-522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content=' 3, 8 Department of Mathematics, The University of Akron, Akron, OH 44325, USA Email address: hungnguyen@uakron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='edu Departamento de Matem´aticas, Universidad de Valencia, 46100 Burjassot, Valencia, Spain Email address: alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfk_jh/content/2301.00446v1.pdf'} +page_content='moreto@uv.' 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Cohu𝑎,𝑏,𝑐 A. Chevalier𝑐 O. Nechyporuk𝑏 A. Franzen𝑏 J. Sauerwald𝑏 J-C. Ianigro𝑎,𝑐 and J. +Marteau 𝑎,𝑐 +𝑎IP2I, IN2P3, CNRS, Université Lyon 1, UMR 5822, 69100 Villeurbanne, France +𝑏ArcelorMittal Maizières Research SA,Voie Romaine, 57280 Maizières-lès-Metz, France +𝑐MUODIM, 31 rue Saint-Maximin, 69003 Lyon, France +E-mail: marteau@ip2i.in2p3.fr +Abstract: The blast furnace (BF) is the fundamental tool used in the iron manufacture. Due +to the difficulty of accessing direct measurements of the inner phenomena, we determined the +density distribution of its internal volume in order to improve its productivity using muography. +This is an imaging technique based on the differential absorption of a flux of incident particles, +muons, by the target under study, similar to clinical X-ray imaging. Muons are elementary particles +that have the property of passing through dense materials, up to hundreds of meters away. Their +relative absorption and deviation allows the generation of density distribution images of an object by +tracking the number of muons received by a detector, before and after passing through a structure. +The incident direction of the detected muons is reconstructed by means of a detector composed of 3 +scintillator panels that we moved on 3 positions around the BF. With this technique, we obtained the +first 3D image of the internal structure of a BF using a Markov Chain Monte Carlo (MCMC) inverse +problem solving algorithm on muon flux data. We were also able to perform a density monitoring +of the BF and some of its operating parameters. We distinguished the position and shape of the +cohesive zone, a key element in the productivity of a furnace, validating this innovative measurement +concept in the application to a BF and opening the field to a series of future experiments to gain +both spatial and temporal resolution. +Keywords: Particle tracking detectors, Scintillators and scintillating fibres and light guides, Com- +puterized Tomography (CT) and Computed Radiography (CR), Image processing. +arXiv:2301.04354v1 [physics.geo-ph] 11 Jan 2023 + +Contents +1 +Introduction +1 +2 +Tomography reconstruction theory +2 +2.1 +Generalities and issues +2 +2.2 +Inverse problem +2 +3 +Application to a blast furnace +3 +3.1 +Acquisition configurations +3 +3.2 +External and internal parameters effects on muon flux +3 +4 +Results +5 +4.1 +2D fields +5 +4.2 +3D-inversion results +7 +4.3 +Analysis of various sensitivities +7 +5 +Conclusion +9 +1 +Introduction +We seek to determine the density distribution of matter inside a blast furnace in order to visualize +the cohesive zone and to carry out a dynamic monitoring of the various phases present in the blast +furnace. To achieve this, muography is applied to make a dynamic image during the operation +of a blast furnace of ArcelorMittal in Bremen, Germany. Muography measures the absorption or +deflection of cosmic muons as they pass through dense materials. Muons are elementary particles +which have the property to pass, in a straight line at first order, up to several kilometers of standard +rocks, and whose relative absorption allows to generate images by contrast densitometry, like a +standard clinical radiography. The acquired muon data allows to follow the density as a function of +time during the operating cycles of the blast furnace. In a second step, the acquisition of 2D images +and then the 3D reconstruction is accomplished by the data inversion from several measurement +points. The final objective is to understand the topological characteristics and the formation rate of +the cohesive zone and the influence of certain loading parameters. +The article is organized in four main parts. The first part explains the theorical points and +algorithms used in tomography reconstructions. The second part presents the simulation parameters +of a blast furnace and explains how we built the 3D image. We have completed our analysis by +monitoring the activity of the blast furnace as function of different environmental parameters +such as atmospheric pressure and temperature. The third part presents the results of real muons +data inversion to visualize the different density zones in the blast furnace. Finally, we report the +conclusions and perspectives of this study. +– 1 – + +2 +Tomography reconstruction theory +2.1 +Generalities and issues +Absorption muography measures the cosmic ray flux deficit in the direction of observation and +determines the integrated density of a structure. Muons are cosmic rays able to cross several meters +of rocks losing energy. The minimum amount of energy that muons must need to penetrate the +structure must be of a value higher than the one lost inside the object, so that the detector can follow +the outgoing muons. The detector position must then be adjusted to optimize the spatial resolution +during field measurements. Muon tomography is limited to the study of a portion of the object only, +because of the limited angular aperture of the detector. In order to take these different points into +consideration, we use the inverse and direct problems jointly. +The direct problem consists, in our case, in predicting the expected muon flux at the exit +of an object. It is necessary to use materials knowledge and physical properties of the object’s +constituents and then to use a density distribution as precise as possible. The parameters sought +during the inversion use the direct problem to estimate the expected measurements. The measured +information is retrieved from a given distribution of 𝑝 parameters in the studied structure. Moreover, +the flux attenuation is estimated from a known law based on the contrasting distribution of zones +(of different density for example). +With the trackers the direction of the muons is reconstructed in order to observe the properties +of the medium on a precise observation axis (see Lesparre et al.[13]). The flux that arrives on +the detector after having crossed the object and the theoretical flux that would reach the detector +in the absence of matter are compared. The contrast between thoses two quantities gives directly +access to the opacity of the matter (in meter equivalent water (mwe)), defined as the integral of the +density along the trajectory of the muon from its entry point to its exit point. Moreover, the only +observable are the particles directions and energy deposits, since the detectors usually do not give +direct information on the particles total energy. In order to solve the data inversion, the measured +flux is coupled to a theoretical flux model and to a flux loss model in matter. +2.2 +Inverse problem +The reconstruction of a muography image is achieved by solving an inverse problem. The goal is to +recover the distribution of properties of the medium (3D density) from measurements of muon rates +and their directions. An inverse problem is a situation in which one tries to determine the parameters +of a model 𝑝 (here the 3D density of the environnement) from experimental measurements 𝑚 (muon +rates) such that 𝑚 = 𝑓 (𝑝) where 𝑓 contains the open muon flux (calibration) and the law governing +the absorption of muons in matter. While a single parameter estimate can be easily obtained by least +squares fitting, the use of Monte Carlo methods allows the maintenance of the stochasticity of the +model during the estimation of these parameters. In order to improve the reliability of the results, +it is good practice to add some informations about the objet to study, other than data, that we call a +priori information [1]. They allow to constrain the solutions space and improve the accuracy of the +statistical answer. +We use the Metropolis-Hastings algorithm which is a particular class of Monte Carlo methods +using Markov chains. It works like a geometrically biased random walk with a data based selection +at each throw [8]. Each new proposal/model can be accepted or rejected if the likelihood of the +– 2 – + +model (regarding the data and the physics of the problem) is greater or smaller than the likelihood of +the previous model. Hence, unlike more general Monte Carlo methods where the sampled values are +statistically independent, in the Metropolis-Hastings algorithm they are statistically auto-correlated. +This auto-correlation is minimized by adding a threshold number of changes that must be accepted +between the recording of two consecutive samples. +We need models whose density values are continuous over finite element discretized volumes. +Here a model stands for any given set of values representing a physical system. The engine that +generates the 3D density models, linked by Markov chain, is made to design density sets per class +forming spatially contiguous voxel sets which share the same density. The inversion algorithm is a +3D adaptation of Mosegaard et al.[11] work, where details can be found in Chevalier et al.[7]. The +method makes selections of the models according to the evolution of the distance 𝐷 : +𝐷 = 𝐹𝑡 − 𝐹𝑚 +𝜎𝑚 +(2.1) +with 𝐹𝑡 the theoretical flux and 𝐹𝑚 the measured flux. 𝐷 is a metric of the distance between the +data and the simulation, expanded or compressed by the degree of uncertainty on the measurement +𝜎𝑚. This distance 𝐷 is recalculated each time the density changes. A new model is accepted as +a solution if the mean difference between the reconstructed signal and the data (evaluated by the +mean square deviation) is lower than the mean noise estimated during the measurement. If the +model is selected, we save its likelihood. This model is then slightly perturbed again by changing +some voxel density values and we recalculate the new likelihood. The total likelihood of a model, +𝐿(𝑚), can be expressed as a product of partial likelihoods, one for each data type. +Our inversion algorithm is able to couple information from several detectors at the same time. +Indeed, each detector measures a flux andtravel length (defined as the thickness of material seen by +the detector in its acquisition configuration) per viewing axis. By taking a set of density distributed +in the voxels, we have the opacity seen by the viewing axis. Finally, the number of registered models +can be modified and it depends on the requirements in terms of accuracy and available computing +time. +3 +Application to a blast furnace +3.1 +Acquisition configurations +In order to obtain a 3D image of the blast furnace, we carried out three muography acquisitions +around the blast furnace, performed between the end of July 2021 and the end of March 2022 with +the 3 planes detector (specifications of the runs are described in table 1). In this way we intersect +a maximum volume of it, common to the fields of view of the three detectors. On figure 1, the +detector virtual lines of sigth at each position are shown. They allow visualization of the field of +view of each location and the common areas that are observed. +3.2 +External and internal parameters effects on muon flux +By counting the muons rate, in a given direction as a function of time for each of the positions +shown on figure 1, we observe temporal variations. The rate of high-energy cosmic ray muons +as measured underground is shown to be strongly correlated with upper-air temperatures during +– 3 – + +Figure 1. Detector virtual lines of sight at each position with different colors : position/run 1 (in red), +position 2 (in blue), position 3 (in green). +Table 1. Specifications of the 3 runs shown in figure 1(Ze=zenith, Az=azimuth angles). +Run +Ze (°) +Az (°) +X (m) +Y (m) +Z (m) +Dates +1 +45 +51 +-7.51 +-7.04 +11.35 +07/29/21 - 20/09/21 +2 +60 +144.33 +-13.49 +9.52 +14.55 +10/08/21- 10/21/21 +& 12/21/21 - 01/24/22 +3 +60 +270 +15.14 +0.24 +14.55 +02/03/22 - 03/31/22 +short-term atmospheric phenomena [2, 3]and with pressure [5]. We have evaluated the effects of +atmospheric parameters and those of the target operation (coke rate estimate at the cohesive zone +and the value of the blast pressure, defined above). Our goal is to substract these different effects +from the measured rate, before realizing the data inversion, to obtain a density distribution 3D image +in the blast furnace. +We chose to compute the pressure/flux correlations for each position separately because the +detectors are not sensitive to the same opacity from one location to another. By performing a linear +fit between the flux and ambient pressure data, we obtain the barometric coefficients 𝛽𝑝 (hPa−1), +such as (𝜙 − 𝜙0) +/ +𝜙0 = 𝛽𝑝 × (𝑃 − 𝑃0) (see Jourde et al.[5]), their error as well as the correlation +coefficient of the fit in the table 2. The correlation values between pressure and flux vary with time +according to the table 2 and the sensitivity of the muon flux to the pressure variation appears more +important in high pressure episodes. Moreover, the muon fluxes do not seem to be sensitive to the +temperature variations of the upper atmosphere over the studied periods and they are thus more +affected by the pressure variations. This result is consistent with the opacity values encountered, +at most 50 mwe (lower than those of Tramontini et al. [2] ∼ 700 mwe and close to Acernese [4] +open sky experiment) which do not allow to filter out the low energy muons. Indeed, only the most +energetic muons are sensitive to temperature variations. +– 4 – + +Table 2. Barometric coefficients of the different acquisitions around the blast furnace. +Run +𝛽𝑝 (hPa−1) +Error (RMS) +Correlation coefficient +1 +-0.0011 +1.3% +0.77 +2 +- 0.0015 +0.9% +0.80 +3 +- 0.0018 +1% +0.47 +A blast furnace is considered to be in operation when air is injected into it, measured by the +value of the so-called blast pressure. At standstill, the density in the blast furnace is greater and +the interior "column" is tighter. In addition, as the blast pressure increases, fine particles of sinter +take the place of gas and the density in the BF increases. Furthermore, a high coke fraction in +the cohesive zone means that the associated density is lower than usual. Several parameters can +therefore affect the muon flux by changing the density inside: the fraction of fine particles, the blast +furnace stop and the addition of coke. +We performed multivariate linear fits (with external and internal parameters) on the relative +muon flux and evaluated the adequacy of our fits with the Pearson linear coefficient of determination. +The pressure appears to be the dominant parameter. Moreover, in October 2021, during the high +pressure period, the coke rate in the cohesive zone seems to be well correlated with the pressure +corrected muon flux. We found 𝛾𝐶𝑅=-0.013 (±1%) with a correlation coefficient of 0.76. This +means that when the coke rate is high, the blast furnace is stopped and the material goes down, as +well as the cohesive zone, so the density in the blast furnace increases and the measured muon flux +behind it decreases. +After solving the direct problem we reconstruct the 3D-average-density model of the blast +furnace by using the measured flux (corrected from parameters that affect it). Markov chain Monte +Carlo method discribed in subsection 2.2 is used. +4 +Results +The results presented in this section were obtained using real data measured by the detector for +three runs. +4.1 +2D fields +On the figure 2 the opacities (left panels) and densities (right panels) seen by the detector at each +of its 3 positions are represented in 2D, before inversion. We can see a slightly denser area (in +yellow) in the middle of the figures. This zone would seem to be a 2D projection of the cohesive +zone. The shells of the blast furnace are clearly visible on density representations. The position 2 +density figure shows a denser area in the center left. As expected, no muons are recorded below +75-90° in the data. Finally, the contained informations of these figures are aggregated and inverted +to reconstruct the 3D blast furnace and the density distribution inside. +– 5 – + +Figure 2. On the left, the opacity (in mwe) and on the right, the density (in g cm−3) apparent for each of +the 3 positions are represented. The axes are the azimuth and zenith in ◦. +– 6 – + +Position 1 : Opacite +40 +35 +30 +1N100 +900 +25 +20° +130° +809 +400 +500 +600 +5( +20 +60 +15 +10 +5Position1:Densite +2.5 +2 +309 +1100 +200 +30° +400 +500 +300 +1.5 +50 +60 +0.5Position 2 : Opacite +35 +30 +25 +190 +20 +1800 +1209 +13060° +1400 +15 +10 +5Position2:Densite +2.5 +2 +100 +1900 +1109 +1209 +f130060° +1700 +1.5 +1400 +1600 +0.5Position3:Opacite +40 +35 +30 +25 +3100 +2509 +20 +15 +10Position 3 : Densite +3 +2.5 +00 +2 +3100 +00 +1.5 +00 +270 +1 +0.54.2 +3D-inversion results +The 3 muon runs were performed at different periods. Therefore, the 3D reconstruction of the +density distribution of the blast furnace gives us an average of what we can find and not a snapshot +of the different zones and their thickness. In figure 3, the distribution of the mean density (at the top) +and its standard deviation (at the bottom) are shown. Real data have been inverted by considering the +CORSIKA [12] theoretical muon flux model as built in Cohu et al. [6] for the calculation of flux loss. +• The shell of the blast furnace is clearly visible. It is very dense : more than 3 g/cm3. The +travel length from the detector in position 2 arrive perpendicular to the shell, so we have +directly the value of the density without having to evaluate an integrated density over the +whole width of the blast furnace. +• A brighter area is noticeable on the shell (bottom left of the average density figures). This +is probably an area where the three positions provide different information on the integrated +density. Perhaps one of the detectors cannot bring measurements from this area, the value of +the associated standard deviations is also high. +• We distinguish, at a height of 15-20 m inside the BF, a sparse zone (<0.5 g/cm3) that would +contain mainly coke/coal. +• From this last zone would leave a slightly denser zone in the shape of a chimney. This +phenomenon appears when a lot of coal is pulverized in the center and little agglomerate. +This is the case in the blast furnace that we have studied. +• The cohesive zone is visible at 20-25 m height, with a density higher than 1.5 g/cm3. It is +close to the shell. +• The superposition of materials in the dry zone is not visible. In this zone, all materials +charged from the top of the furnace are in the solid state. The iron charge and the coke are +descending and maintain a layered structure. There is a superposition of coke and sinter +sublayers with a density of 0.7 g/cm3 and 2 g/cm3 respectively. However one can see some +" lines" especially at the top of the image, they are the "travel length" due to the acceptance +effects of the detector. +The results of the 3D inversion obtained here are very satisfactory and the cohesive zone +could be highlighted. The algorithm and the inversion method have been successfully tested on the +internal structure of the BF. We will test the robustness of the inversion in the next subsection. +4.3 +Analysis of various sensitivities +We studied, first, the differences observed on our 3D reconstructions as a function of the muon +flux model. We compare Tang et al. [10] and CORSIKA to calculate the muon flux loss. Tang is +an analytical model commonly used and the CORSIKA flux allows to adapt to the environmental +conditions and to the location of an experiment [6]. We analyzed the performances and uncertainties +of the reconstruction engine : what are the differences caused by the randomness of the inversion? +– 7 – + +Then we looked at the consequences of the number of models registered in the MCMC algorithm +and the randomness of the algorithm itself. We also tested what is implied by a different density +model in the input (different density at the cohesive zone). In all cases, the areas with the largest +density differences are located at the bottom of the blast furnace : where a few data is collected +which is obviously a source of noise. +Figure 3. Results of real data inversion using a theoretical flux modeled with CORSIKA in Bremen for the +calculation of flux loss. The axes (𝑋𝑌𝑍) are in meters and the density in g cm−3. +- at the top : distribution of the average density in the blast furnace, +- at the bottom : distribution of the standard deviation of the density in the blast furnace. +– 8 – + +40、 +40 +40 +35 +35、 +30. +30~ +30 、 +25 ~ +1.5 +15 +1.5 +20~ +20. +20~ +15. +15 +15、 +10 +10、 +1040、 +40 +40 +35. +35 +30. +30 、 +25 、 +25 ~ +1.2 +20~ +20. +20 ~ +15. +15 +15、 +0.8 +0.8 +0.8 +10. +10、 +105 +Conclusion +We performed a muography experiment on a blast furnace of ArcelorMittal and obtained the first 3D +image of it. We hoped to be able to clearly distinguish the location of the cohesive zone, which turns +out to be the key of the furnace’s productivity. The 3D reconstruction of the density distribution +was a great success. We used an inversion program on our measured muon data using Markov +chain Monte Carlo (MCMC). This algorithm is stable (few variations between 2 identical models) +and rather fast even with a large number of recorded models. The results of the 3D inversion with +the use of CORSIKA or Tang [10] models of theorical flux show that the opacity estimates are +strongly influenced in the regions of zenith angle between 70 and 90° and especially for the areas +of low opacity. As a reminder, the theoretical flux of CORSIKA is also dependent on atmospheric +conditions and must be adapted to the season when the acquisitions were made. An uncertainty of +10% on the flux leads to an error of 4% on the opacity. The 3D images obtained are only an average +of the density distribution but are quite realistic and validated by the operators of the blast furnaces +studied. The density contrasts are obvious, especially for the shell and the cohesive zone which are +clearly distinguishable. We could monitor the activity inside the BF. Indeed, we have evaluated the +effect of the atmospheric pressure variation on the measured muon flux and we are able to correct +it for this impact. The measured and corrected flux seems to be sensitive to the coke rate variations +in the cohesive zone too. All these elements related to the composition or the shape of the cohesive +zone may allow the blast furnace operators to adapt their material loading according to the state of +this zone. Further improvements to the method are under study. +Acknowledgments +This work was the subject of a CIFRE agreement between the ArcelorMittal Maizières Research +SA and IP2I (Lyon). +References +[1] Tarantola, Albert, Inverse problem theory and methods for model parameter estimation, SIAM(2005). +[2] Tramontini, Matias and Rosas-Carbajal, Marina and Nussbaum, Christophe and Gibert, Dominique +and Marteau, Jacques, Middle-atmosphere dynamics observed with a portable muon detector, Wiley +Online Library 6 (2019) 1865–1876. +[3] Adamson, P and Anghel, I and Aurisano, A and Barr, G and Bishai, M and Blake, A and Bock, GJ +and Bogert, D and Cao, SV and Castromonte, CM and others, Observation of muon intensity +variations by season with the MINOS near detector, APS (2014) 012010. +[4] Acernese, F and Agathos, M and Ain, A and Albanesi, S and Allocca, A and Amato, A and Andrade, +T and Andres, N and Andrés-Carcasona, M and Andrić, T and others, The Virgo O3 run and the +impact of the environment, arXiv preprint arXiv:2203.04014 (2022) . +[5] Jourde, K and Gibert, D and Marteau, J and de Bremond d’Ars, J and Gardien, S and Girerd, C and +Ianigro, JC, Monitoring temporal opacity fluctuations of large structures with muon radiography: a +calibration experiment using a water tower, Scientific Reports 6 (2016). +– 9 – + +[6] Cohu, Amélie and Tramontini, Matias and Chevalier, Antoine and Ianigro, Jean-Christophe and +Marteau, Jacques, Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for +Muography Applications, Instruments, MDPI 6 (2022) 24. +[7] Chevalier, A and Legchenko, Anatoli and Girard, J-F and Descloitres, Marc, Monte Carlo inversion of +3-D magnetic resonance measurements, Geophysical Journal International, Oxford University Press +198 (2014), 216–228. +[8] Sambridge, M and Mosegaard, K, Monte Carlo methods in geophysical inverse problems, Reviews of +Geophysics, Wiley Online Library 40 (2002) 3–1. +[9] Marteau, J. and Gibert, D. and Lesparre, N. and Nicollin, F. and Noli, P. and Giacoppo, F., Muons +tomography applied to geosciences and volcanology, Nuclear Instruments and Methods in Physics +Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Science +Direct(2012). +[10] Tang, Alfred and Horton-Smith, Glenn and Kudryavtsev, Vitaly A. and Tonazzo, Alessandra, Muon +simulations for Super-Kamiokande, KamLAND, and CHOOZ, Physical Review D 74 (2006) . +[11] Mosegaard, Klaus and Tarantola, Albert, Monte Carlo sampling of solutions to inverse problems, +Journal of Geophysical Research: Solid Earth, Wiley Online Library 100 (1995) 12431–12447. +[12] Heck, Dieter and Schatz, G and Knapp, J and Thouw, T and Capdevielle, JN, CORSIKA: a Monte +Carlo code to simulate extensive air showers (1998) . +[13] Lesparre, N and Marteau, J and Déclais, Y and Gibert, Dominique and Carlus, B and Nicollin, +Florence and Kergosien, Bruno, Design and operation of a field telescope for cosmic ray geophysical +tomography, Geoscientific Instrumentation, Methods and Data Systems, Copernicus GmbH 1 (2012) +11. Geoscientific Instrumentation, Methods and Data Systems, +– 10 – + diff --git a/mtE3T4oBgHgl3EQfKglz/content/tmp_files/load_file.txt b/mtE3T4oBgHgl3EQfKglz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db99f933e96b659fb52e1498b53e8882a4d2ed83 --- /dev/null +++ b/mtE3T4oBgHgl3EQfKglz/content/tmp_files/load_file.txt @@ -0,0 +1,262 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf,len=261 +page_content='Prepared for submission to JINST First 3D reconstruction of a blast furnace using muography A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Cohu𝑎,𝑏,𝑐 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Chevalier𝑐 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Nechyporuk𝑏 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Franzen𝑏 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Sauerwald𝑏 J-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Ianigro𝑎,𝑐 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Marteau 𝑎,𝑐 𝑎IP2I, IN2P3, CNRS, Université Lyon 1, UMR 5822, 69100 Villeurbanne, France 𝑏ArcelorMittal Maizières Research SA,Voie Romaine, 57280 Maizières-lès-Metz, France 𝑐MUODIM, 31 rue Saint-Maximin, 69003 Lyon, France E-mail: marteau@ip2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='fr Abstract: The blast furnace (BF) is the fundamental tool used in the iron manufacture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Due to the difficulty of accessing direct measurements of the inner phenomena, we determined the density distribution of its internal volume in order to improve its productivity using muography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This is an imaging technique based on the differential absorption of a flux of incident particles, muons, by the target under study, similar to clinical X-ray imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Muons are elementary particles that have the property of passing through dense materials, up to hundreds of meters away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Their relative absorption and deviation allows the generation of density distribution images of an object by tracking the number of muons received by a detector, before and after passing through a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The incident direction of the detected muons is reconstructed by means of a detector composed of 3 scintillator panels that we moved on 3 positions around the BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' With this technique, we obtained the first 3D image of the internal structure of a BF using a Markov Chain Monte Carlo (MCMC) inverse problem solving algorithm on muon flux data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We were also able to perform a density monitoring of the BF and some of its operating parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We distinguished the position and shape of the cohesive zone, a key element in the productivity of a furnace, validating this innovative measurement concept in the application to a BF and opening the field to a series of future experiments to gain both spatial and temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Keywords: Particle tracking detectors, Scintillators and scintillating fibres and light guides, Com- puterized Tomography (CT) and Computed Radiography (CR), Image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='04354v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='geo-ph] 11 Jan 2023 Contents 1 Introduction 1 2 Tomography reconstruction theory 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 Generalities and issues 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 Inverse problem 2 3 Application to a blast furnace 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 Acquisition configurations 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 External and internal parameters effects on muon flux 3 4 Results 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 2D fields 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 3D-inversion results 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='3 Analysis of various sensitivities 7 5 Conclusion 9 1 Introduction We seek to determine the density distribution of matter inside a blast furnace in order to visualize the cohesive zone and to carry out a dynamic monitoring of the various phases present in the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' To achieve this, muography is applied to make a dynamic image during the operation of a blast furnace of ArcelorMittal in Bremen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Muography measures the absorption or deflection of cosmic muons as they pass through dense materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Muons are elementary particles which have the property to pass, in a straight line at first order, up to several kilometers of standard rocks, and whose relative absorption allows to generate images by contrast densitometry, like a standard clinical radiography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The acquired muon data allows to follow the density as a function of time during the operating cycles of the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In a second step, the acquisition of 2D images and then the 3D reconstruction is accomplished by the data inversion from several measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The final objective is to understand the topological characteristics and the formation rate of the cohesive zone and the influence of certain loading parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The article is organized in four main parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The first part explains the theorical points and algorithms used in tomography reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The second part presents the simulation parameters of a blast furnace and explains how we built the 3D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We have completed our analysis by monitoring the activity of the blast furnace as function of different environmental parameters such as atmospheric pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The third part presents the results of real muons data inversion to visualize the different density zones in the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Finally, we report the conclusions and perspectives of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 1 – 2 Tomography reconstruction theory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 Generalities and issues Absorption muography measures the cosmic ray flux deficit in the direction of observation and determines the integrated density of a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Muons are cosmic rays able to cross several meters of rocks losing energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The minimum amount of energy that muons must need to penetrate the structure must be of a value higher than the one lost inside the object, so that the detector can follow the outgoing muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The detector position must then be adjusted to optimize the spatial resolution during field measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Muon tomography is limited to the study of a portion of the object only, because of the limited angular aperture of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In order to take these different points into consideration, we use the inverse and direct problems jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The direct problem consists, in our case, in predicting the expected muon flux at the exit of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' It is necessary to use materials knowledge and physical properties of the object’s constituents and then to use a density distribution as precise as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The parameters sought during the inversion use the direct problem to estimate the expected measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The measured information is retrieved from a given distribution of 𝑝 parameters in the studied structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Moreover, the flux attenuation is estimated from a known law based on the contrasting distribution of zones (of different density for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' With the trackers the direction of the muons is reconstructed in order to observe the properties of the medium on a precise observation axis (see Lesparre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='[13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The flux that arrives on the detector after having crossed the object and the theoretical flux that would reach the detector in the absence of matter are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The contrast between thoses two quantities gives directly access to the opacity of the matter (in meter equivalent water (mwe)), defined as the integral of the density along the trajectory of the muon from its entry point to its exit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Moreover, the only observable are the particles directions and energy deposits, since the detectors usually do not give direct information on the particles total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In order to solve the data inversion, the measured flux is coupled to a theoretical flux model and to a flux loss model in matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 Inverse problem The reconstruction of a muography image is achieved by solving an inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The goal is to recover the distribution of properties of the medium (3D density) from measurements of muon rates and their directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' An inverse problem is a situation in which one tries to determine the parameters of a model 𝑝 (here the 3D density of the environnement) from experimental measurements 𝑚 (muon rates) such that 𝑚 = 𝑓 (𝑝) where 𝑓 contains the open muon flux (calibration) and the law governing the absorption of muons in matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' While a single parameter estimate can be easily obtained by least squares fitting, the use of Monte Carlo methods allows the maintenance of the stochasticity of the model during the estimation of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In order to improve the reliability of the results, it is good practice to add some informations about the objet to study, other than data, that we call a priori information [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' They allow to constrain the solutions space and improve the accuracy of the statistical answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We use the Metropolis-Hastings algorithm which is a particular class of Monte Carlo methods using Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' It works like a geometrically biased random walk with a data based selection at each throw [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Each new proposal/model can be accepted or rejected if the likelihood of the – 2 – model (regarding the data and the physics of the problem) is greater or smaller than the likelihood of the previous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Hence, unlike more general Monte Carlo methods where the sampled values are statistically independent, in the Metropolis-Hastings algorithm they are statistically auto-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This auto-correlation is minimized by adding a threshold number of changes that must be accepted between the recording of two consecutive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We need models whose density values are continuous over finite element discretized volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Here a model stands for any given set of values representing a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The engine that generates the 3D density models, linked by Markov chain, is made to design density sets per class forming spatially contiguous voxel sets which share the same density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The inversion algorithm is a 3D adaptation of Mosegaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' [11] work, where details can be found in Chevalier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The method makes selections of the models according to the evolution of the distance 𝐷 : 𝐷 = 𝐹𝑡 − 𝐹𝑚 𝜎𝑚 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1) with 𝐹𝑡 the theoretical flux and 𝐹𝑚 the measured flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 𝐷 is a metric of the distance between the data and the simulation, expanded or compressed by the degree of uncertainty on the measurement 𝜎𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This distance 𝐷 is recalculated each time the density changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' A new model is accepted as a solution if the mean difference between the reconstructed signal and the data (evaluated by the mean square deviation) is lower than the mean noise estimated during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' If the model is selected, we save its likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This model is then slightly perturbed again by changing some voxel density values and we recalculate the new likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The total likelihood of a model, 𝐿(𝑚), can be expressed as a product of partial likelihoods, one for each data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Our inversion algorithm is able to couple information from several detectors at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Indeed, each detector measures a flux andtravel length (defined as the thickness of material seen by the detector in its acquisition configuration) per viewing axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' By taking a set of density distributed in the voxels, we have the opacity seen by the viewing axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Finally, the number of registered models can be modified and it depends on the requirements in terms of accuracy and available computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 3 Application to a blast furnace 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 Acquisition configurations In order to obtain a 3D image of the blast furnace, we carried out three muography acquisitions around the blast furnace, performed between the end of July 2021 and the end of March 2022 with the 3 planes detector (specifications of the runs are described in table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In this way we intersect a maximum volume of it, common to the fields of view of the three detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' On figure 1, the detector virtual lines of sigth at each position are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' They allow visualization of the field of view of each location and the common areas that are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 External and internal parameters effects on muon flux By counting the muons rate, in a given direction as a function of time for each of the positions shown on figure 1, we observe temporal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The rate of high-energy cosmic ray muons as measured underground is shown to be strongly correlated with upper-air temperatures during – 3 – Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Detector virtual lines of sight at each position with different colors : position/run 1 (in red), position 2 (in blue), position 3 (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Specifications of the 3 runs shown in figure 1(Ze=zenith, Az=azimuth angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Run Ze (°) Az (°) X (m) Y (m) Z (m) Dates 1 45 51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='04 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='35 07/29/21 - 20/09/21 2 60 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='33 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='49 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='52 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='55 10/08/21- 10/21/21 & 12/21/21 - 01/24/22 3 60 270 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='24 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='55 02/03/22 - 03/31/22 short-term atmospheric phenomena [2, 3]and with pressure [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We have evaluated the effects of atmospheric parameters and those of the target operation (coke rate estimate at the cohesive zone and the value of the blast pressure, defined above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Our goal is to substract these different effects from the measured rate, before realizing the data inversion, to obtain a density distribution 3D image in the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We chose to compute the pressure/flux correlations for each position separately because the detectors are not sensitive to the same opacity from one location to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' By performing a linear fit between the flux and ambient pressure data, we obtain the barometric coefficients 𝛽𝑝 (hPa−1), such as (𝜙 − 𝜙0) / 𝜙0 = 𝛽𝑝 × (𝑃 − 𝑃0) (see Jourde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' [5]), their error as well as the correlation coefficient of the fit in the table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The correlation values between pressure and flux vary with time according to the table 2 and the sensitivity of the muon flux to the pressure variation appears more important in high pressure episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Moreover, the muon fluxes do not seem to be sensitive to the temperature variations of the upper atmosphere over the studied periods and they are thus more affected by the pressure variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This result is consistent with the opacity values encountered, at most 50 mwe (lower than those of Tramontini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' [2] ∼ 700 mwe and close to Acernese [4] open sky experiment) which do not allow to filter out the low energy muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Indeed, only the most energetic muons are sensitive to temperature variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 4 – Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Barometric coefficients of the different acquisitions around the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Run 𝛽𝑝 (hPa−1) Error (RMS) Correlation coefficient 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='0011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='77 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='80 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='0018 1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='47 A blast furnace is considered to be in operation when air is injected into it, measured by the value of the so-called blast pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' At standstill, the density in the blast furnace is greater and the interior "column" is tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In addition, as the blast pressure increases, fine particles of sinter take the place of gas and the density in the BF increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Furthermore, a high coke fraction in the cohesive zone means that the associated density is lower than usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Several parameters can therefore affect the muon flux by changing the density inside: the fraction of fine particles, the blast furnace stop and the addition of coke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We performed multivariate linear fits (with external and internal parameters) on the relative muon flux and evaluated the adequacy of our fits with the Pearson linear coefficient of determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The pressure appears to be the dominant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Moreover, in October 2021, during the high pressure period, the coke rate in the cohesive zone seems to be well correlated with the pressure corrected muon flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We found 𝛾𝐶𝑅=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='013 (±1%) with a correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This means that when the coke rate is high, the blast furnace is stopped and the material goes down, as well as the cohesive zone, so the density in the blast furnace increases and the measured muon flux behind it decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' After solving the direct problem we reconstruct the 3D-average-density model of the blast furnace by using the measured flux (corrected from parameters that affect it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Markov chain Monte Carlo method discribed in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 4 Results The results presented in this section were obtained using real data measured by the detector for three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='1 2D fields On the figure 2 the opacities (left panels) and densities (right panels) seen by the detector at each of its 3 positions are represented in 2D, before inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We can see a slightly denser area (in yellow) in the middle of the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This zone would seem to be a 2D projection of the cohesive zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The shells of the blast furnace are clearly visible on density representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The position 2 density figure shows a denser area in the center left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' As expected, no muons are recorded below 75-90° in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Finally, the contained informations of these figures are aggregated and inverted to reconstruct the 3D blast furnace and the density distribution inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 5 – Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' On the left, the opacity (in mwe) and on the right, the density (in g cm−3) apparent for each of the 3 positions are represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The axes are the azimuth and zenith in ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 6 – Position 1 : Opacite 40 35 30 1N100 900 25 20° 130° 809 400 500 600 5( 20 60 15 10 5Position1:Densite 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 2 309 1100 200 30° 400 500 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5Position 2 : Opacite 35 30 25 190 20 1800 1209 13060° 1400 15 10 5Position2:Densite 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 2 100 1900 1109 1209 f130060° 1700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 1400 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5Position3:Opacite 40 35 30 25 3100 2509 20 15 10Position 3 : Densite 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 00 2 3100 00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 00 270 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 3D-inversion results The 3 muon runs were performed at different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Therefore, the 3D reconstruction of the density distribution of the blast furnace gives us an average of what we can find and not a snapshot of the different zones and their thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In figure 3, the distribution of the mean density (at the top) and its standard deviation (at the bottom) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Real data have been inverted by considering the CORSIKA [12] theoretical muon flux model as built in Cohu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' [6] for the calculation of flux loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The shell of the blast furnace is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' It is very dense : more than 3 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The travel length from the detector in position 2 arrive perpendicular to the shell, so we have directly the value of the density without having to evaluate an integrated density over the whole width of the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' A brighter area is noticeable on the shell (bottom left of the average density figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This is probably an area where the three positions provide different information on the integrated density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Perhaps one of the detectors cannot bring measurements from this area, the value of the associated standard deviations is also high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We distinguish, at a height of 15-20 m inside the BF, a sparse zone (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 g/cm3) that would contain mainly coke/coal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' From this last zone would leave a slightly denser zone in the shape of a chimney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This phenomenon appears when a lot of coal is pulverized in the center and little agglomerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This is the case in the blast furnace that we have studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The cohesive zone is visible at 20-25 m height, with a density higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' It is close to the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The superposition of materials in the dry zone is not visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In this zone, all materials charged from the top of the furnace are in the solid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The iron charge and the coke are descending and maintain a layered structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' There is a superposition of coke and sinter sublayers with a density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='7 g/cm3 and 2 g/cm3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' However one can see some " lines" especially at the top of the image, they are the "travel length" due to the acceptance effects of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The results of the 3D inversion obtained here are very satisfactory and the cohesive zone could be highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The algorithm and the inversion method have been successfully tested on the internal structure of the BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We will test the robustness of the inversion in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='3 Analysis of various sensitivities We studied, first, the differences observed on our 3D reconstructions as a function of the muon flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We compare Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' [10] and CORSIKA to calculate the muon flux loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Tang is an analytical model commonly used and the CORSIKA flux allows to adapt to the environmental conditions and to the location of an experiment [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We analyzed the performances and uncertainties of the reconstruction engine : what are the differences caused by the randomness of the inversion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 7 – Then we looked at the consequences of the number of models registered in the MCMC algorithm and the randomness of the algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We also tested what is implied by a different density model in the input (different density at the cohesive zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' In all cases, the areas with the largest density differences are located at the bottom of the blast furnace : where a few data is collected which is obviously a source of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Results of real data inversion using a theoretical flux modeled with CORSIKA in Bremen for the calculation of flux loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The axes (𝑋𝑌𝑍) are in meters and the density in g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' at the top : distribution of the average density in the blast furnace, at the bottom : distribution of the standard deviation of the density in the blast furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' – 8 – 40、 40 40 35 35、 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 30~ 30 、 25 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='5 20~ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 20~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 15 15、 10 10、 1040、 40 40 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 35 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 30 、 25 、 25 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='2 20~ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 20 ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 15 15、 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' 10、 105 Conclusion We performed a muography experiment on a blast furnace of ArcelorMittal and obtained the first 3D image of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We hoped to be able to clearly distinguish the location of the cohesive zone, which turns out to be the key of the furnace’s productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The 3D reconstruction of the density distribution was a great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We used an inversion program on our measured muon data using Markov chain Monte Carlo (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' This algorithm is stable (few variations between 2 identical models) and rather fast even with a large number of recorded models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The results of the 3D inversion with the use of CORSIKA or Tang [10] models of theorical flux show that the opacity estimates are strongly influenced in the regions of zenith angle between 70 and 90° and especially for the areas of low opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' As a reminder, the theoretical flux of CORSIKA is also dependent on atmospheric conditions and must be adapted to the season when the acquisitions were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' An uncertainty of 10% on the flux leads to an error of 4% on the opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The 3D images obtained are only an average of the density distribution but are quite realistic and validated by the operators of the blast furnaces studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The density contrasts are obvious, especially for the shell and the cohesive zone which are clearly distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' We could monitor the activity inside the BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Indeed, we have evaluated the effect of the atmospheric pressure variation on the measured muon flux and we are able to correct it for this impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' The measured and corrected flux seems to be sensitive to the coke rate variations in the cohesive zone too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' All these elements related to the composition or the shape of the cohesive zone may allow the blast furnace operators to adapt their material loading according to the state of this zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Further improvements to the method are under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' Acknowledgments This work was the subject of a CIFRE agreement between the ArcelorMittal Maizières Research SA and IP2I (Lyon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' References [1] Tarantola, Albert, Inverse problem theory and methods for model parameter estimation, SIAM(2005).' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' and Lesparre, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' and Nicollin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' and Noli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=' and Giacoppo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE3T4oBgHgl3EQfKglz/content/2301.04354v1.pdf'} +page_content=', Muons tomography applied to geosciences and volcanology, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Science Direct(2012).' metadata={'source': 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0000000000000000000000000000000000000000..e58112770376d6ca2323cb4ea35f087e25259c14 --- /dev/null +++ b/n9FLT4oBgHgl3EQfgC--/content/tmp_files/2301.12098v1.pdf.txt @@ -0,0 +1,1995 @@ +Turbulence control in plane Couette flow using low-dimensional +neural ODE-based models and deep reinforcement learning +Alec J. Linota, Kevin Zenga, Michael D. Graham1a +aDepartment of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI +53706, USA +Abstract +The high dimensionality and complex dynamics of turbulent flows remain an obstacle to +the discovery and implementation of control strategies. Deep reinforcement learning (RL) is +a promising avenue for overcoming these obstacles, but requires a training phase in which the +RL agent iteratively interacts with the flow environment to learn a control policy, which can +be prohibitively expensive when the environment involves slow experiments or large-scale +simulations. We overcome this challenge using a framework we call “DManD-RL” (data- +driven manifold dynamics-RL), which generates a data-driven low-dimensional model of our +system that we use for RL training. With this approach, we seek to minimize drag in a +direct numerical simulation (DNS) of a turbulent minimal flow unit of plane Couette flow at +Re = 400 using two slot jets on one wall. We obtain, from DNS data with O(105) degrees of +freedom, a 25-dimensional DManD model of the dynamics by combining an autoencoder and +neural ordinary differential equation. Using this model as the environment, we train an RL +control agent, yielding a 440-fold speedup over training on the DNS, with equivalent control +performance. The agent learns a policy that laminarizes 84% of unseen DNS test trajectories +within 900 time units, significantly outperforming classical opposition control (58%), despite +the actuation authority being much more restricted. The agent often achieves laminarization +through a counterintuitive strategy that drives the formation of two low-speed streaks, with +a spanwise wavelength that is too small to be self-sustaining. The agent demonstrates the +same performance when we limit observations to wall shear rate. +1. Introduction +Energy loss due to turbulent drag is ubiquitous in many industrial and commercial pro- +cesses, ranging from air flowing over a plane wing, a ship in the ocean, or oil pumped through +a pipe. In total, turbulent drag accounts for 25% of the energy used in industry and com- +merce, resulting in 5% of all man-made CO2 emissions [1]. Even small reductions in this +drag can yield massive savings in energy, which has long motivated the search for better flow +control strategies. +Many control types for reducing turbulent drag exist, including, but not limited to, +polymer/surfactant drag reduction [2, 3], riblets [4], wall oscillations [5], plasma actuators +1Corresponding author: mdgraham@wisc.edu +Preprint submitted to Elsevier +January 31, 2023 +arXiv:2301.12098v1 [physics.flu-dyn] 28 Jan 2023 + +[6], and synthetic jets [7]. Due to the complexity of reducing drag, it has been most common +to apply these control methods in an open-loop manner where the control policy at any +given time is independent of the flow state [8]. However, application of feedback control on +a turbulent system could yield far better performance in controlling drag. +Unfortunately, the complexity of the problem has typically limited applications of feed- +back control to methods based on heuristics. A well-studied heuristic method is opposition +control [9]. Here the wall-normal velocity at the wall is set to have the opposite sign as the +wall-normal velocity at some detection plane in the channel, a straightforward actuation in +simulations. This method has been applied in simulations [9, 10, 11, 12] and experiments +(with some modifications) [13, 14], and extensions exist to use just wall observations [15, 16]. +When heuristics are replaced with methods from optimal control theory, like model predic- +tive control (MPC), the drag reduction far outperforms opposition control [17] while using +the same actuation scheme. However, the real-time implementation of MPC on DNS still re- +mains infeasible because it involves solving the DNS forward over a time horizon (preferably +a long one) and then solving an adjoint backwards in time for every actuation [17]. +A potential approach to overcome the high computational cost of real-time optimization +of a control strategy is deep reinforcement learning (RL) [18]. Deep RL gained significant +traction when it was used to defeat the best professional players in GO [19], DOTA II [20], +and Starcraft II [21], in addition to the best engines in GO, Chess, and Shogi [22]. In deep +RL, a neural network (NN) control agent is trained through iterative interactions with the +environment (i.e. the system to be controlled) to maximize a scalar total reward (i.e. control +objective) that includes present as well as discounted future reward values. Once trained, +the control agent can be deployed in real time without the need for online optimization. +In recent years, RL has been applied in fluids simulations to reduce the drag experienced +in flow around a cylinder [23, 24, 25], to optimize jets on an airfoil [26], and to find efficient +swimming strategies [27]. RL has even been applied to experimental flow systems [28]. Re- +cently multi-agent deep RL has been explored for the control of pressure-driven turbulent +channel flow [29, 30] in a problem formulation similar to opposition control [9]. In these +works, an RL policy is trained to map local detection plane observables to a wall-normal +velocity response at the walls to reduce drag. Notably, the same RL policy is locally im- +plemented at each wall grid point. We differentiate the control problem addressed from the +previously mentioned works in that we limit the control authority to just two spatially local- +ized jets on a single wall, with a zero-net flux constraint, as opposed to full spatial control of +both walls. We feel that this is much closer to experimental realizability than an approach +with control authority everywhere on the wall. A recent review of the application of deep +RL applied to fluid mechanics problems is presented in Viquerat et al. [31]. +In these active flow control problems, deep RL possesses the advantageous property of +being completely data-driven, allowing it to discover novel and nontrivial control strategies +in complex systems from just data alone without the need to analytically derive or hard- +code system properties into the method. However, the training portion of RL is a major +bottleneck, requiring a tremendous number of interactions with the target environment to +find an approximately optimal policy [32]. +Practically speaking, this can correspond to +running an enormous number of high-resolution simulations or flow experiments, both of +which may be prohibitively expensive. +In the present work, we apply RL to control a minimal flow unit (MFU) (the smallest +2 + +Figure 1: Schematic of the Couette flow domain with two slot jets on one wall. +domain that sustains turbulence) [33] of plane Couette flow at Re = 400 using a pair of +streamwise-aligned slot jets at one wall, with a no-net-flux constraint. Therefore there is +only one degree of freedom for actuation. We select this system because the unactuated flow +isolates the self-sustaining regeneration cycle of wall-bounded turbulence [34, 35]. This case +is well-studied for tasks such as reduced-order modeling [36, 37, 38, 39], finding invariant +solutions [40, 41], and applying opposition control [12]. +In order to overcome the high computational cost of RL training in this environment, in +this work we replace the high-resolution simulation with an accurate low-dimensional sur- +rogate model, aiming to dramatically reduce the time required to train the control policy. +We showed in [42] that this data-driven model-based RL approach, which we refer to as +“Data-Driven Manifold Dynamics” RL (DManD-RL), works well for controlling spatiotem- +poral chaotic dynamics in the Kuramoto-Sivashinksy Equation. For further discussion on +the various types of model-based RL, we refer the reader to Zeng et al. [42]. +In Sec. 2 +we introduce the control environment and the DManD-RL framework. Then, in Sec. 3 we +describe the data used for training the DManD model, the performance of the model, and +the results of applying RL to the DManD model and to the DNS environment. Finally, we +conclude in Sec. 4 with a summary of the key results. +2. Framework +2.1. Navier-Stokes Equation with Slot Jets +The environment we consider is a direct numerical simulation (DNS) of the Navier-Stokes +Equations (NSE) +∂u +∂t + u · ∇u = −∇p + Re−1∇2u, +∇ · u = 0. +(1) +The velocities in the streamwise x ∈ [0, Lx], wall-normal y ∈ [−Ly/2, Ly/2], and spanwise +z ∈ [0, Lz] directions are defined as u = [ux, uy, uz], and the pressure is p. Here we have +3 + +nondimensionalized velocity by the speed U of the walls, length by the channel half-height +(h = Ly/2), time with h/U and pressure with ρU 2, where ρ is the fluid density. +The +Reynolds number is Re = Uh/ν, where ν is the kinematic viscosity. The boundary conditions +for this setup are periodic in x and z (u(0, y, z) = u(Lx, y, z), u(x, y, 0) = u(x, y, Lz)), +no-slip boundary conditions at the walls (ux(x, ±Ly/2, z) = ±1, uz(x, ±Ly/2, z) = 0), no +penetration at the top wall (uy(x, Ly/2, z) = 0), and finally, the actuation on the bottom +wall (uy(x, −Ly/2, z) = fa(x, z)), as we now describe. +The actuation on the bottom wall is in the form of two slot jets that are Gaussian in z +and travel the length of the channel: +uy(x, −Ly/2, z) = fa(x, z) = a(t)Vmax +� +exp +� +−(z − Lz/4)2 +2σ2 +� +− exp +� +−(z − 3Lz/4)2 +2σ2 +�� +. +(2) +We set σ ≈ 0.16 so that the jets act “locally”, and the velocity of the jet is dictated by +a(t)Vmax, where a(t) ∈ [−1, 1] is the instantaneous actuation amplitude scaled by a maximum +velocity Vmax = 0.05. For perspective, the root-mean-squared wall-normal velocity at the +channel centerline for turbulent unactuated flow is ∼ 0.063. We chose this small velocity +to evaluate how the agent performs with limited control authority. In Fig. 1 we show a +schematic illustrating this system. +The complexity of the flow increases as the Reynolds number and the domain size Lx +and Lz increase. +Here we chose the same setup as Hamilton et al. [34], Re = 400 and +[Lx, Ly, Lz] = [1.75π, 2, 1.2π]. These parameters isolate the “self-sustaining process” (SSP) +that drives wall-bounded turbulence. In the SSP, low-speed streaks that have been lifted +from the wall become wavy, this waviness leads to the breakdown of the streaks, generating +streamwise rolls, and, finally, these rolls lift low-speed fluid off the wall to regenerate streaks, +completing the cycle. By working in this well-studied domain that is dominated by the SSP, +we can better identify the means by which a control strategy can disrupt or suppress this +process. +In this work, the control strategy is to minimize the turbulent drag averaged between +both walls +D = 1 +2 +� Lx +0 +� Lz +0 +� +∂ux +∂y +���� +y=1 +− 1 +� ++ +� +∂ux +∂y +���� +y=−1 +− 1 +� +dxdz, +(3) +subject to a quadratic penalty on actuation amplitude a. (If the relation between the pressure +drop and actuation velocity for pumping fluid into/out of the domain is linear, then this +penalty is proportional to the power consumption of the actuation.) Further details are +described in Sec. 2.2. We report drag in this fashion because this quantity goes to 0 when +the flow laminarizes. +We simulate the flow using a Fourier-Chebyshev pseudo-spectral code we implemented in +Python [43], which is based on the Channelflow code developed by Gibson et al. ([44, 45]). +Linear terms are treated implicitly and the nonlinear term explicitly. The specific time inte- +gration schemes we use are the multistage SMRK2 scheme [46] for the first two timesteps after +every actuation, and the multistep Adams-Bashforth Backward-Differentiation 3 scheme [47] +until the next actuation. The multistep scheme is more computationally efficient, but, be- +cause actuations change instantaneously, using previous steps with the incorrect boundary +condition would lead to incorrect results. For all trials we evolve solutions forward using +4 + +∆t = 0.02 on a grid of [Nx, Ny, Nz] = [32, 35, 32] in x, y, and z from random divergence-free +initial conditions that we evolve forward 100 time units so initial conditions are near the +turbulent attractor. +While most of this approach is standard, here we include some details on the simulation +procedure to highlight explicitly how we set the jet actuation boundary condition. At each +time step, the approach involves solving the expression +Re−1d2ˆui+1 +kx,kz +dy2 +− λˆui+1 +kx,kz − ˆ∇ˆpi+1 +kx,kz = − ˆRi +kx,kz, +(4) +where i is the timestep and ˆ· = Fx,z(·) denotes the Fourier transform in x and z. +The +variable λ includes the timestep ∆t and the x and z components of the diffusive term and +R encompasses all the remaining explicit terms (for a multistep method this includes ˆukx,kz +multiple steps back). We refer the reader to [44] for a more detailed discussion. Upon taking +the divergence of Eq. 4, and accounting for incompressibility, we isolate the problem down +to 4 sets of one-dimensional Helmholtz equations (for conciseness we suppress indices kx, kz, +and i): +Re−1d2ˆux +dy2 − λˆux − 2πikx +Lx +ˆp = − ˆRx +ˆux(±1) = ±δkx,0δkz,0 +(5) +Re−1d2ˆuz +dy2 − λˆuz − 2πikz +Lz +ˆp = − ˆRz +ˆuz(±1) = 0 +(6) +Re−1d2ˆuy +dy2 − λˆuy − dˆp +dy = − ˆRy +ˆuy(−1) = Fx,z(fa), +ˆuy(1) = 0 +(7) +d2ˆp +dy2 − 4π2 +� k2 +x +L2 +x ++ k2 +z +L2 +z +� +ˆp = ˆ∇ · ˆR +dˆuy +dy (±1) = 0. +(8) +These equations can be solved for every wavenumber pair kx and kz. The challenge in +solving these equations is due to the coupling in Eq. 7 and Eq. 8. The pressure is coupled +to the wall-normal velocity because an explicit boundary condition is unknown. Instead, +from incompressibility, we know dˆuy/dy(±1) = 0, which we substitute for the pressure +boundary condition. To solve these coupled equations we use the influence matrix method +and tau correction developed by Kleiser and Schumann [48]. Although we set the wall-normal +boundary condition in Eq. 7 by the slot jets in Eq. 2, we note that it is simple to replace +this boundary condition with any shape of actuation. +2.2. Data-driven framework +The objective in deep RL is to train an agent, commonly a neural network, to approximate +the optimal control policy a = π∗(s), which given a state observation s, outputs the optimal +control action a. The optimal policy seeks to maximize the expected long time discounted +cumulative reward, +π∗ = arg max +π +E +� ∞ +� +l=0 +γl(rt+lτ) +� +, +(9) +5 + +where 0 < γ < 1 is the discount factor, τ is the time between control actions, and rt is the +reward, a scalar-valued control objective function decided by the user evaluated at time t. +As our objective in this work is to minimize the drag of our turbulent Couette system while +simultaneously avoiding the use of superfluous control actions, we define the reward function +as the following, +rt = − +� +D(t) + c∥a(t)∥2� +τ , +(10) +where c is a scalar and ⟨·⟩τ is the average from t to t + τ. We note here that our actuation +penalty is proportional to the power required for actuation. In deep RL π∗ is learned via +repeated cyclic interactions between the agent and the environment i.e. the target system. +A typical cycle consists of the following: given a state observation of the system at time +t, st, the control agent outputs its estimated best control response at. This control action +is then applied to the environment. The system is allowed to evolve for τ time units, and +then the impact of the action is quantified by observing the resulting system state, st+τ, as +well as the reward signal, rt. This iterate of data, [st,at,rt,st+τ], is then stored and used for +updating the control agent for the next time interval. +2.3. DManD Modeling Framework +Applications of deep RL often require repeating this cycle O(106+) times. Because deep +RL conventionally requires an online realization of the target system during training, the +practicality of training an RL agent for systems that are computationally or experimentally +expensive to realize online, e.g. a DNS of turbulent channel flow, is especially bottlenecked +by the expense of the environment itself [32]. +To circumvent this bottleneck, we employ a method denoted “Data-driven Manifold +Dynamics for RL” [42], or “DManD-RL” for short, with some modification. This framework +consists of two main learning objectives, which can be broken down into five steps, illustrated +in Fig. 2. The first objective is to obtain an efficient and accurate low-dimensional surrogate +model of the underlying dynamics of the turbulent DNS, which we refer to as the DManD +model. This objective is achieved via the first three steps outlined in Fig. 2: 1) collect data +tuples of the target system experiencing random control actions, 2) obtain a low-dimensional +representation of the environment’s dynamics, 3) model the dynamics of the environment +and its response to control inputs. +The second objective is to use this DManD model for RL training to quickly and effi- +ciently obtain an effective control agent. This objective is achieved via the remaining two +steps outlined in Fig. 2: 4) perform deep RL with the DManD model, and 5) deploy the +control agent to the original environment. In the following sections we discuss the details +for generating the DManD model in Sec. 2.3 and the method for training and deploying the +DManD-RL agent in Sec. 2.4. +As this framework is completely data-driven, the first step involves collecting sufficient +data to learn an accurate surrogate model. This model must capture the underlying flow +system, the response of the dynamics to control inputs, and the impact the control inputs +have on the objective. Generating this model requires that we have a large dataset that +includes the cycle of data described above ([st,at,rt,st+τ]). In RL training actions are chosen +by the policy, however, for training the model we do not necessarily have any policy to +generate this data. As such, we instead chose to randomly actuate the flow to generate the +6 + +Figure 2: Schematic of the DManD-RL framework. After step 3, ˜· is omitted for clarity. +original dataset used in training the DManD model. Details on the specifics of the dataset +are included in Sec. 3.1. +With this data, the second step of the DManD-RL framework involves finding a low- +dimensional representation of the state. +For many dissipative systems, there is either +proof or evidence that the long-time dynamics collapse onto a finite-dimensional invariant +manifold[49, 50, 51, 52, 53]. We can define a mapping to coordinates parameterizing this +manifold +ht = χ(st), +(11) +where ht ∈ Rdh is the manifold coordinate system and an inverse mapping back to the state +st = ˇχ(ht). +(12) +When the data lies on a finite-dimensional invariant manifold then the finite-dimensional +manifold coordinate representation ht contains the same information as the state st. Thus, +if we know χ and ˇχ we can simply use ht in place of st for training the RL agent, which +requires far fewer degrees of freedom. One subtlety that we gloss over here is that a dM- +dimensional manifold may require a set of overlapping local representations called charts if +one wants to represent the manifold with dM parameters [54, 55, 56]. However, a manifold +with a topological dimension dM can be embedded in R2dM [57, 58]. So, in the worst case, +as long as dh ≥ 2dM, a single global coordinate representation can be used, as we do here. +In this work we will approximate χ and ˇχ using an undercomplete autoencoder. This +consists of two NNs: an encoder (χ) that reduces the dimension and a decoder (ˇχ) that +7 + +1: Generate Training Data with Random Actuations +4: Train Agent with the Manifold Model +Encode +Environment +ht = x(St; QE) +(DNS) +DManD Model +Sts at, It St+t) +Random +h = g(h, at; 0g) + Ah +Actuations +Reward +rt = R(ht, at; Or) +2: Learn Manifold Coordinates +at +h +Encode +ht +Decode +ht = x(St: QE) +St = x(ht; Op) +Agent +at = π(ht; 0A) +S,seRd +h E Rdh<J +JWe find that our DManD-RL agent can consistently drive unseen turbulent initial condi- +tions in the original DNS to the laminar state, despite never having any direct observations +or interactions with the DNS. We also demonstrate that there exists a mapping between wall +observables and the manifold state, which allowed us to apply the DManD-RL agent with +equal effectiveness using only wall shear rate observations. +When investigating the mechanistic nature of the learned control strategy, we observed +multiple control strategies executed by the DManD-RL agent. One novel strategy the agent +appears to employ consists of manipulating the low-speed streak to a preferred location, +causing the breakdown of the streak, and in the wake of the break-down forming two low- +speed streaks in its place. These two low-speed streaks are unsustainable within the domain, +breaking the SSP and resulting in laminarization. +When comparing the ensemble behavior of our control agent to opposition control, we +find that the controller behaves anti-oppositionally at the commonly-used detection plane +location of y+ = 10. At a detection plane of y+ = 33, we find that the ensemble behavior +is opposition-like, however, the broadness of the control action distribution, as well as the +observation of the two-streak structure, leads us to conclude that the learned controller +behavior is much more complex and diverse than a simple opposition feedback rule. We also +compare our DManD-RL agent’s control performance to that of opposition control and we +find that our control agent out-performs opposition control by a notable margin (16% vs. +42% probability of remaining turbulent after 900 time units of control) despite our control +set-up and agent possessing much greater restrictions on it spatial and temporal control +authority (fixed time intervals of control and localized jets on only the lower wall) compared +to opposition control. +Acknowledgments +This work was supported by AFOSR FA9550-18-1-0174 and ONR N00014-18-1-2865 (Van- +nevar Bush Faculty Fellowship). +References +[1] J. +Jim´enez, +A. +Lozano-Dur´an, +Coherent +structures +in +wall-bounded +turbu- +lence, +Journal of Fluid Mechanics 842 (2018). doi:10.1007/978-3-319-20388-1_3. +arXiv:1710.07493. +[2] P. +S. +Virk, +Drag +reduction +fundamentals, +AIChE +Journal +21 +(1975) +625–656. +URL: +https://aiche.onlinelibrary.wiley.com/doi/abs/10. +1002/aic.690210402. +doi:https://doi.org/10.1002/aic.690210402. +arXiv:https://aiche.onlinelibrary.wiley.com/doi/pdf/10.1002/aic.690210402. +[3] M. 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Metzler, The characteristics of low-speed streaks in the near-wall +region of a turbulent boundary layer, Journal of Fluid Mechanics 129 (1983) 27–54. +doi:10.1017/S0022112083000634. +27 + diff --git a/n9FLT4oBgHgl3EQfgC--/content/tmp_files/load_file.txt b/n9FLT4oBgHgl3EQfgC--/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..535d87a35fb18d94d6d52e3423b863bf3942f709 --- /dev/null +++ b/n9FLT4oBgHgl3EQfgC--/content/tmp_files/load_file.txt @@ -0,0 +1,1533 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf,len=1532 +page_content='Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learning Alec J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Linota, Kevin Zenga, Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Graham1a aDepartment of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI 53706, USA Abstract The high dimensionality and complex dynamics of turbulent flows remain an obstacle to the discovery and implementation of control strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Deep reinforcement learning (RL) is a promising avenue for overcoming these obstacles, but requires a training phase in which the RL agent iteratively interacts with the flow environment to learn a control policy, which can be prohibitively expensive when the environment involves slow experiments or large-scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We overcome this challenge using a framework we call “DManD-RL” (data- driven manifold dynamics-RL), which generates a data-driven low-dimensional model of our system that we use for RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' With this approach, we seek to minimize drag in a direct numerical simulation (DNS) of a turbulent minimal flow unit of plane Couette flow at Re = 400 using two slot jets on one wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We obtain, from DNS data with O(105) degrees of freedom, a 25-dimensional DManD model of the dynamics by combining an autoencoder and neural ordinary differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Using this model as the environment, we train an RL control agent, yielding a 440-fold speedup over training on the DNS, with equivalent control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The agent learns a policy that laminarizes 84% of unseen DNS test trajectories within 900 time units, significantly outperforming classical opposition control (58%), despite the actuation authority being much more restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The agent often achieves laminarization through a counterintuitive strategy that drives the formation of two low-speed streaks, with a spanwise wavelength that is too small to be self-sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The agent demonstrates the same performance when we limit observations to wall shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Introduction Energy loss due to turbulent drag is ubiquitous in many industrial and commercial pro- cesses, ranging from air flowing over a plane wing, a ship in the ocean, or oil pumped through a pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In total, turbulent drag accounts for 25% of the energy used in industry and com- merce, resulting in 5% of all man-made CO2 emissions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Even small reductions in this drag can yield massive savings in energy, which has long motivated the search for better flow control strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Many control types for reducing turbulent drag exist, including, but not limited to, polymer/surfactant drag reduction [2, 3], riblets [4], wall oscillations [5], plasma actuators 1Corresponding author: mdgraham@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='edu Preprint submitted to Elsevier January 31, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='12098v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='flu-dyn] 28 Jan 2023 [6], and synthetic jets [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Due to the complexity of reducing drag, it has been most common to apply these control methods in an open-loop manner where the control policy at any given time is independent of the flow state [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' However, application of feedback control on a turbulent system could yield far better performance in controlling drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Unfortunately, the complexity of the problem has typically limited applications of feed- back control to methods based on heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' A well-studied heuristic method is opposition control [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Here the wall-normal velocity at the wall is set to have the opposite sign as the wall-normal velocity at some detection plane in the channel, a straightforward actuation in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This method has been applied in simulations [9, 10, 11, 12] and experiments (with some modifications) [13, 14], and extensions exist to use just wall observations [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' When heuristics are replaced with methods from optimal control theory, like model predic- tive control (MPC), the drag reduction far outperforms opposition control [17] while using the same actuation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' However, the real-time implementation of MPC on DNS still re- mains infeasible because it involves solving the DNS forward over a time horizon (preferably a long one) and then solving an adjoint backwards in time for every actuation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' A potential approach to overcome the high computational cost of real-time optimization of a control strategy is deep reinforcement learning (RL) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Deep RL gained significant traction when it was used to defeat the best professional players in GO [19], DOTA II [20], and Starcraft II [21], in addition to the best engines in GO, Chess, and Shogi [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In deep RL, a neural network (NN) control agent is trained through iterative interactions with the environment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' the system to be controlled) to maximize a scalar total reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' control objective) that includes present as well as discounted future reward values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Once trained, the control agent can be deployed in real time without the need for online optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In recent years, RL has been applied in fluids simulations to reduce the drag experienced in flow around a cylinder [23, 24, 25], to optimize jets on an airfoil [26], and to find efficient swimming strategies [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' RL has even been applied to experimental flow systems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Re- cently multi-agent deep RL has been explored for the control of pressure-driven turbulent channel flow [29, 30] in a problem formulation similar to opposition control [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In these works, an RL policy is trained to map local detection plane observables to a wall-normal velocity response at the walls to reduce drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Notably, the same RL policy is locally im- plemented at each wall grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We differentiate the control problem addressed from the previously mentioned works in that we limit the control authority to just two spatially local- ized jets on a single wall, with a zero-net flux constraint, as opposed to full spatial control of both walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We feel that this is much closer to experimental realizability than an approach with control authority everywhere on the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' A recent review of the application of deep RL applied to fluid mechanics problems is presented in Viquerat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In these active flow control problems, deep RL possesses the advantageous property of being completely data-driven, allowing it to discover novel and nontrivial control strategies in complex systems from just data alone without the need to analytically derive or hard- code system properties into the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' However, the training portion of RL is a major bottleneck, requiring a tremendous number of interactions with the target environment to find an approximately optimal policy [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Practically speaking, this can correspond to running an enormous number of high-resolution simulations or flow experiments, both of which may be prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In the present work, we apply RL to control a minimal flow unit (MFU) (the smallest 2 Figure 1: Schematic of the Couette flow domain with two slot jets on one wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' domain that sustains turbulence) [33] of plane Couette flow at Re = 400 using a pair of streamwise-aligned slot jets at one wall, with a no-net-flux constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Therefore there is only one degree of freedom for actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We select this system because the unactuated flow isolates the self-sustaining regeneration cycle of wall-bounded turbulence [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This case is well-studied for tasks such as reduced-order modeling [36, 37, 38, 39], finding invariant solutions [40, 41], and applying opposition control [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In order to overcome the high computational cost of RL training in this environment, in this work we replace the high-resolution simulation with an accurate low-dimensional sur- rogate model, aiming to dramatically reduce the time required to train the control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We showed in [42] that this data-driven model-based RL approach, which we refer to as “Data-Driven Manifold Dynamics” RL (DManD-RL), works well for controlling spatiotem- poral chaotic dynamics in the Kuramoto-Sivashinksy Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' For further discussion on the various types of model-based RL, we refer the reader to Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2 we introduce the control environment and the DManD-RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 3 we describe the data used for training the DManD model, the performance of the model, and the results of applying RL to the DManD model and to the DNS environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Finally, we conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 4 with a summary of the key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Navier-Stokes Equation with Slot Jets The environment we consider is a direct numerical simulation (DNS) of the Navier-Stokes Equations (NSE) ∂u ∂t + u · ∇u = −∇p + Re−1∇2u, ∇ · u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' (1) The velocities in the streamwise x ∈ [0, Lx], wall-normal y ∈ [−Ly/2, Ly/2], and spanwise z ∈ [0, Lz] directions are defined as u = [ux, uy, uz], and the pressure is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Here we have 3 nondimensionalized velocity by the speed U of the walls, length by the channel half-height (h = Ly/2), time with h/U and pressure with ρU 2, where ρ is the fluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The Reynolds number is Re = Uh/ν, where ν is the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The boundary conditions for this setup are periodic in x and z (u(0, y, z) = u(Lx, y, z), u(x, y, 0) = u(x, y, Lz)), no-slip boundary conditions at the walls (ux(x, ±Ly/2, z) = ±1, uz(x, ±Ly/2, z) = 0), no penetration at the top wall (uy(x, Ly/2, z) = 0), and finally, the actuation on the bottom wall (uy(x, −Ly/2, z) = fa(x, z)), as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The actuation on the bottom wall is in the form of two slot jets that are Gaussian in z and travel the length of the channel: uy(x, −Ly/2, z) = fa(x, z) = a(t)Vmax � exp � −(z − Lz/4)2 2σ2 � − exp � −(z − 3Lz/4)2 2σ2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' (2) We set σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='16 so that the jets act “locally”, and the velocity of the jet is dictated by a(t)Vmax, where a(t) ∈ [−1, 1] is the instantaneous actuation amplitude scaled by a maximum velocity Vmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' For perspective, the root-mean-squared wall-normal velocity at the channel centerline for turbulent unactuated flow is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We chose this small velocity to evaluate how the agent performs with limited control authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 1 we show a schematic illustrating this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The complexity of the flow increases as the Reynolds number and the domain size Lx and Lz increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Here we chose the same setup as Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' [34], Re = 400 and [Lx, Ly, Lz] = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='75π, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' These parameters isolate the “self-sustaining process” (SSP) that drives wall-bounded turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In the SSP, low-speed streaks that have been lifted from the wall become wavy, this waviness leads to the breakdown of the streaks, generating streamwise rolls, and, finally, these rolls lift low-speed fluid off the wall to regenerate streaks, completing the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' By working in this well-studied domain that is dominated by the SSP, we can better identify the means by which a control strategy can disrupt or suppress this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In this work, the control strategy is to minimize the turbulent drag averaged between both walls D = 1 2 � Lx 0 � Lz 0 � ∂ux ∂y ���� y=1 − 1 � + � ∂ux ∂y ���� y=−1 − 1 � dxdz, (3) subject to a quadratic penalty on actuation amplitude a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' (If the relation between the pressure drop and actuation velocity for pumping fluid into/out of the domain is linear, then this penalty is proportional to the power consumption of the actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=') Further details are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We report drag in this fashion because this quantity goes to 0 when the flow laminarizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We simulate the flow using a Fourier-Chebyshev pseudo-spectral code we implemented in Python [43], which is based on the Channelflow code developed by Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' ([44, 45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Linear terms are treated implicitly and the nonlinear term explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The specific time inte- gration schemes we use are the multistage SMRK2 scheme [46] for the first two timesteps after every actuation, and the multistep Adams-Bashforth Backward-Differentiation 3 scheme [47] until the next actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The multistep scheme is more computationally efficient, but, be- cause actuations change instantaneously, using previous steps with the incorrect boundary condition would lead to incorrect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' For all trials we evolve solutions forward using 4 ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='02 on a grid of [Nx, Ny, Nz] = [32, 35, 32] in x, y, and z from random divergence-free initial conditions that we evolve forward 100 time units so initial conditions are near the turbulent attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' While most of this approach is standard, here we include some details on the simulation procedure to highlight explicitly how we set the jet actuation boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' At each time step, the approach involves solving the expression Re−1d2ˆui+1 kx,kz dy2 − λˆui+1 kx,kz − ˆ∇ˆpi+1 kx,kz = − ˆRi kx,kz, (4) where i is the timestep and ˆ· = Fx,z(·) denotes the Fourier transform in x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The variable λ includes the timestep ∆t and the x and z components of the diffusive term and R encompasses all the remaining explicit terms (for a multistep method this includes ˆukx,kz multiple steps back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We refer the reader to [44] for a more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Upon taking the divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 4, and accounting for incompressibility, we isolate the problem down to 4 sets of one-dimensional Helmholtz equations (for conciseness we suppress indices kx, kz, and i): Re−1d2ˆux dy2 − λˆux − 2πikx Lx ˆp = − ˆRx ˆux(±1) = ±δkx,0δkz,0 (5) Re−1d2ˆuz dy2 − λˆuz − 2πikz Lz ˆp = − ˆRz ˆuz(±1) = 0 (6) Re−1d2ˆuy dy2 − λˆuy − dˆp dy = − ˆRy ˆuy(−1) = Fx,z(fa), ˆuy(1) = 0 (7) d2ˆp dy2 − 4π2 � k2 x L2 x + k2 z L2 z � ˆp = ˆ∇ · ˆR dˆuy dy (±1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' (8) These equations can be solved for every wavenumber pair kx and kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The challenge in solving these equations is due to the coupling in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 7 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The pressure is coupled to the wall-normal velocity because an explicit boundary condition is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Instead, from incompressibility, we know dˆuy/dy(±1) = 0, which we substitute for the pressure boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' To solve these coupled equations we use the influence matrix method and tau correction developed by Kleiser and Schumann [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Although we set the wall-normal boundary condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 7 by the slot jets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2, we note that it is simple to replace this boundary condition with any shape of actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Data-driven framework The objective in deep RL is to train an agent, commonly a neural network, to approximate the optimal control policy a = π∗(s), which given a state observation s, outputs the optimal control action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The optimal policy seeks to maximize the expected long time discounted cumulative reward, π∗ = arg max π E � ∞ � l=0 γl(rt+lτ) � , (9) 5 where 0 < γ < 1 is the discount factor, τ is the time between control actions, and rt is the reward, a scalar-valued control objective function decided by the user evaluated at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' As our objective in this work is to minimize the drag of our turbulent Couette system while simultaneously avoiding the use of superfluous control actions, we define the reward function as the following, rt = − � D(t) + c∥a(t)∥2� τ , (10) where c is a scalar and ⟨·⟩τ is the average from t to t + τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We note here that our actuation penalty is proportional to the power required for actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In deep RL π∗ is learned via repeated cyclic interactions between the agent and the environment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' the target system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' A typical cycle consists of the following: given a state observation of the system at time t, st, the control agent outputs its estimated best control response at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This control action is then applied to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The system is allowed to evolve for τ time units, and then the impact of the action is quantified by observing the resulting system state, st+τ, as well as the reward signal, rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This iterate of data, [st,at,rt,st+τ], is then stored and used for updating the control agent for the next time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' DManD Modeling Framework Applications of deep RL often require repeating this cycle O(106+) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Because deep RL conventionally requires an online realization of the target system during training, the practicality of training an RL agent for systems that are computationally or experimentally expensive to realize online, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' a DNS of turbulent channel flow, is especially bottlenecked by the expense of the environment itself [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' To circumvent this bottleneck, we employ a method denoted “Data-driven Manifold Dynamics for RL” [42], or “DManD-RL” for short, with some modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This framework consists of two main learning objectives, which can be broken down into five steps, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The first objective is to obtain an efficient and accurate low-dimensional surrogate model of the underlying dynamics of the turbulent DNS, which we refer to as the DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This objective is achieved via the first three steps outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2: 1) collect data tuples of the target system experiencing random control actions, 2) obtain a low-dimensional representation of the environment’s dynamics, 3) model the dynamics of the environment and its response to control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' The second objective is to use this DManD model for RL training to quickly and effi- ciently obtain an effective control agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This objective is achieved via the remaining two steps outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2: 4) perform deep RL with the DManD model, and 5) deploy the control agent to the original environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In the following sections we discuss the details for generating the DManD model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='3 and the method for training and deploying the DManD-RL agent in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' As this framework is completely data-driven, the first step involves collecting sufficient data to learn an accurate surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This model must capture the underlying flow system, the response of the dynamics to control inputs, and the impact the control inputs have on the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Generating this model requires that we have a large dataset that includes the cycle of data described above ([st,at,rt,st+τ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In RL training actions are chosen by the policy, however, for training the model we do not necessarily have any policy to generate this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' As such, we instead chose to randomly actuate the flow to generate the 6 Figure 2: Schematic of the DManD-RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' After step 3, ˜· is omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' original dataset used in training the DManD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Details on the specifics of the dataset are included in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' With this data, the second step of the DManD-RL framework involves finding a low- dimensional representation of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' For many dissipative systems, there is either proof or evidence that the long-time dynamics collapse onto a finite-dimensional invariant manifold[49, 50, 51, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' We can define a mapping to coordinates parameterizing this manifold ht = χ(st), (11) where ht ∈ Rdh is the manifold coordinate system and an inverse mapping back to the state st = ˇχ(ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' (12) When the data lies on a finite-dimensional invariant manifold then the finite-dimensional manifold coordinate representation ht contains the same information as the state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Thus, if we know χ and ˇχ we can simply use ht in place of st for training the RL agent, which requires far fewer degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' One subtlety that we gloss over here is that a dM- dimensional manifold may require a set of overlapping local representations called charts if one wants to represent the manifold with dM parameters [54, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' However, a manifold with a topological dimension dM can be embedded in R2dM [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' So, in the worst case, as long as dh ≥ 2dM, a single global coordinate representation can be used, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' In this work we will approximate χ and ˇχ using an undercomplete autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' This consists of two NNs: an encoder (χ) that reduces the dimension and a decoder (ˇχ) that 7 1: Generate Training Data with Random Actuations 4: Train Agent with the Manifold Model Encode Environment ht = x(St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' QE) (DNS) DManD Model Sts at, It St+t) Random h = g(h, at;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 0g) + Ah Actuations Reward rt = R(ht, at;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Or) 2: Learn Manifold Coordinates at h Encode ht Decode ht = x(St: QE) St = x(ht;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' Op) Agent at = π(ht;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FLT4oBgHgl3EQfgC--/content/2301.12098v1.pdf'} +page_content=' 0A) S,seRd h E Rdh< κ, which +overtakes the probe κ-soliton, shifts it per unit time back- +wards by distance +− +� ∞ +κ +2 +κ ln η + κ +η − κ · [s(η) − s(κ))]f(η)dη = += +� ∞ +κ +2 +κ ln +���� +κ + η +κ − η +���� · [s(κ) − s(η)]f(η)dη, +which can be obtained by multiplying shift − 2 +κ ln η+κ +η−κ +during a single collision by the number of such collisions +[s(η) − s(κ))]f(η) and integrating with respect to val- +ues of η > κ. Adding these corrections to the unmod- +ified value of velocity κ2, we obtain the following self- +consistent equation: +s(κ) = κ2 + 2 +κ +� ∞ +0 +ln +���� +κ + η +κ − η +���� [s(κ) − s(η)]f(η)dη. +(4) +Therefore, the distribution function for solitons with pa- +rameter κ is transferred along the x axis with velocity +s(κ), defined by integral equation (4), and the condition +of the spectrum conservation during the evolution of the +wave in accordance with the KdV equation can be writ- +ten in the form of conservation law +∂f(κ, x, t) +∂t ++ ∂[s(κ)f(κ, x, t)] +∂x += 0. +(5) +This equation, supplemented with integral equation (4), +is known as the KdV kinetic equation for a soliton gas; +this equation was obtained by El in [6] using a different +method. It should be noted that the inclusion in Eq. (4) +of pair collisions alone for a dense soliton gas is justi- +fied by the aforementioned addition of shifts for multiple +soliton collisions. +If the soliton gas is rarefied, i.e., +� +f(κ)dκ ≪ κ0, where +κ0 is the characteristic value of parameter κ in distribu- +tion f(κ), the correction term in formula (4) is small, and +we can substitute unmodified value of s(κ) ≈ κ2 into it: +s(κ) = κ2 + 2 +κ +� ∞ +0 +ln +���� +κ + η +κ − η +���� (κ2 − η2)f(η)dη. +(6) +This expression defines the function s(κ) by a closed ex- +pression in contrast to integral equation (4) for a dense +soliton gas. System (5), (6) was obtained by Zakharov in +[5] together with the approach to the kinetics of the soli- +ton gas formulated here. Clearly, such kinetic equations +can also be obtained for other completely integrable equa- +tions with the isospectral evolution of nonlinear waves. +III. +TWO-STREAM FLOWS OF SOLITON +GASES +To get an idea of the dynamics of soliton gases based on +kinetic equation (4), (5), we assume that the distribution +function has two very narrow peaks near κ1 and κ2, i.e., +this function can be written in form +f(κ, x, t) = f1(x, t)δ(κ − κ1) + f2(x, t)δ(κ − κ2). +(7) +This means that we consider the dynamics of interacting +gases over times, such that we can disregard the collisions +of solitons of the same species with very close velocities +and take into account only collisions between solitons of +different species. Then the substitution of distribution +function (7) into Eqs. (5) and (6) gives conservation laws +∂f1 +∂t + ∂(s1f1) +∂x += 0, +∂f2 +∂t + ∂(s2f2) +∂x += 0, +(8) + +3 +where velocities s1, s2 satisfy equations +s1 = κ2 +1 + α1f2(s1 − s2), +s2 = κ2 +2 + α2f1(s2 − s1), (9) +and we have introduced the following notation for conve- +nience: +α1 = 2 +κ1 +ln +���� +κ1 + κ2 +κ1 − κ2 +���� , +α2 = 2 +κ2 +ln +���� +κ1 + κ2 +κ1 − κ2 +���� . +(10) +Equations (9) make it possible to express the renormal- +ized velocities in terms of the densities of soliton gases: +s1 = κ2 +1(1 − α2f1) − κ2 +2α1f2 +1 − α1f2 − α2f1 +, +s2 = κ2 +2(1 − α1f2) − κ2 +1α2f1 +1 − α1f2 − α2f1 +. +(11) +If, however, we express densities f1,2 from Eqs. (9) in +terms of velocities s1,2, +f1 = +s2 − κ2 +2 +α2(s2 − s1), +f2 = +κ2 +1 − s1 +α1(s2 − s1), +(12) +and substitute these expressions into Eq. (8), this system +can be reduced to a remarkably simple diagonal form: +∂s1 +∂t + s2 +∂s1 +∂x = 0, +∂s2 +∂t + s1 +∂s2 +∂x = 0, +(13) +where renormalized velocities s1, s2 are Riemann invari- +ants of this system of equations of the hydrodynamic +type. +Formally, Eqs. (13) resemble the gas dynamic equa- +tions in the Riemann form, but physical properties of +the soliton gas differ significantly from the properties of +a conventional gas. +It should be noted above all that +although the soliton velocities are renormalized in the +overlap region of soliton clouds, such a change in veloc- +ity cannot be interpreted as the acceleration of solitons +under the action of pressure: after the clouds leave the +overlap region, their velocities restore their initial values. +Further, the gases flow freely through each other without +experiencing any dissipation processes. In this case, dy- +namic equations (8) have the form of conservation laws +and can have discontinuous solutions like in the theory +of viscous shock waves. Finally, we can assume in our +case that soliton gases have zero temperature so that a +transition through a discontinuity is not associated with +an increase of entropy; i.e., the Jouguet-Zempl´en theo- +rem (see, for example, [16]) is inapplicable, and the dis- +continuity can have any sign. This also means that the +existence of such formally discontinuous flows is not asso- +ciated with soliton pulse breaking followed by formation +of a narrow transition layer connecting both flows with +different parameters like in the theory of viscous shock +waves. +This situation is clarified with the help of the interre- +lation between Eqs. (13) and the Chaplygin gas theory. +Chaplygin [17] noted that the equation of state of a gas, +p = A − B +ρ +(14) +(p is the gas pressure, ρ is its density, and A, B are con- +stant parameters) can serve as a convenient approxima- +tion of small segments of the Poisson adiabat, where the +formulas of the theory are simplified significantly. (Chap- +lygin noticed also the connection between the gas dy- +namic equations for this case and the theory of minimal +surfaces.) However, we approach the Chaplygin gas the- +ory from a different point of view. +Already at the initial stage of development of the the- +ory of shock waves, Stokes and Kelvin discussed whether +gas dynamic equations +ρt + (ρu)x = 0, +ut + uux + c2 +ρ ρx = 0, +(15) +(c2 = dp/dρ, c being the velocity of sound) permit so- +lutions in the form of a traveling stationary wave. With +such a formulation of the problem, both density ρ and the +flow velocity u in the stationary solution are obviously +functions of only the coordinate ξ = x − V t, these quan- +tities are connected by a one-to-one dependence. Conse- +quently, such a solution must be a simple wave expressed +by the Poisson relation (see [16]) +ρ = ρ0[x − (c + u)t], +(16) +ρ0(x) being the density profile in such a wave propagating +with velocity c + u. In this case, the relation between +ρ and u is expressed by the condition of constancy of +the Riemann invariant: u − +� +cdρ/ρ = r− = const (see +[16]). A wave with a stationary profile can exist only if +condition c+u = const is satisfied. Substituting into this +expression relations +c = +� +dp +dρ, +u = +� ρ +0 +� +dp +dρ +dρ +ρ + const, +and differentiating with respect to ρ, we obtain the fol- +lowing equation for gas equation of state p = p(ρ) which +permits a stationary wave: +d2p +dρ2 + 2 +ρ +dp +dρ = 0. +This equation can be solved easily and gives exactly +above dependence (14). +Setting for simplicity B = 1, +we obtain c2 = dp/dρ = 1/ρ2, c = 1/ρ and the Riemann +invariants (see [16]) turn out to be +s1 = u + +� cdρ +ρ += u − 1 +ρ, +s2 = u − +� cdρ +ρ += u + 1 +ρ, +(17) +where u and ρ obey the Euler equations for the Chaplygin +gas: +ρt + (ρu)x = 0, +ut + uux + ρx +ρ3 = 0, +(18) + +4 +x +c−t +c+t +f20 +f1c + f2c +f10 +f1c +f2c +Figure 1: +Coordinate dependence of the densities of soli- +ton gases interacting after a collision of soliton clouds with +constant initial densities and velocities. +and can be expressed in terms of the Riemann invariants: +ρ = +2 +s2 − s1 +, +u = 1 +2(s2 + s1). +(19) +If we write Eqs. (18) in terms of Riemann invariants (17), +simple calculations lead to the dynamic equations for a +Chaplygin gas in the diagonal Riemann form coinciding +with Eqs. (13) which describe the dynamics of two inter- +acting soliton gases. +If we now seek the solution to Eq. (18) in the form of +a wave traveling with constant velocity V (ρ = ρ(ξ), u = +u(ξ), ξ = x − V t), we can easily see that these equations +are satisfied for any function ρ(ξ), if the Chaplygin gas +flow velocity can be expressed in terms of ρ(ξ) as +u(ξ) = V + +1 +ρ(ξ). +(20) +In this case, the Riemann invariants are given by +s1 = V, +s2 = V + +2 +ρ(ξ). +(21) +These expressions obviously give a solution to Eq. (13) +in the form of a simple wave: if we set, for example, +s1 = V = const, the second equation is transformed into +s2,t + V s2,x = 0 with general solution s2 = F(x − V t), +which coincides with (21), if we write function F(ξ) in +form F(ξ) = V +2/ρ(ξ). In particular, the function F(ξ) +can include a discontinuity propagating without change +of form in the overlap region of the two gases. +The relation between the equations for a Chaplygin gas +and the kinetic equation describing the dynamics of two +interacting soliton clouds makes it possible to construct +instructive examples of solutions to the kinetic equation. +IV. +COLLISION OF TWO SOLITON GASES +Equations (13) have an obvious degenerate solution, +in which s1 and s2 are constants depending neither on +x, nor t. +Despite the triviality of this solution, it has +a clear physical meaning: it can serve as a “plateau” +connecting two aforementioned solutions in the form of +simple waves. In particular, during the collision of mutu- +ally penetrating gases with constant densities f1, f2 and, +hence, with Riemann invariants s1, s2, a region of a two- +stream flow is formed, in which the velocities of solitons +of one species are renormalized because of their interac- +tion with solitons of the other species. Thus, the problem +of collision of two gases is reduced to the determination +of densities and velocities of solitons in the region of their +“mixing”, as well as the velocities of the edges of this re- +gion. Although this problem has already been analyzed +in [7, 18], we will briefly consider this problem because +the relevant results will be used in further analysis. We +assume that at the initial instant, there is a discontinuity +in the density distributions for solitons of two species: +� +f2(x, 0) = f20, +f1(x, 0) = 0, +x < 0, +f2(x, 0) = 0, +f1(x, 0) = f10, +x > 0, +(22) +here, we assume that κ2 > κ1, so that the faster soli- +ton cloud with density f20 overtakes at instant t = 0 +the slower cloud with density f10 at point x = 0 and +their mutual penetration begins. As usual, in the case +of initial discontinuity the solution must depend only on +self-similar variable ζ = x/t. Therefore, the initial dis- +continuity leads to the formation of a plateau between +two simple waves, each of which is a discontinuity with a +constant value of one of the Riemann invariants (the ex- +istence of such solutions is determined by the aforemen- +tioned properties of the Chaplygin gas). In the whole, +the solution is a sequence of three flows with constant +densities, which are separated by discontinuities: +f(x, t) = +� +� +� +f20, +x < c−t, +f1c + f2c, +c−t < x < c+t, +f10, +x > c+t. +(23) +It can clearly be seen from Fig. 1 that the leading edge +of the plateau moves with renormalized velocity s2c of +the fast gas, while the rear edge of the plateau moves +with renormalized velocity s1c of the slow gas because +the densities at these edges vanish: +c− = s1c, +c+ = s2c. +(24) +In the two-flow region c−t < x < c+t of the plateau, the +soliton velocities are renormalized by their interaction; in +accordance with relations (11) we have +s1c = κ2 +1(1 − α2f1c) − κ2 +2α1f2c +1 − α1f2c − α2f1c +, +s2c = κ2 +2(1 − α1f2c) − κ2 +1α2f1c +1 − α1f2c − α2f1c +. +(25) +The slow gas flows through the right discontinuity into +the plateau region; equating the expressions for its flux +on different sides of the discontinuity in the reference + +5 +frame in which it is at rest, we obtain f1c(s1c − c+) = +f10(κ2 +1−c+). Fast gas flows through the left discontinuity +into the plateau region, and analogous calculation gives +f20(κ2 +2 − c−) = f2c(s2c − c−). With account for relations +(24) and (25) the resulting equalities lead to equations +f1c = f10 +s2c − κ2 +1 +s2c − s1c += f10(1 − α1f2c), +f2c = f20 +κ2 +2 − s1c +s2c − s1c += f20(1 − α2f1c), +which gives expressions +f1c = f10(1 − α1f20) +1 − α1α2f10f20 +, +f2c = f20(1 − α2f10) +1 − α1α2f10f20 +, +(26) +for the densities of the soliton gases in the two-flow region +of the plateau. Clearly, these expressions have sense only +when the following inequality holds: +α1f1c + α2f2c < 1, +(27) +when renormalized velocities (25) are positive, i.e., the +densities of soliton gases cannot be too high. Actually, +more stringent limitations on the soliton density are im- +posed by the condition that variation u2 − u2 of the fluc- +tuating wave field in a soliton gas must be positive (see +[19]). The relations derived above are in good agreement +with the results of numerical solution of the KdV equa- +tion with the initial data in the form of a large number +of solitons of two different species, which are located on +different sides of the coordinate origin (see [18]). +V. +FLOW OF TWO SOLITON GASES IN THE +FORM OF A SIMPLE WAVE +Expressions (21) give a more general form of the so- +lution for the flow of two soliton gases in the form of a +simple wave. +Let us suppose that the densities of the +soliton gases at infinity tend to constant values: +f1 → f10, +f2 → f20 +as +x → ±∞. +(28) +Then the constant value of Riemann invariant s1 is equal +to the wave velocity of the soliton gas (see (25)) +V = s1 = κ2 +1(1 − α2f10) − κ2 +2α1f20 +1 − α1f20 − α2f10 +, +(29) +which coincides with the renormalized velocity of the slow +gas, while formulas (12) give expressions for the densities +of soliton gases: +f1(ξ) = 1 +α2 +� +1 + 1 +2(V − κ2 +2)ρ(ξ) +� +, +f2(ξ) = +1 +2α1 +(κ2 +1 − V )ρ(ξ). +(30) +x +f1, f2 +f2 +f1 +−4 +−2 +0 +2 +4 +0.02 +0.04 +0.06 +0.08 +0.10 +(a) +x +s1, s2 +s1 +s2 +−4 +−2 +0 +2 +4 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +(b) +Figure 2: +(a) Distributions of densities of soliton gases and +(b) of their renormalized velocities as functions of the coor- +dinate for the wave profile defined by formula (34). Soliton +gas parameters: κ1 = 1, κ2 = 1.8, f10 = 0.05, f20 = 0.1. +The wave velocity coinciding with the flow velocity of slow +solitons is V = s1 = 0.174, while the flow velocity for the +fast soliton gas at infinity is s20 = 3.469, which is higher than +nonrenormalized velocity κ2 +2 = 3.24. +It can easily be seen that these densities are connected +by relation +1 − α1f2 − α2f1 = 1 +2(κ2 +2 − κ2 +1)ρ, +(31) +which implies, in particular, that +ρ0 = 2(1 − α1f20 − α2f10) +κ2 +2 − κ2 +1 +. +(32) +Density distributions (30) contain arbitrary function ρ(ξ) +and the distribution of the renormalized velocity of the +fast soliton gas can also be expressed in terms of this +function in accordance with the second expression in (21). +At infinity, s2 tends to the value (see expressions (25)) +s20 = κ2 +2(1 − α1f20) − κ2 +1α2f10 +1 − α1f20 − α2f10 +. +(33) +The resulting solution has physical sense for such val- +ues of parameters κ1, κ2, f10, f20, that the distribution +of densities f1, f2 of the soliton gases, as well as their +renormalized velocities s1, s2 are positive everywhere. + +6 +Figure 2 illustrates the flow for two soliton gases for +function ρ(ξ) given by +ρ(ξ) = ρ0 + +a +cosh ξ . +(34) +In this case, the profile of density f1 of the slow gas moves +together with this component with velocity V = s1. The +fast gas flows through the slow gas with local velocity s2, +which decreases in the region of the “well” in distribution +f1, this leads to an increase in density f2 in this region. +VI. +GENERAL SOLUTION FOR THE FLOW OF +TWO SOLITON GASES +The hodograph method, which is well known in gas +dynamics (see, for example, [16]) makes it possible to +easily obtain the general solution to Eqs. (13), in the +case when both velocities s1 and s2 vary in space and +time. +Performing standard hodograph transformation +t = t(s1, s2), x = x(s1, s2), we reduce these equations +to the form +∂x +∂s1 +− s1 +∂t +∂s1 += 0, +∂x +∂s2 +− s2 +∂t +∂s2 += 0. +(35) +Eliminating x, we obtain +∂2t +∂s1∂s2 += 1 +s1 +∂2x +∂s1∂s2 += 1 +s1 +∂ +∂s1 +� +s2 +∂t +∂s2 +� += s2 +s1 +∂2t +∂s1∂s2 +. +Since s1 ̸= s2, we arrive at equation +∂2t +∂s1∂s2 += 0, +(36) +the general solution to which can be written in terms of +two arbitrary functions (it is convenient to denote them +as F ′′(s1) and G′′(s2)), in form +t(s1, s2) = F ′′(s1) + G′′(s2). +(37) +Equations (35) readily give +x(s1, s2) = s1F ′′(s1) + s2G′′(s2) − F ′(s1) − G′(s2). (38) +This solution can be transform to +x − s1t = (s2 − s1)G′′(s2) − F ′(s1) − G′(s2), +x − s2t = −(s2 − s1)F ′′(s1) − F ′(s1) − G′(s2). +(39) +In this form, the solution has been obtained in [20] by a +different method. +For completeness of analysis, we consider other repre- +sentations of the general solution, which may turn out to +be more convenient for solving specific problems. If we +turn to the hodograph method in the Tsarev form [21], +the general solution can be written in the form +x − s2t = ∂W +∂s1 +, +x − s1t = ∂W +∂s2 +, +(40) +where function W = W(s1, s2) must satisfy the Euler- +Poisson equation: +∂2W +∂s1∂s2 ++ +1 +s1 − s2 +�∂W +∂s1 +− ∂W +∂s2 +� += 0. +(41) +It can easily be verified that the solutions to this equation +can be expressed in terms of arbitrary functions F(s1) +and G(s2) as follows: +W(s1, s2) = (s1 −s2)[F ′(s1)−G′(s2)]−2[F(s1)+G(s2)]. +(42) +Substituting this expression into (40), we obtain the so- +lution in form (39). Functions F(s1) and G(s2) should +be determined from the initial conditions to the problem. +As was shown by Riemann (see, for example, [22]), in- +stead of using this general solution expressed in terms +of arbitrary functions, it is often convenient to employ +the method in which the solution to the problem is ex- +pressed directly in terns of initial conditions with the +help of the so-called Riemann function. In our case, the +Riemann function can be obtained from the general ex- +pression for the polytropic gas dynamics with adiabatic +exponent γ = −1. The hypergeometric function appear- +ing in this expression reduces to F(−1, 2; 1; z) = 1 − 2z +and the Riemann function for Eqs. (13) takes an espe- +cially simple form: +R(r1, r2; s1, s2) = (r1 + r2)(s1 + s2) − 2(r1r2 − s1s2) +(r1 − r2)2 +, +(43) +where (r1, r2) are the current coordinates on the hodo- +graph plane and (s1, s2) are the coordinates of point P, +at which the value of function W = W(P) = W(s1, s2) is +sought. We will not write here rather cumbersome gen- +eral expressions that do not add much for understanding +the dynamics of interacting soliton gases; instead of this, +we consider the solution to one of typical gas dynamic +problems, viz., the problem of expansion of a gas layer. +The solution can easily be obtained in elementary form, +and its difference from the solutions to the same problem +for other physical systems demonstrates well the substan- +tial difference between the conventional gas dynamics and +the dynamics of soliton gases. +VII. +EXPANSION OF A LAYER FROM A +MIXTURE OF TWO SOLITON GASES +One of typical problems in which a domain of the gen- +eral solution appears is the problem of expansion of a +gas layer. This problem was considered in [23] for a clas- +sical monatomic gas; in [24] for the Bose-Einstein con- +densate; in [25] for a two-temperature plasma, and in +[26–28] for an ultrarelativistic gas. In all cases, rarefac- +tion waves propagated from the edges to the bulk of the +layer, and after their collision, the general solution do- +main bordering the rarefaction waves at the edges was +formed. However, in the case of a soliton gas, there is + +7 +x +− l +2 + s2t +− l +2 + κ2 +1t +l +2 + s1t +l +2 + κ2 +2t +f1, κ2 +1 +f2, κ2 +2 +f10, s1 +f20, s2 +Figure 3: +Coordinate dependencies of the densities of two +soliton gas clouds during the evolution of the initial layer of +two gases before the instant of their separation. Bold dashed +line shows the density of the slow gas, while solid line corre- +sponds to the density of the fast gas. +no solution in the form of rarefaction waves, and instead +of this, discontinuous solutions separated by a plateau +with constant values of the Riemann invariants appears. +The solution constructed in this way makes it possible to +draw certain general conclusions about the evolution of +overlapping clouds of soliton gases. +If we assume that a single soliton gas with parameter +κ1 at the initial moment of time has a density distribution +in the form of a plateau with value f10 for −l/2 ≤ x ≤ +l/2, its motion at subsequent instants is obvious: this +distribution is transferred along the x axis with velocity +κ2 +1 without change of its shape. If, however, there is a +mixture of soliton gases with parameters κ1 and κ2, (κ1 < +κ2) and densities f10 and f20 at the initial instant in +domain −l/2 ≤ x ≤ l/2, then the evolution of these gases +is more complex: in the course of separation of these +gases into two clouds moving with velocities κ2 +1 and κ2 +2, a +two-flow plateau appears. Our aim is the description of +the process of spatial separation of soliton clouds and the +determination of their parameters after the separation. +On the plateau formed during the separation, the gases +obviously move with renormalized velocities s1, s2, pre- +serving their initial densities f10, f20. +In front of this +layer, there appears a fast gas layer with density f2, +while behind this layer, a slow gas layer with density +f1 is formed. The left edge of the overlap region moves +with velocity s2, while its right edge moves with velocity +s1. Consequently, the overlap region exists during time +T = +l +s2 − s1 +(44) +and the gases are separated at t > T. It can easily be +seen from Fig. 3, that the lengths of the clouds of slow +and fast gases at the instant of separation are given by +l1 = l · s2 − κ2 +1 +s2 − s1 +, +l2 = l · κ2 +2 − s1 +s2 − s1 +, +(45) +respectively, both of them being smaller than the initial +length l, because s1 < κ2 +1 and s2 > κ2 +2. From the conser- +vation of the number of solitons in each component, we +find amplitudes +f1 = f10 · s2 − s1 +s2 − κ2 +1 +, +f2 = f20 · s2 − s1 +κ2 +2 − s2 +. +(46) +At the instant of separation of the clouds, the centers of +mass of the slow soliton cloud and the fast soliton cloud +are, respectively, at points +x1(T) = l +2 · s1 + κ2 +1 +s2 − s1 +, +x2(T) = l +2 · s2 + κ2 +2 +s2 − s1 +. +(47) +If solitons did not interact with one another, these clouds +would move with nonrenormalized velocities and would +be at points x10(T) = κ2 +1l/(s2 − s1), x20(T) = κ2 +2l/(s2 − +s1) at this instant; i.e., because of the interaction, the +clouds are shifted through distances +∆x1 = x1(T) − x10(T) = l +2 · s1 − κ2 +1 +s2 − s1 +< 0, +∆x2 = x2(T) − x20(T) = l +2 · s2 − κ2 +2 +s2 − s1 +> 0. +(48) +Therefore, because of their interaction, the clouds are +narrowed, the soliton densities in both clouds increase, +and the slow cloud is shifted backwards, while the fast +cloud is shifted in the forward direction. A transition to +these values of the shifts are illustrated in Fig. 4, which +is obtained from the numerical solution for the Chaply- +gin gas. +At the initial instant, the centers of mass of +both gases were at the origin of coordinates, and in the +course of separation of the clouds, the center of mass of +the fast gas moved more rapidly than it would do with +its nonrenormalized velocity, while the center of mass of +the slow gas would lag behind the analogous motion with +its nonrenormalized velocity. After the instant of sepa- +ration, the centers of mass of both gases continue their +motion with nonrenormalized velocities, acquiring addi- +tional shifts due to the interaction with solitons of other +species. +Good agreement between the values of shifts +(48) predicted by the theory and the result of numerical +solution means that our idea of the formation of simple +waves with discontinuities from the initial discontinuities +corresponds to the actual dynamics of interacting soliton +clouds in accordance with the kinetic equation. +VIII. +CONCLUSION +In this study, it is shown that a fundamental property +of the dynamics of two interacting soliton gases is the ex- +istence of simple waves propagating without a change in +form. In particular, such simple waves can have discon- +tinuities; therefore, in problems of the “dam breaking”, +instead of rarefaction waves in the dynamic of conven- +tional gases, simple waves appear with discontinuities, +which are connected (instead of the general solution in +which both Riemann invariants change) by a plateau re- +gion with constant Riemann invariants. This property of + +8 +0 +0.5 +1 +1.5 +2 +2.5 +0 +0.2 +0.4 +0.6 +0.8 +t +|∆x| +Figure 4: +Shifts of the centers of mass of clouds from their +positions in the absence of interaction, which are obtained by +numerical solution of equations for the Chaplygin gas for the +initial values of parameters f10 = 0.05, f20 = 0.1, κ1 = 1.3, +κ2 = 2.5, l = 10. At instant T = 1.6 these parameters assume +the theoretical values given by formulas (48). +the dynamics of soliton gases differ drastically from the +properties of gas dynamics of conventional gases. At the +same time, based on the aforementioned pattern, simple +analytic solutions to typical problems, which are con- +firmed by exact numerical solutions of the kinetic equa- +tion reduced to the equivalent form of equations for the +dynamics of the Chaplygin gas, can be constructed. The +equations constructed in this way lead to an intuitively +clear pattern of the behavior of soliton gas clouds: as +a result of the interaction, the cloud of fast solitons is +shifted in the forward direction, while the cloud of slow +solitons is shifted backwards, both clouds becoming nar- +rower and soliton densities increase in both of them. The +resulting expressions make it possible to estimate these +effects in current experimental investigations of soliton +gases in waves on the water surface, in nonlinear optics, +and in the physics of cold atoms (see, for example, [12– +14]). +The authors are grateful to G. A. El for fruitful discus- +sions. 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Phys. 129, 607 +(2019). + diff --git a/otE3T4oBgHgl3EQfLQmx/content/tmp_files/load_file.txt b/otE3T4oBgHgl3EQfLQmx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..904782bde60552c007b44f05be69a0bd0d7ef7f4 --- /dev/null +++ b/otE3T4oBgHgl3EQfLQmx/content/tmp_files/load_file.txt @@ -0,0 +1,460 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf,len=459 +page_content='Dynamics of Interaction of Two Soliton Clouds∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Kamchatnov1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Shaykin2 1Institute of Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Russian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Troitsk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 108840,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Russia 2Moscow Institute of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Institutsky lane 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Dolgoprudny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Moscow region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 141700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Russia On the basis of relationship between the kinetic equation for two soliton clouds in the theory of the Korteweg-de Vries equation and equations of the Chaplygin gas dynamics it is shown that the existence of waves propagating without a change in their form is a fundamental property of the nonlinear dynamics of soliton gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The solutions of several typical problems in the soliton gas dynamics are considered and characteristic features of such dynamics, which make it possible to estimate the effects of interaction of soliton gases, are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' INTRODUCTION It is well known that the term “solitons” has been introduced [1] analogously to the names of elementary particles (electron, proton, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=') in view of their elastic interaction with one another for an important class of nonlinear wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Namely, two solitary waves spaced by a large distance prior to their “collision”, re- turn to their initial form after their passage through the interaction stage without forming any additional waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Nevertheless, the event of their interaction is not quite traceless: as a result of this interaction, the trajectories of solitons acquire additional shifts as compared to their initial trajectories [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' For example, in the case of the Korteweg-de Vries (KdV) equation, which is a univer- sal wave equation for describing waves with account for weak nonlinearity and dispersion effects and which can be written in terms of standard dimensionless variables in form ut + 6uux + uxxx = 0, (1) the soliton solution is given by the expression u(x, t) = κ2 2 · 1 cosh2[κ(x − κ2t − x0)/2], (2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', velocity s = κ2 of a soliton is proportional to its amplitude a = κ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If the wave excitation at the initial instant can be represented to a high degree of accuracy in the form of two soliton pulses(2) separated by a large distance (the faster soliton with parameter κ = κ2 has initial coordinate x02 on the left of coordinate x01 of the slower soliton with parameter κ1 (κ1 < κ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' their initial trajectories x = κ2 1t + x01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' x = κ2 2t + x02 acquire after the collision the shifts x = κ2 1t + x01 + ∆x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' x = κ2 2t + x02 + ∆x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' where ∆x1 = − 2 κ1 ln κ2 + κ1 κ2 − κ1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' ∆x2 = 2 κ2 ln κ2 + κ1 κ2 − κ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (3) Therefore, the fast soliton with a large amplitude is shifted in the forward direction, while the slow and lower ∗JETP, 135, 768 (2022) soliton is shifted in the backward direction, the shift of the lower soliton being larger (in absolute value) than the shift of the higher soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Remarkably, for an important class of integrable equations, solutions can be obtained with any number of solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In this case, as a result of simultaneous collision of three or more solitons, the to- tal shift of each soliton is equal to the sum of shifts of form (3) for pairwise collisions (see, for example, [3, 4]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', multiple simultaneous collisions of solitons in some spatial region do not differ in this sense from sequential spatially separated pair collisions of solitons with one an- other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Let us now suppose that in our wave system, a very large number of solitons, each is characterized by its parameter κ, are excited in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In this case, we can speak of a soliton gas and use the concepts of gas kinetics for its description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If we denote by f(κ, x, t)dxdκ the number of solitons with coordinates from interval (x, x + dx), which have parameter κ from interval (κ, κ + dκ) at instant t, the evolution of such a gas is characterized by the time dependence of distri- bution function f(κ, x, t) in the coordinate and parame- ter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The problem of obtaining the kinetic equation for such evolution was formulated by Zakharov in [5] and was solved by him for a rarefied gas of solitons when their mu- tual collisions change velocities s = κ2 very little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Later, El, who investigated the special limit of Whitham mod- ulation equations for the infinite-phase solution of the KdV equation, obtained in [6] a generalization of the Za- kharov kinetic equation to a dense soliton gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' A simpler derivation of this equation based on the self-consistent determination of the mean velocity of a soliton moving through the gas was given in [7] where the simplest solu- tion to these equations, which described the collision of two soliton gas clouds with close parameters κ was also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The formal solutions for any number of such clouds were analyzed in [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At present, the theory of kinetic equations for solitons has become a part of ac- tively developing “generalized hydrodynamics”, which is applicable to any integrable models of systems of many interacting particles (see, for example, reviews [10, 11] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' However, despite such a remark- able formal progress, the physical explanation of the be- havior of soliton gases appears as insufficient since the number of solved problems that give an idea of char- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='04362v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='PS] 11 Jan 2023 2 acteristic features of the dynamics of interacting soliton gases is very scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At the same time, experimental in- vestigations of soliton gas dynamics in various physical systems including waves in water and cold atoms have been launched (see, for example, [12–14]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' this requires the development of well-elaborated theory of the soliton gas dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In this article, we give several examples of such typical dynamics, which clarify some essential fea- tures of the soliton gas dynamics, distinguishing it from the dynamics of conventional gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' KINETIC EQUATION FOR A SOLITON GAS Let us first consider for completeness the brief deriva- tion of the KdV kinetic equation for a soliton gas follow- ing the method proposed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' It should be noted above all that in view of the integrability of the KdV equation discovered in [15], the evolution of wave u(x, t) occurs so that the spectrum of the eigenvalue problem for the time- independent Schr¨odinger equation ψxx = −(u(x, t)+λ)ψ, which is associated with the KdV equation, is indepen- dent of time t and each soliton corresponds to a certain value λ < 0 of discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' More convenient pa- rameter κ, which has been used in the soliton solution in form (2), is connected with λ by relation κ = √ −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Consequently, during the evolution of the wave, both the spectrum and the set of κ-values for the motion of soli- tons accompanied with their collisions remain unchanged in accordance with the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If solitons did not experience shifts (3) during their collisions, their velocities could be expressed by formula s = κ2 for all solitons with parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' However, col- lisions modify this velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In each collision of a soliton with slower solitons characterized by parameter η < κ, the “probe” κ-soliton advances in the forward direction through additional distance 2 κ ln κ+η κ−η, and the number of such collisions per unit time is equal to relative velocity s(κ) − s(η), multiplied by the density of η-solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Con- sequently, owing to such collisions, the κ-soliton acquires additional velocity � κ 0 2 κ ln κ + η κ − η · [s(κ) − s(η)]f(η)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Analogously, each soliton with parameter η > κ, which overtakes the probe κ-soliton, shifts it per unit time back- wards by distance − � ∞ κ 2 κ ln η + κ η − κ · [s(η) − s(κ))]f(η)dη = = � ∞ κ 2 κ ln ���� κ + η κ − η ���� · [s(κ) − s(η)]f(η)dη, which can be obtained by multiplying shift − 2 κ ln η+κ η−κ during a single collision by the number of such collisions [s(η) − s(κ))]f(η) and integrating with respect to val- ues of η > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Adding these corrections to the unmod- ified value of velocity κ2, we obtain the following self- consistent equation: s(κ) = κ2 + 2 κ � ∞ 0 ln ���� κ + η κ − η ���� [s(κ) − s(η)]f(η)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (4) Therefore, the distribution function for solitons with pa- rameter κ is transferred along the x axis with velocity s(κ), defined by integral equation (4), and the condition of the spectrum conservation during the evolution of the wave in accordance with the KdV equation can be writ- ten in the form of conservation law ∂f(κ, x, t) ∂t + ∂[s(κ)f(κ, x, t)] ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (5) This equation, supplemented with integral equation (4), is known as the KdV kinetic equation for a soliton gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' this equation was obtained by El in [6] using a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' It should be noted that the inclusion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (4) of pair collisions alone for a dense soliton gas is justi- fied by the aforementioned addition of shifts for multiple soliton collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If the soliton gas is rarefied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', � f(κ)dκ ≪ κ0, where κ0 is the characteristic value of parameter κ in distribu- tion f(κ), the correction term in formula (4) is small, and we can substitute unmodified value of s(κ) ≈ κ2 into it: s(κ) = κ2 + 2 κ � ∞ 0 ln ���� κ + η κ − η ���� (κ2 − η2)f(η)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (6) This expression defines the function s(κ) by a closed ex- pression in contrast to integral equation (4) for a dense soliton gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' System (5), (6) was obtained by Zakharov in [5] together with the approach to the kinetics of the soli- ton gas formulated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Clearly, such kinetic equations can also be obtained for other completely integrable equa- tions with the isospectral evolution of nonlinear waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' TWO-STREAM FLOWS OF SOLITON GASES To get an idea of the dynamics of soliton gases based on kinetic equation (4), (5), we assume that the distribution function has two very narrow peaks near κ1 and κ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', this function can be written in form f(κ, x, t) = f1(x, t)δ(κ − κ1) + f2(x, t)δ(κ − κ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (7) This means that we consider the dynamics of interacting gases over times, such that we can disregard the collisions of solitons of the same species with very close velocities and take into account only collisions between solitons of different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Then the substitution of distribution function (7) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (5) and (6) gives conservation laws ∂f1 ∂t + ∂(s1f1) ∂x = 0, ∂f2 ∂t + ∂(s2f2) ∂x = 0, (8) 3 where velocities s1, s2 satisfy equations s1 = κ2 1 + α1f2(s1 − s2), s2 = κ2 2 + α2f1(s2 − s1), (9) and we have introduced the following notation for conve- nience: α1 = 2 κ1 ln ���� κ1 + κ2 κ1 − κ2 ���� , α2 = 2 κ2 ln ���� κ1 + κ2 κ1 − κ2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (10) Equations (9) make it possible to express the renormal- ized velocities in terms of the densities of soliton gases: s1 = κ2 1(1 − α2f1) − κ2 2α1f2 1 − α1f2 − α2f1 , s2 = κ2 2(1 − α1f2) − κ2 1α2f1 1 − α1f2 − α2f1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (11) If, however, we express densities f1,2 from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (9) in terms of velocities s1,2, f1 = s2 − κ2 2 α2(s2 − s1), f2 = κ2 1 − s1 α1(s2 − s1), (12) and substitute these expressions into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (8), this system can be reduced to a remarkably simple diagonal form: ∂s1 ∂t + s2 ∂s1 ∂x = 0, ∂s2 ∂t + s1 ∂s2 ∂x = 0, (13) where renormalized velocities s1, s2 are Riemann invari- ants of this system of equations of the hydrodynamic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Formally, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13) resemble the gas dynamic equa- tions in the Riemann form, but physical properties of the soliton gas differ significantly from the properties of a conventional gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' It should be noted above all that although the soliton velocities are renormalized in the overlap region of soliton clouds, such a change in veloc- ity cannot be interpreted as the acceleration of solitons under the action of pressure: after the clouds leave the overlap region, their velocities restore their initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Further, the gases flow freely through each other without experiencing any dissipation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In this case, dy- namic equations (8) have the form of conservation laws and can have discontinuous solutions like in the theory of viscous shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Finally, we can assume in our case that soliton gases have zero temperature so that a transition through a discontinuity is not associated with an increase of entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', the Jouguet-Zempl´en theo- rem (see, for example, [16]) is inapplicable, and the dis- continuity can have any sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' This also means that the existence of such formally discontinuous flows is not asso- ciated with soliton pulse breaking followed by formation of a narrow transition layer connecting both flows with different parameters like in the theory of viscous shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' This situation is clarified with the help of the interre- lation between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13) and the Chaplygin gas theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Chaplygin [17] noted that the equation of state of a gas, p = A − B ρ (14) (p is the gas pressure, ρ is its density, and A, B are con- stant parameters) can serve as a convenient approxima- tion of small segments of the Poisson adiabat, where the formulas of the theory are simplified significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (Chap- lygin noticed also the connection between the gas dy- namic equations for this case and the theory of minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=') However, we approach the Chaplygin gas the- ory from a different point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Already at the initial stage of development of the the- ory of shock waves, Stokes and Kelvin discussed whether gas dynamic equations ρt + (ρu)x = 0, ut + uux + c2 ρ ρx = 0, (15) (c2 = dp/dρ, c being the velocity of sound) permit so- lutions in the form of a traveling stationary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' With such a formulation of the problem, both density ρ and the flow velocity u in the stationary solution are obviously functions of only the coordinate ξ = x − V t, these quan- tities are connected by a one-to-one dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Conse- quently, such a solution must be a simple wave expressed by the Poisson relation (see [16]) ρ = ρ0[x − (c + u)t], (16) ρ0(x) being the density profile in such a wave propagating with velocity c + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In this case, the relation between ρ and u is expressed by the condition of constancy of the Riemann invariant: u − � cdρ/ρ = r− = const (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' A wave with a stationary profile can exist only if condition c+u = const is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Substituting into this expression relations c = � dp dρ, u = � ρ 0 � dp dρ dρ ρ + const, and differentiating with respect to ρ, we obtain the fol- lowing equation for gas equation of state p = p(ρ) which permits a stationary wave: d2p dρ2 + 2 ρ dp dρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' This equation can be solved easily and gives exactly above dependence (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Setting for simplicity B = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' we obtain c2 = dp/dρ = 1/ρ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' c = 1/ρ and the Riemann invariants (see [16]) turn out to be s1 = u + � cdρ ρ = u − 1 ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' s2 = u − � cdρ ρ = u + 1 ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (17) where u and ρ obey the Euler equations for the Chaplygin gas: ρt + (ρu)x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' ut + uux + ρx ρ3 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (18) 4 x c−t c+t f20 f1c + f2c f10 f1c f2c Figure 1: Coordinate dependence of the densities of soli- ton gases interacting after a collision of soliton clouds with constant initial densities and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' and can be expressed in terms of the Riemann invariants: ρ = 2 s2 − s1 , u = 1 2(s2 + s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (19) If we write Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (18) in terms of Riemann invariants (17), simple calculations lead to the dynamic equations for a Chaplygin gas in the diagonal Riemann form coinciding with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13) which describe the dynamics of two inter- acting soliton gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If we now seek the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (18) in the form of a wave traveling with constant velocity V (ρ = ρ(ξ), u = u(ξ), ξ = x − V t), we can easily see that these equations are satisfied for any function ρ(ξ), if the Chaplygin gas flow velocity can be expressed in terms of ρ(ξ) as u(ξ) = V + 1 ρ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (20) In this case, the Riemann invariants are given by s1 = V, s2 = V + 2 ρ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (21) These expressions obviously give a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13) in the form of a simple wave: if we set, for example, s1 = V = const, the second equation is transformed into s2,t + V s2,x = 0 with general solution s2 = F(x − V t), which coincides with (21), if we write function F(ξ) in form F(ξ) = V +2/ρ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In particular, the function F(ξ) can include a discontinuity propagating without change of form in the overlap region of the two gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The relation between the equations for a Chaplygin gas and the kinetic equation describing the dynamics of two interacting soliton clouds makes it possible to construct instructive examples of solutions to the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' COLLISION OF TWO SOLITON GASES Equations (13) have an obvious degenerate solution, in which s1 and s2 are constants depending neither on x, nor t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Despite the triviality of this solution, it has a clear physical meaning: it can serve as a “plateau” connecting two aforementioned solutions in the form of simple waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In particular, during the collision of mutu- ally penetrating gases with constant densities f1, f2 and, hence, with Riemann invariants s1, s2, a region of a two- stream flow is formed, in which the velocities of solitons of one species are renormalized because of their interac- tion with solitons of the other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Thus, the problem of collision of two gases is reduced to the determination of densities and velocities of solitons in the region of their “mixing”, as well as the velocities of the edges of this re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Although this problem has already been analyzed in [7, 18], we will briefly consider this problem because the relevant results will be used in further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' We assume that at the initial instant, there is a discontinuity in the density distributions for solitons of two species: � f2(x, 0) = f20, f1(x, 0) = 0, x < 0, f2(x, 0) = 0, f1(x, 0) = f10, x > 0, (22) here, we assume that κ2 > κ1, so that the faster soli- ton cloud with density f20 overtakes at instant t = 0 the slower cloud with density f10 at point x = 0 and their mutual penetration begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' As usual, in the case of initial discontinuity the solution must depend only on self-similar variable ζ = x/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Therefore, the initial dis- continuity leads to the formation of a plateau between two simple waves, each of which is a discontinuity with a constant value of one of the Riemann invariants (the ex- istence of such solutions is determined by the aforemen- tioned properties of the Chaplygin gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In the whole, the solution is a sequence of three flows with constant densities, which are separated by discontinuities: f(x, t) = � � � f20, x < c−t, f1c + f2c, c−t < x < c+t, f10, x > c+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (23) It can clearly be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 1 that the leading edge of the plateau moves with renormalized velocity s2c of the fast gas, while the rear edge of the plateau moves with renormalized velocity s1c of the slow gas because the densities at these edges vanish: c− = s1c, c+ = s2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (24) In the two-flow region c−t < x < c+t of the plateau, the soliton velocities are renormalized by their interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' in accordance with relations (11) we have s1c = κ2 1(1 − α2f1c) − κ2 2α1f2c 1 − α1f2c − α2f1c , s2c = κ2 2(1 − α1f2c) − κ2 1α2f1c 1 − α1f2c − α2f1c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (25) The slow gas flows through the right discontinuity into the plateau region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' equating the expressions for its flux on different sides of the discontinuity in the reference 5 frame in which it is at rest, we obtain f1c(s1c − c+) = f10(κ2 1−c+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Fast gas flows through the left discontinuity into the plateau region, and analogous calculation gives f20(κ2 2 − c−) = f2c(s2c − c−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' With account for relations (24) and (25) the resulting equalities lead to equations f1c = f10 s2c − κ2 1 s2c − s1c = f10(1 − α1f2c), f2c = f20 κ2 2 − s1c s2c − s1c = f20(1 − α2f1c), which gives expressions f1c = f10(1 − α1f20) 1 − α1α2f10f20 , f2c = f20(1 − α2f10) 1 − α1α2f10f20 , (26) for the densities of the soliton gases in the two-flow region of the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Clearly, these expressions have sense only when the following inequality holds: α1f1c + α2f2c < 1, (27) when renormalized velocities (25) are positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', the densities of soliton gases cannot be too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Actually, more stringent limitations on the soliton density are im- posed by the condition that variation u2 − u2 of the fluc- tuating wave field in a soliton gas must be positive (see [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The relations derived above are in good agreement with the results of numerical solution of the KdV equa- tion with the initial data in the form of a large number of solitons of two different species, which are located on different sides of the coordinate origin (see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' FLOW OF TWO SOLITON GASES IN THE FORM OF A SIMPLE WAVE Expressions (21) give a more general form of the so- lution for the flow of two soliton gases in the form of a simple wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Let us suppose that the densities of the soliton gases at infinity tend to constant values: f1 → f10, f2 → f20 as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (28) Then the constant value of Riemann invariant s1 is equal to the wave velocity of the soliton gas (see (25)) V = s1 = κ2 1(1 − α2f10) − κ2 2α1f20 1 − α1f20 − α2f10 , (29) which coincides with the renormalized velocity of the slow gas, while formulas (12) give expressions for the densities of soliton gases: f1(ξ) = 1 α2 � 1 + 1 2(V − κ2 2)ρ(ξ) � , f2(ξ) = 1 2α1 (κ2 1 − V )ρ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (30) x f1, f2 f2 f1 −4 −2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='10 (a) x s1, s2 s1 s2 −4 −2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 (b) Figure 2: (a) Distributions of densities of soliton gases and (b) of their renormalized velocities as functions of the coor- dinate for the wave profile defined by formula (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Soliton gas parameters: κ1 = 1, κ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='8, f10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='05, f20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The wave velocity coinciding with the flow velocity of slow solitons is V = s1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='174, while the flow velocity for the fast soliton gas at infinity is s20 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='469, which is higher than nonrenormalized velocity κ2 2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' It can easily be seen that these densities are connected by relation 1 − α1f2 − α2f1 = 1 2(κ2 2 − κ2 1)ρ, (31) which implies, in particular, that ρ0 = 2(1 − α1f20 − α2f10) κ2 2 − κ2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (32) Density distributions (30) contain arbitrary function ρ(ξ) and the distribution of the renormalized velocity of the fast soliton gas can also be expressed in terms of this function in accordance with the second expression in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At infinity, s2 tends to the value (see expressions (25)) s20 = κ2 2(1 − α1f20) − κ2 1α2f10 1 − α1f20 − α2f10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (33) The resulting solution has physical sense for such val- ues of parameters κ1, κ2, f10, f20, that the distribution of densities f1, f2 of the soliton gases, as well as their renormalized velocities s1, s2 are positive everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 6 Figure 2 illustrates the flow for two soliton gases for function ρ(ξ) given by ρ(ξ) = ρ0 + a cosh ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (34) In this case, the profile of density f1 of the slow gas moves together with this component with velocity V = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The fast gas flows through the slow gas with local velocity s2, which decreases in the region of the “well” in distribution f1, this leads to an increase in density f2 in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' GENERAL SOLUTION FOR THE FLOW OF TWO SOLITON GASES The hodograph method, which is well known in gas dynamics (see, for example, [16]) makes it possible to easily obtain the general solution to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13), in the case when both velocities s1 and s2 vary in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Performing standard hodograph transformation t = t(s1, s2), x = x(s1, s2), we reduce these equations to the form ∂x ∂s1 − s1 ∂t ∂s1 = 0, ∂x ∂s2 − s2 ∂t ∂s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (35) Eliminating x, we obtain ∂2t ∂s1∂s2 = 1 s1 ∂2x ∂s1∂s2 = 1 s1 ∂ ∂s1 � s2 ∂t ∂s2 � = s2 s1 ∂2t ∂s1∂s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Since s1 ̸= s2, we arrive at equation ∂2t ∂s1∂s2 = 0, (36) the general solution to which can be written in terms of two arbitrary functions (it is convenient to denote them as F ′′(s1) and G′′(s2)), in form t(s1, s2) = F ′′(s1) + G′′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (37) Equations (35) readily give x(s1, s2) = s1F ′′(s1) + s2G′′(s2) − F ′(s1) − G′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (38) This solution can be transform to x − s1t = (s2 − s1)G′′(s2) − F ′(s1) − G′(s2), x − s2t = −(s2 − s1)F ′′(s1) − F ′(s1) − G′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (39) In this form, the solution has been obtained in [20] by a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' For completeness of analysis, we consider other repre- sentations of the general solution, which may turn out to be more convenient for solving specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If we turn to the hodograph method in the Tsarev form [21], the general solution can be written in the form x − s2t = ∂W ∂s1 , x − s1t = ∂W ∂s2 , (40) where function W = W(s1, s2) must satisfy the Euler- Poisson equation: ∂2W ∂s1∂s2 + 1 s1 − s2 �∂W ∂s1 − ∂W ∂s2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (41) It can easily be verified that the solutions to this equation can be expressed in terms of arbitrary functions F(s1) and G(s2) as follows: W(s1, s2) = (s1 −s2)[F ′(s1)−G′(s2)]−2[F(s1)+G(s2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (42) Substituting this expression into (40), we obtain the so- lution in form (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Functions F(s1) and G(s2) should be determined from the initial conditions to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' As was shown by Riemann (see, for example, [22]), in- stead of using this general solution expressed in terms of arbitrary functions, it is often convenient to employ the method in which the solution to the problem is ex- pressed directly in terns of initial conditions with the help of the so-called Riemann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In our case, the Riemann function can be obtained from the general ex- pression for the polytropic gas dynamics with adiabatic exponent γ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The hypergeometric function appear- ing in this expression reduces to F(−1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' z) = 1 − 2z and the Riemann function for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (13) takes an espe- cially simple form: R(r1, r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' s1, s2) = (r1 + r2)(s1 + s2) − 2(r1r2 − s1s2) (r1 − r2)2 , (43) where (r1, r2) are the current coordinates on the hodo- graph plane and (s1, s2) are the coordinates of point P, at which the value of function W = W(P) = W(s1, s2) is sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' We will not write here rather cumbersome gen- eral expressions that do not add much for understanding the dynamics of interacting soliton gases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' instead of this, we consider the solution to one of typical gas dynamic problems, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', the problem of expansion of a gas layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The solution can easily be obtained in elementary form, and its difference from the solutions to the same problem for other physical systems demonstrates well the substan- tial difference between the conventional gas dynamics and the dynamics of soliton gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' EXPANSION OF A LAYER FROM A MIXTURE OF TWO SOLITON GASES One of typical problems in which a domain of the gen- eral solution appears is the problem of expansion of a gas layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' This problem was considered in [23] for a clas- sical monatomic gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' in [24] for the Bose-Einstein con- densate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' in [25] for a two-temperature plasma, and in [26–28] for an ultrarelativistic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In all cases, rarefac- tion waves propagated from the edges to the bulk of the layer, and after their collision, the general solution do- main bordering the rarefaction waves at the edges was formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' However, in the case of a soliton gas, there is 7 x − l 2 + s2t − l 2 + κ2 1t l 2 + s1t l 2 + κ2 2t f1, κ2 1 f2, κ2 2 f10, s1 f20, s2 Figure 3: Coordinate dependencies of the densities of two soliton gas clouds during the evolution of the initial layer of two gases before the instant of their separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Bold dashed line shows the density of the slow gas, while solid line corre- sponds to the density of the fast gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' no solution in the form of rarefaction waves, and instead of this, discontinuous solutions separated by a plateau with constant values of the Riemann invariants appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The solution constructed in this way makes it possible to draw certain general conclusions about the evolution of overlapping clouds of soliton gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If we assume that a single soliton gas with parameter κ1 at the initial moment of time has a density distribution in the form of a plateau with value f10 for −l/2 ≤ x ≤ l/2, its motion at subsequent instants is obvious: this distribution is transferred along the x axis with velocity κ2 1 without change of its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' If, however, there is a mixture of soliton gases with parameters κ1 and κ2, (κ1 < κ2) and densities f10 and f20 at the initial instant in domain −l/2 ≤ x ≤ l/2, then the evolution of these gases is more complex: in the course of separation of these gases into two clouds moving with velocities κ2 1 and κ2 2, a two-flow plateau appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Our aim is the description of the process of spatial separation of soliton clouds and the determination of their parameters after the separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' On the plateau formed during the separation, the gases obviously move with renormalized velocities s1, s2, pre- serving their initial densities f10, f20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In front of this layer, there appears a fast gas layer with density f2, while behind this layer, a slow gas layer with density f1 is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The left edge of the overlap region moves with velocity s2, while its right edge moves with velocity s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Consequently, the overlap region exists during time T = l s2 − s1 (44) and the gases are separated at t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' It can easily be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 3, that the lengths of the clouds of slow and fast gases at the instant of separation are given by l1 = l · s2 − κ2 1 s2 − s1 , l2 = l · κ2 2 − s1 s2 − s1 , (45) respectively, both of them being smaller than the initial length l, because s1 < κ2 1 and s2 > κ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' From the conser- vation of the number of solitons in each component, we find amplitudes f1 = f10 · s2 − s1 s2 − κ2 1 , f2 = f20 · s2 − s1 κ2 2 − s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (46) At the instant of separation of the clouds, the centers of mass of the slow soliton cloud and the fast soliton cloud are, respectively, at points x1(T) = l 2 · s1 + κ2 1 s2 − s1 , x2(T) = l 2 · s2 + κ2 2 s2 − s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (47) If solitons did not interact with one another, these clouds would move with nonrenormalized velocities and would be at points x10(T) = κ2 1l/(s2 − s1), x20(T) = κ2 2l/(s2 − s1) at this instant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=', because of the interaction, the clouds are shifted through distances ∆x1 = x1(T) − x10(T) = l 2 · s1 − κ2 1 s2 − s1 < 0, ∆x2 = x2(T) − x20(T) = l 2 · s2 − κ2 2 s2 − s1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' (48) Therefore, because of their interaction, the clouds are narrowed, the soliton densities in both clouds increase, and the slow cloud is shifted backwards, while the fast cloud is shifted in the forward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' A transition to these values of the shifts are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 4, which is obtained from the numerical solution for the Chaply- gin gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At the initial instant, the centers of mass of both gases were at the origin of coordinates, and in the course of separation of the clouds, the center of mass of the fast gas moved more rapidly than it would do with its nonrenormalized velocity, while the center of mass of the slow gas would lag behind the analogous motion with its nonrenormalized velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' After the instant of sepa- ration, the centers of mass of both gases continue their motion with nonrenormalized velocities, acquiring addi- tional shifts due to the interaction with solitons of other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Good agreement between the values of shifts (48) predicted by the theory and the result of numerical solution means that our idea of the formation of simple waves with discontinuities from the initial discontinuities corresponds to the actual dynamics of interacting soliton clouds in accordance with the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' CONCLUSION In this study, it is shown that a fundamental property of the dynamics of two interacting soliton gases is the ex- istence of simple waves propagating without a change in form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' In particular, such simple waves can have discon- tinuities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' therefore, in problems of the “dam breaking”, instead of rarefaction waves in the dynamic of conven- tional gases, simple waves appear with discontinuities, which are connected (instead of the general solution in which both Riemann invariants change) by a plateau re- gion with constant Riemann invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' This property of 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='8 t |∆x| Figure 4: Shifts of the centers of mass of clouds from their positions in the absence of interaction, which are obtained by numerical solution of equations for the Chaplygin gas for the initial values of parameters f10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='05, f20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='1, κ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='3, κ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='5, l = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At instant T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content='6 these parameters assume the theoretical values given by formulas (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' the dynamics of soliton gases differ drastically from the properties of gas dynamics of conventional gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' At the same time, based on the aforementioned pattern, simple analytic solutions to typical problems, which are con- firmed by exact numerical solutions of the kinetic equa- tion reduced to the equivalent form of equations for the dynamics of the Chaplygin gas, can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The equations constructed in this way lead to an intuitively clear pattern of the behavior of soliton gas clouds: as a result of the interaction, the cloud of fast solitons is shifted in the forward direction, while the cloud of slow solitons is shifted backwards, both clouds becoming nar- rower and soliton densities increase in both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The resulting expressions make it possible to estimate these effects in current experimental investigations of soliton gases in waves on the water surface, in nonlinear optics, and in the physics of cold atoms (see, for example, [12– 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' The authors are grateful to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' El for fruitful discus- sions.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Kamchatnov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} +page_content=' 129, 607 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE3T4oBgHgl3EQfLQmx/content/2301.04362v1.pdf'} diff --git a/pdE2T4oBgHgl3EQfKQak/content/tmp_files/2301.03701v1.pdf.txt b/pdE2T4oBgHgl3EQfKQak/content/tmp_files/2301.03701v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bce78954734d45fba9757048c6212ea45384cace --- /dev/null +++ b/pdE2T4oBgHgl3EQfKQak/content/tmp_files/2301.03701v1.pdf.txt @@ -0,0 +1,773 @@ +Medical Image Analysis (2023) +Contents lists available at ScienceDirect +Medical Image Analysis +journal homepage: www.elsevier.com/locate/media +MOC-AE: An Anatomically-Pathological-Based model for Clinical Decision Support +System of tumoural brain images +Guillermo Iglesiasa,b,∗, Edgar Talaveraa,1, Alberto D´ıaz- ´Alvareza, Miguel Garc´ıa-Remesalc +aDepartamento de Sistemas Inform´aticos, Escuela T´ecnica Superior de Ingenier´ıa de Sistemas Inform´aticos, Universidad Polit´ecnica de Madrid, Spain +bCentro Universitario de Tecnolog´ıa y Arte Digital, Spain +cBiomedical Informatics Group, Departamento de Inteligencia Artificial, Escuela T´ecnica Superior de Ingenieros Inform´aticos, Universidad Polit´ecnica de +Madrid, Spain +A R T I C L E I N F O +Article history: +2000 MSC: +68T45, +68U10, +94A08, +65D17 +Keywords: +content-based +image +re- +trieval, +deep +learning, +autoencoder, +feature +extraction, +clinical +decision +support system, comparative diagnostic, +magnetic resonance imaging +A B S T R A C T +The present work proposes a Multi-Output Classification Autoencoder (MOC-AE) al- +gorithm to extract features from brain tumour images. The proposed algorithm is able +to focus on both the normal features of the patient and the pathological features present +in the case, resulting in a compact and significant representation of each image. The ar- +chitecture of MOC-AE combines anatomical information from the patient’s scan using +an Autoencoder (AE) with information related to a specific pathology using a classi- +fication output with the same image descriptor. This combination of goals forces the +network to maintain a balance between anatomical and pathological features of the case +while maintaining the low cost of the labels being used. +The results obtained are compared with those of similar studies and the strengths and +limitations of each approach are discussed. The results demonstrate that the proposed +algorithm is capable of achieving state-of-the-art results in terms of both the anatomical +and tumor characteristics of the recommended cases. +© 2023 Elsevier B. V. All rights reserved. +1. Introduction +The techniques used in medicine to treat pathologies are be- +coming more advanced each day and are capable of performing +less invasive and more efficient treatments. However, before a +patient can apply proper treatment, a correct diagnosis of the +disease is necessary, which becomes an important task in this +process. +⋆This is an example for title footnote coding. +∗Corresponding author +e-mail: guillermo.iglesias.hernandez@alumnos.upm.es +(Guillermo Iglesias), e.talavera@upm.es (Edgar Talavera), +alberto.diaz@upm.es (Alberto D´ıaz- ´Alvarez), mgremesal@fi.upm.es +(Miguel Garc´ıa-Remesal) +Although medicine has radically evolved, diagnosis is still +mostly a human process and the expert must be able to thor- +oughly evaluate the patient’s evidence and avoid making mis- +takes during the process. A late or incorrect diagnosis can lead +to an increase in pathologies that, in cases such as cancer, can +be fatal and irreversible (Tan and Fielding, 2006). +Therefore, any improvement that a doctor can make during +the diagnostic process may be vital and necessary to greatly im- +prove treatment results, because early diagnosis improves treat- +ment results, as it improves the result of the procedure (Kimber- +Trojnar et al., 2021; Sullivan et al., 2021; Haq et al., 2021). +Preprint submitted to Medical Image Analysis +January 11, 2023 +arXiv:2301.03701v1 [eess.IV] 9 Jan 2023 + +EISEVIERN13-AI5 +MEDICAL +IMAGE +ANALYSIS2 +Guillermo Iglesias et al. / Medical Image Analysis (2023) +At this point, computer algorithms are specially relevant, since +they can be used as an additional tool for decision making, +which is known as Clinical Decision Support System (CDSS). +It is precisely in the current paradigm of health information sci- +ence where these CDSSs can be used to process in real time +large amounts of data. +In this sense, there are two approaches to process large +amounts of medical images, e.g: X-rays, computed tomogra- +phy scanners, magnetic resonance images, nuclear medicine +images. The first solution are Content-Based Image Retrieval +(CBIR) systems which retrieve the most similar images from +a database using the information of the query image itself, ex- +tracting the most relevant features of the input and recommend- +ing the most similar images stored. While other approximation +are the so-called Concept-Based Image Indexing systems, that +use meta-information of the images to extract its content and be +able of recommending similar samples from the database. +CBIR systems are focused on processing large amounts of +data, Artificial Intelligence (AI) algorithms arise as the best so- +lution to this problem. These frameworks are capable of obtain- +ing the most similar images from a dataset given a certain query. +Different solutions have been proposed during the last years +combining CBIR with Deep Learning (DL) techniques (Sir- +adjuddin et al., 2019; Kobayashi et al., 2021). These systems +make possible to use large amounts of information in an ordered +manner, taking advantage of the information available without +the necessity to manually use all information. +However, it is not widely considered that AI should replace +health professionals in tasks they normally perform, but instead +assist them in diagnosis and decision making. It should be taken +into account that the human professional will always make the +final decision, it should never be substituted by an AI. In the +same way that surgical robots need an operator to perform their +tasks, these tools based on AI should be considered diagnostic +support systems, instead of decision makers. Aspects such as +lack of explainability or precision (Ossa et al., 2022), or inabil- +ity to deal with outliers suggest that it is probably too early to +completely delegate critical tasks to an AI, at least in the fore- +seeable future (Krittanawong, 2018). +One of the techniques in which artificial intelligence has re- +cently achieved better results is in Computer Vision (CV) (Guo +et al., 2022). Here, medical imaging data can be used by these +algorithms to extract and process its information to help profes- +sionals perform certain tasks (Esteva et al., 2021; Ward et al., +2021). Thus, this relationship between CV and medical imag- +ing has led to the emergence of works that combine the latest +advances of artificial intelligence in image feature extraction +along with medical data (Ji et al., 2021; Karimi et al., 2021). +Autoencoders (AEs) (Rumelhart et al., 1985) are machine +learning models used in dimensionality reduction and feature +extraction processes. In this work, it is proposed to use a novel +AE variant, named as Multi-Output Classification Autoencoder +(MOC-AE), that factorizes brain tumoural images, to then use +the generated image descriptors to recommend cases with a +similar pathology. +The architecture presented focuses on improving the retrieval +accuracy of a standard AE without the need to use costly infor- +mation, such as tumour area segmentation. MOC-AE is able +to balance the normal anatomical features of each patient with +the tumour features in a single descriptor. Thus, the model is +able to recommend interesting cases taking into account rele- +vant medical information, in this case, the tumour area. +The proposed MOC-AE architecture has two main advan- +tages over previous similar CBIR models. First, the information +required to develop the model training is much less costly com- +pared to other approximations, due to the fact that the model +only uses binary labels to learn the pathologies features of the +images. This reduction in terms of cost does not imply worse +results in image recommendation; as is studied in Section 5 the +model results show improved performance in tumour similarity +and a better balance between normal and abnormal features of +each case. +The present work describes the process of training and evalu- +ation of MOC-AE to extract features of the tumoural brain and +its performance is demonstrated in the task of recommending +cases with similar pathologies. The results obtained are com- + +Guillermo Iglesias et al. / Medical Image Analysis (2023) +3 +pared with similar works and the strengths and drawbacks of +each approximation are discussed. +The source code of the project, along with trained models +and results are publicly available1. +2. Related works +Different works published in the last decade use machine +learning techniques to improve the diagnosis of doctors (Fa- +tima et al., 2017; Ralbovsky and Lednev, 2020; Haq et al., +2021; Quellec et al., 2010). Taking advantage of the latest AI +research and applying it to the medical field, different works +have obtained impressive results in tasks related to the medical +field (Gardezi et al., 2019). +Regarding medical imaging algorithms, CDSSs is a pro- +lific area where in recent years many articles have been pub- +lished (Musen et al., 2021; Rani et al., 2021; Rama Krishna and +Sirajuddin, 2022; Tuppad and Patil, 2022). These techniques +combine the potential AI can achieve in extracting the most im- +portant features of medical images along with support systems +that provide the doctor with the most relevant information in +each case (Jiang et al., 2016). +In this sense, CBIR arises as a solution where it is possible +to make use of all data stored in large databases that, in an- +other context, it would be impossible for a human to fully use. +There are different works in this area, such as Lehmann et al. +(2005) where the categorisation of medical images from differ- +ent sources is sought through CBIR using traditional algorithms +to generate image descriptors. The recommendation of a simi- +lar descriptor is made using the nearest neighbour algorithm, as +was done in other research based on descriptors CBIR, indepen- +dently of medical imaging (Siradjuddin et al., 2019; Shakarami +and Tarrah, 2020; Lehmann et al., 2005). +The work of +Tarjoman et al. (2013) proposes a sim- +ple feature extraction method using Support Vector Machine +(SVM) (Cortes and Vapnik, 1995) and the Grey Level Co- +occurrence Matrix as the main input of each image. A simi- +lar approximation was followed in Kumar et al. (2016) where +1https://purl.com/mocae_brats +SVM is used as the descriptor generator mechanism, in this case +using different features of the liver images. Finally, using the +weighted nearest neighbour (Cover and Hart, 1967) a classifi- +cation of the query is produced. +The work presented in (Kobayashi et al., 2021) proposes a +CBIR scheme based on AEs to extract the most important fea- +tures of brain tumour images. This work uses three different +AE that generate three different image descriptors, one focused +on the healthy features of the image, the other on the tumour +area, and the last one uses the information from the entire im- +age. Using these different outputs, researchers can disentangle +the normal and abnormal characteristics of the query to pro- +vide a controlled recommendation. This work combines DL +techniques along with traditional medical CBIR to recommend +similar images of the database given a certain query. The au- +thors then evaluated the accuracy of the recommendation us- +ing the Sorensen-Dice coefficient (Dice, 1945; Sorensen, 1948), +achieving a score of 0.695 in normal anatomy and 0.201 in ab- +normal. +Compared to previous research, the present work proposes +a CBIR system for tumour segmentation using DL algorithms. +The proposed MOC-AE improves the results of previous work +while using less costly information to train the model. +3. Multimodal +Brain +Tumor +Segmentation +Challenge +dataset +Multimodal Brain Tumor Segmentation Challenge (BraTS) +2020 dataset (Menze et al., 2014; Bakas et al., 2017, 2018) +was used to perform CBIR, which contains two different di- +visions. MICCAI BraTS Training contains information on 369 +different cases; this subset also contains manually segmented +regions of the tumoural areas of each case. Additionally, MIC- +CAI BraTS Validation contains 125 non-labelled scanners. +Each scanner is available as four different Neuroimaging In- +formatics Technology Initiative (NIfTI) files: Native scanner +(T1), Post-contrast T1 weighted (T1Gd), T2 weighted (T2) and +T2 Fluid Attenuated Inversion Recovery (T2-Flair). Segmen- +tation of each scan is divided into three labels, Gd-enhancing +tumour (ET), Pleritumoural edema (ED) and Non-enhancing + +4 +Guillermo Iglesias et al. / Medical Image Analysis (2023) +tumour core (NET), manually segmented and approved by neu- +roradiologists. +Due to the fact that for the training and evaluation of the pro- +posed algorithm, the labelled information for each scanner is +necessary, it will only be used in the MICCAI BraTS Training +partition. This dataset will be divided into a training and evalu- +ation set. +Figure 1 contains samples of the information present in the +dataset. It must be noted that each image represents a full 3 +dimensional scan of a patient. +Fig. 1. Sample images from unprocessed BraTS dataset. +To test the cases, we study the similarity of segmentation be- +tween the query and the retrieved images will be studied. In the +case of tumoural characteristics of each patient, similarity mea- +surement is performed using the tumour segmentation informa- +tion present in the original dataset. However, to compare the +similarity of the healthy areas of each patient, we have the same +approximation of (Kobayashi et al., 2021) where the brain of +each patient is divided anatomically. This process must be per- +formed to compare the results of the proposed model with those +of the work of (Kobayashi et al., 2021) and, to obtain the same +information they used, each scanner must be preprocessed. +3.1. Dataset preprocessing +Each 3 dimensional scanner consists on a scanner of +240x240x155 pixels of information, but the input of the pro- +posed method is a two dimensional image. In order to obtain +the images from the 3 dimensional data that are stored in the +BraTS 2020 dataset the scanners must be sliced in layers. Orig- +inally, the data dimension was 240x240x155, which is sliced on +the third axis to generate 240x240 images by taking the infor- +mation about each layer separately. Furthermore, each image is +normalized between the range [-1, 1] to be treated with Artifi- +cial Neural Networks (ANNs). +In addition, each healthy image is labelled with six normal +anatomical labels: left and right cerebrum, left and right cere- +bellum and left and right ventricle. This division is achieved us- +ing BrainSuite 19a software (Shattuck and Leahy, 2002). This +program is able to obtain a voxel segmentation of the cerebrum, +cerebellum and ventricle of each case, making it possible to use +this information to evaluate the similarity between the query +and the retrieved cases. +Figure 2 has a sample of the new labels generated for each +image. As can be seen after segmentation of the anatomical +labels, each brain is divided into six different areas, as was done +in Kobayashi et al. (2021). +Fig. 2. Sample images of the labelled dataset. +Figure 3 shows a brief scheme of the preprocessing process, +from the NIfTI files to a 2 dimensional images of each case, +obtaining in addition a segmentation of the anatomical labels of +each patient. +4. Multi-Output Classification Autoencoder +In Figure 4 the schematic of the proposed method can be +seen. According to the figure, the architecture presented com- +bines two different approaches: an AE network that extracts +the structural information of each image and a binary classifier + +ET +ED +NETET +ED +NET +Right cerebrum +Left cerebrum +Right cerebellum +Left cerebellum +Right ventricle +Left ventricleGuillermo Iglesias et al. / Medical Image Analysis (2023) +5 +Fig. 3. Data preparation scheme. (*) show that the anatomical labels were +obtained using BrainSuite 19a software (Shattuck and Leahy, 2002). (**) +shows that each slice corresponds to a certain depth in z axis. +that is responsible for extracting the tumour information from +each case. This dual-objective architecture enhances the fea- +tures represented in the descriptor, which is obtained from the +latent vector. +Fig. 4. MOC-AE model definition. +On the one hand, in MOC-AE an AE is used following the +same approximation as (Siradjuddin et al., 2019) where a CBIR +system is designed using the latent space of an AE as image +descriptors. This simple scheme makes it possible to extract +the composition features of an image by forcing the network to +reduce the dimensionality of the input images. One of the main +advantages of this method is that it does not require any label +to work, as a result of the self-supervised learning scheme of +AEs (Krishnan et al., 2022). Using an AE as the main base +for image descriptor generation, the network can learn latent +representations of the input image. +The main drawback of using the latent space of an AE as a +descriptor is that it considers with the same importance every +portion of the input image. Furthermore, there is no control +over the information represented in the latent space. +To solve this problem, it is proposed to add an auxiliary clas- +sifier that shares the descriptor with AE. This addition to the +network will attempt to maintain the information from the pa- +tient’s tumour. Here, it is important to note that this learning +scheme is focused on trying to keep certain features of the input +information, in order to disentangle the healthy and tumoural +information of the patient in the image descriptor. This solution +is based on the work of (Kobayashi et al., 2021) where three +different AEs were used to disentangle tumour and normal in- +formation from each case. +Using the classifier, the network is forced to learn the infor- +mation of the tumour to be able to classify the cases where a +tumour is present. At the same time, AE forces the descriptor +to maintain the structural characteristics of each patient. This +dual-objective forces the network to focus on some features that +are present in this case. +This architecture is focused on medical CBIR due to the fact +that it makes possible to make the network focus on the pos- +sible pathologies of each image. With regard to the work of +Kobayashi et al. (2021), this scheme does not require a seg- +mented dataset. In the proposed method, the binary classifier +only needs information about the presence of a tumour in the +image, but segmentation of the tumoural region of the image is +not necessary. +As stated in Krishnan et al. (2022), in the field of medicine, +the process of annotating the data is particularly costly because +manual annotation must be performed by specialists in the field. +Therefore, the possibility of developing a CBIR system capable +of focusing in the most relevant areas of the image while main- +taining the relatively low cost of the labels used is a character- +istic that is particularly desired. +4.1. Model training scheme +The proposed architecture must combine two different learn- +ing processes using the same parameters. As stated above, this +shared information forces the network to combine the normal +and pathological features of each image. +Regarding the training of the model, two different outputs +will use two different losses, which must be combined to train +the model. The AE output is responsible for reconstructing the + +6 +Guillermo Iglesias et al. / Medical Image Analysis (2023) +information in the input image, while the classifier must dif- +ferentiate between healthy and regions with the presence of a +tumour in it. +First, the input image x must be reconstructed by the AE gen- +erating ˆx, both the input and the reconstructed image will be +compared using the L2 norm. +Lr(x, ˆx) = ||x − ˆx||2 +2 +(1) +This reconstruction loss function Lr will force the latent +space to learn the spatial features of the input image. +At the same time, the classification head of the proposed +model will be trained using the binary cross-entropy loss func- +tion, formulated as follows: +Lc(x) = − 1 +N +N +� +i=1 +yi · log(ˆyi) + (1 − yi) · log(1 − ˆyi) +(2) +The head loss classification function Lc focuses on differen- +tiating healthy and tumoural images, focusing on detecting the +presence of tumours in the input image. When seeking this ob- +jective, the model will be forced to maintain tumoural features +in the latent space. +Both losses are combined to train the model using the follow- +ing equation: +Lt = γ1 ∗ Lr + γ2 ∗ Lc +(3) +where both γ terms are normalization coefficients to balance +both losses. +The loss function is minimised using the Adam (Kingma and +Ba, 2014) optimizer. The best values of the γ parameters were +found using the grid search in the training phase. The range of +values tested varied between 0 and 1, requiring that the sum of +both γ must be of 1. The best performance was found to be +achieved with γ1 = 0.2 and γ2 = 0.8. +4.2. Network architecture definition +The proposed network composition is Convolutional Neu- +ral Network (CNN) using the Residual blocks presented in the +ResNet architecture (He et al., 2016a). The same principles +as those used in ResNet were used to design the Encoder and +Decoder networks of the model. But in particular, it was de- +cided to use complete pre-activation blocks as was proposed in +(He et al., 2016b) that generally produce the best results. The +AE generates the latent vector by reducing the information to a +space of 500 dimensions that corresponds to the image descrip- +tor. +Regarding the classifier, it uses the latent vector information +of 500 positions and generates a binary classification, using an +intermediate dense layer of 64 neurons. +Figure 5 shows details of the model architecture. The input +data that the Encoder receives are 4 slices of each case, cor- +responding to the different scanners (T1, T1Gd, T2, T2-Flair). +Each residual block contains two separable convolutions (Chol- +let, 2017) along with Batch Normalization (Ioffe and Szegedy, +2015) and Dropout (Hinton et al., 2012) layers. The Rectified +Linear Unit (ReLU) activation function was used in the hidden +layers, while the hyperbolic tangent and the sigmoid activation +were used in the reconstruction and classification outputs, re- +spectively. +4.3. CBIR algorithm +CBIR system aims to obtain, from the database of docu- +mented clinical cases, the patient most similar to the query. In +other words, each time a scan is received, the algorithm CBIR +finds in the database the most similar cases in the database by +comparing the healthy and tumoural structures of the query. +To compare each query with the rest of the documented +cases, the image descriptor is generated. This descriptor cor- +responds to the latent space of MOC-AE generated using the +trained Encoder network. To compare the query image with the +rest of the database, we use the Euclidean distance, following +the same approximation as (Kobayashi et al., 2021; Tarjoman +et al., 2013; Shakarami and Tarrah, 2020; Siradjuddin et al., +2019). +The recommendation system will also use the classification +learning of the network in order to retrieve similar cases from +the database. Because of the inherent ability of the network of +classifying a certain query as tumoural or not this information +will be used to enhance the recommendation. If the query is +classified as tumoural by the MOC-AE with certain reliability +it will only search among other tumoural images. This way, + +Guillermo Iglesias et al. / Medical Image Analysis (2023) +7 +Fig. 5. MOC-AE architecture. +when the network is sure that a case contains a tumour it will +only search on other tumoural cases. In particular, it is decided +to put a threshold of 90% of confidence from where search on +tumoural cases. +5. Results +The experimental results of the proposed system CBIR are +shown in this section. The proposed algorithm performance +will be analysed by its training results and by comparing it +with similar works, this way evaluating its individual perfor- +mance and measuring how it performs with respect state-of-the- +art similar systems. +As mentioned in Section 3 the dataset from the input of the +experiment consists of 369 cases divided into 155 slices each. +Thus, a total of 57.195 images were used. However, 10% of the +whole dataset (5.720 images) is reserved for testing purposes. +5.1. Empirical evaluation of the CBIR system +It is decided to empirically evaluate the results of the CBIR +recommendation. To compare the results of the model, Figure +6 shows the results of different queries used as input to the sys- +tem. As can be seen, the model is capable of retrieving similar +images in both anatomical and tumoural aspects. +5.2. Comparison results +Table 1 shows the results of the proposed algorithm com- +pared with the results obtained in (Kobayashi et al., 2021) in +terms of the Sorensen-Dice coefficient (Dice, 1945; Sorensen, +1948). In particular, the table measures the consistency between +Fig. 6. Two different experiments of CBIR of the trained MOC-AE. +the anatomy of the healthy and tumoural features of the im- +age. This coefficient varies between 1 and 0, where the higher +the value, the more similar are the query and the retrieved im- +ages. To calculate the Sorensen-Dice coefficient of the proposed +method, the same procedure will be followed as (Kobayashi +et al., 2021), where the queries used are the slices of each case +with the largest tumoural area. Three different coefficients are +analysed, first, the normal Dice measures the similarity of the +healthy features of each patient, using the six anatomical labels +obtained with the procedure explained in Section 3.1 and calcu- +lating the multi-label Sorensen-Dice coefficient, the tumoural +Dice uses the segmentation information of each image, measur- +ing the similarity of the tumoural sections of each case and fi- +nally the entire Dice combines the normal and tumoural Dice to +balance the result and provide a more complete measurement. + +OOOOO +5x15x256 +30x30x128 +30x30x128 +57600 +60x60x64 +57600 +60x60x64 +120x120x32 +120x120x32 +240x240x16 +240x240x16 +240x240xET +ED +NET +Right cerebrum +Left cerebrum +Right cerebellum + Left cerebellum +Right ventricle + Left ventricle8 +Guillermo Iglesias et al. / Medical Image Analysis (2023) +Model +Normal Dice +Tumoural Dice +Entire Dice +MOC-AE +0.634±0.220 +0.310±0.276 +0.472±0.175 +Kobayashi et al. Normal latent space +0.730±0.196 +0.072±0.098 +0.401±0.108 +Kobayashi et al. Abnormal latent space +0.505±0.235 +0.289±0.120 +0.397±0.137 +Kobayashi et al. Entire latent space +0.695±0.208 +0.201±0.158 +0.448±0.123 +Table 1. Sorensen-Dice coefficient values comparing MOC-AE and the work of Kobayashi et al. (2021). +The presented results outperform the results of (Kobayashi +et al., 2021), especially in terms of tumour detection. Thus, +it is shown that MOC-AE can achieve state-of-the-art results +in tumour recommendation. In addition, it should be remem- +bered that the main strength of MOC-AE is that it does not +require segmentation information of the cases to train the net- +work. With respect to the compared work, MOC-AE shows a +global improvement in terms of patient tumoural features and a +performance comparable to that of Kobayashi et al. in anatom- +ical features. The best tumoural performance is achieved with +the proposed method, while the best anatomical similarity is ob- +tained in (Kobayashi et al., 2021). Finally, the best balance be- +tween normal and abnormal features of each patient is achieved +with MOC-AE. Therefore, the proposed MOC-AE is able to +balance the anatomical and tumour areas of each patient, to re- +trieve more similar images from the database. +6. Discussion +CDSSs represent critical algorithms that could significantly +improve the diagnostic tasks of doctors. The proposed MOC- +AE achieves state-of-the-art results in both the anatomical and +tumoural features of recommended cases. +When factorizing +medical images using MOC-AE, the model can focus on both +the normal features of the patient and the pathologies present in +the case, generating a compact and significant representation of +each image. These image descriptors can be used to recommend +similar cases from a database, helping the doctor diagnose the +task. +The architecture presented in the current work is able to com- +bine features present in the image with labels annotated by pro- +fessionals. One of the main strengths that differentiates MOC- +AE from similar CBIR models is that it does not require costly +information on the label such as tumour segmentation. The +MOC-AE learn characteristics of each patient combining rec- +ommendation and classification outputs, thus generating an en- +riched image descriptor using only binary label information. In +an area where the cost of producing high-quality labels is espe- +cially costly, it is considered crucial to develop a model that is +able to produce state-of-the-art results with a low-cost associ- +ated. +The results of the model improve the previous work by +obtaining better results in both retrieving cases with similar +pathologies and balancing both anatomical and abnormal fea- +tures of each case. In view of the results, MOC-AE is con- +sidered the best alternative to develop a CDSS specialized in +medical imaging recommendation. +7. Conclusions +In the present paper, a novel CBIR neural network called +MOC-AE is presented to recommend the most similar cases +given a query, taking particular attention to the anatomical and +pathological features of the patient at the same time. The pro- +posed model combines in an unique descriptor the most relevant +characteristics of the case, being able to extract from a database +the most similar cases. +This solution is capable of improving the results of state-of- +the-art similar approaches, improving the results regarding the +similarity of the tumoural area and balancing better healthy and +abnormal features of the patient. Thus, the proposed system +arises as the best solution to create a CDSS in tumoural brain +recommendation. +At the same time, with respect to other works, the presented +MOC-AE reduces the cost of the training process, only being +necessary labels for the presence of the tumour. This cost re- + +Guillermo Iglesias et al. / Medical Image Analysis (2023) +9 +duction is particularly relevant when treating with medical im- +ages, where the cost of manually annotating costly information +requires the time and effort of specialised professionals in the +area. 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The authors alone +are responsible for the content and writing of this paper +Acknowledgements +The authors thank the research group KNOwledge Discov- +ery and Information Systems (KNODIS) for the compute in- +frastructure. +Supplementary Material +The source code of the project, along with trained models and +results are publicly available and can be consulted in https: +//purl.com/mocae_brats. + diff --git a/pdE2T4oBgHgl3EQfKQak/content/tmp_files/load_file.txt b/pdE2T4oBgHgl3EQfKQak/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f1a9a0e271667896348cd4148627ec2434e835d --- /dev/null +++ b/pdE2T4oBgHgl3EQfKQak/content/tmp_files/load_file.txt @@ -0,0 +1,758 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf,len=757 +page_content='Medical Image Analysis (2023) Contents lists available at ScienceDirect Medical Image Analysis journal homepage: www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Alberto D´ıaz- ´Alvareza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Miguel Garc´ıa-Remesalc aDepartamento de Sistemas Inform´aticos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Escuela T´ecnica Superior de Ingenier´ıa de Sistemas Inform´aticos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Universidad Polit´ecnica de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Spain bCentro Universitario de Tecnolog´ıa y Arte Digital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Spain cBiomedical Informatics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Departamento de Inteligencia Artificial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Escuela T´ecnica Superior de Ingenieros Inform´aticos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Universidad Polit´ecnica de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Spain A R T I C L E I N F O Article history: 2000 MSC: 68T45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 68U10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 94A08,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 65D17 Keywords: content-based image re- trieval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' deep learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' autoencoder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' feature extraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' clinical decision support system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' comparative diagnostic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' magnetic resonance imaging A B S T R A C T The present work proposes a Multi-Output Classification Autoencoder (MOC-AE) al- gorithm to extract features from brain tumour images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The proposed algorithm is able to focus on both the normal features of the patient and the pathological features present in the case, resulting in a compact and significant representation of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The ar- chitecture of MOC-AE combines anatomical information from the patient’s scan using an Autoencoder (AE) with information related to a specific pathology using a classi- fication output with the same image descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This combination of goals forces the network to maintain a balance between anatomical and pathological features of the case while maintaining the low cost of the labels being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The results obtained are compared with those of similar studies and the strengths and limitations of each approach are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The results demonstrate that the proposed algorithm is capable of achieving state-of-the-art results in terms of both the anatomical and tumor characteristics of the recommended cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' © 2023 Elsevier B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Introduction The techniques used in medicine to treat pathologies are be- coming more advanced each day and are capable of performing less invasive and more efficient treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' However, before a patient can apply proper treatment, a correct diagnosis of the disease is necessary, which becomes an important task in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' ⋆This is an example for title footnote coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' ∗Corresponding author e-mail: guillermo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='iglesias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='hernandez@alumnos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='es (Guillermo Iglesias), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='talavera@upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='es (Edgar Talavera), alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='diaz@upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='es (Alberto D´ıaz- ´Alvarez), mgremesal@fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='es (Miguel Garc´ıa-Remesal) Although medicine has radically evolved, diagnosis is still mostly a human process and the expert must be able to thor- oughly evaluate the patient’s evidence and avoid making mis- takes during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' A late or incorrect diagnosis can lead to an increase in pathologies that, in cases such as cancer, can be fatal and irreversible (Tan and Fielding, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Therefore, any improvement that a doctor can make during the diagnostic process may be vital and necessary to greatly im- prove treatment results, because early diagnosis improves treat- ment results, as it improves the result of the procedure (Kimber- Trojnar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Preprint submitted to Medical Image Analysis January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='03701v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='IV] 9 Jan 2023 EISEVIERN13-AI5 MEDICAL IMAGE ANALYSIS2 Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) At this point, computer algorithms are specially relevant, since they can be used as an additional tool for decision making, which is known as Clinical Decision Support System (CDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' It is precisely in the current paradigm of health information sci- ence where these CDSSs can be used to process in real time large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In this sense, there are two approaches to process large amounts of medical images, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='g: X-rays, computed tomogra- phy scanners, magnetic resonance images, nuclear medicine images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The first solution are Content-Based Image Retrieval (CBIR) systems which retrieve the most similar images from a database using the information of the query image itself, ex- tracting the most relevant features of the input and recommend- ing the most similar images stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' While other approximation are the so-called Concept-Based Image Indexing systems, that use meta-information of the images to extract its content and be able of recommending similar samples from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' CBIR systems are focused on processing large amounts of data, Artificial Intelligence (AI) algorithms arise as the best so- lution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' These frameworks are capable of obtain- ing the most similar images from a dataset given a certain query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Different solutions have been proposed during the last years combining CBIR with Deep Learning (DL) techniques (Sir- adjuddin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' These systems make possible to use large amounts of information in an ordered manner, taking advantage of the information available without the necessity to manually use all information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' However, it is not widely considered that AI should replace health professionals in tasks they normally perform, but instead assist them in diagnosis and decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' It should be taken into account that the human professional will always make the final decision, it should never be substituted by an AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In the same way that surgical robots need an operator to perform their tasks, these tools based on AI should be considered diagnostic support systems, instead of decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Aspects such as lack of explainability or precision (Ossa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2022), or inabil- ity to deal with outliers suggest that it is probably too early to completely delegate critical tasks to an AI, at least in the fore- seeable future (Krittanawong, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' One of the techniques in which artificial intelligence has re- cently achieved better results is in Computer Vision (CV) (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Here, medical imaging data can be used by these algorithms to extract and process its information to help profes- sionals perform certain tasks (Esteva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Ward et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Thus, this relationship between CV and medical imag- ing has led to the emergence of works that combine the latest advances of artificial intelligence in image feature extraction along with medical data (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Autoencoders (AEs) (Rumelhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 1985) are machine learning models used in dimensionality reduction and feature extraction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In this work, it is proposed to use a novel AE variant, named as Multi-Output Classification Autoencoder (MOC-AE), that factorizes brain tumoural images, to then use the generated image descriptors to recommend cases with a similar pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The architecture presented focuses on improving the retrieval accuracy of a standard AE without the need to use costly infor- mation, such as tumour area segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' MOC-AE is able to balance the normal anatomical features of each patient with the tumour features in a single descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Thus, the model is able to recommend interesting cases taking into account rele- vant medical information, in this case, the tumour area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The proposed MOC-AE architecture has two main advan- tages over previous similar CBIR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' First, the information required to develop the model training is much less costly com- pared to other approximations, due to the fact that the model only uses binary labels to learn the pathologies features of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This reduction in terms of cost does not imply worse results in image recommendation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' as is studied in Section 5 the model results show improved performance in tumour similarity and a better balance between normal and abnormal features of each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The present work describes the process of training and evalu- ation of MOC-AE to extract features of the tumoural brain and its performance is demonstrated in the task of recommending cases with similar pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The results obtained are com- Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) 3 pared with similar works and the strengths and drawbacks of each approximation are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The source code of the project, along with trained models and results are publicly available1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Related works Different works published in the last decade use machine learning techniques to improve the diagnosis of doctors (Fa- tima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Ralbovsky and Lednev, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Haq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Quellec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Taking advantage of the latest AI research and applying it to the medical field, different works have obtained impressive results in tasks related to the medical field (Gardezi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Regarding medical imaging algorithms, CDSSs is a pro- lific area where in recent years many articles have been pub- lished (Musen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Rama Krishna and Sirajuddin, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Tuppad and Patil, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' These techniques combine the potential AI can achieve in extracting the most im- portant features of medical images along with support systems that provide the doctor with the most relevant information in each case (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In this sense, CBIR arises as a solution where it is possible to make use of all data stored in large databases that, in an- other context, it would be impossible for a human to fully use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' There are different works in this area, such as Lehmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2005) where the categorisation of medical images from differ- ent sources is sought through CBIR using traditional algorithms to generate image descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The recommendation of a simi- lar descriptor is made using the nearest neighbour algorithm, as was done in other research based on descriptors CBIR, indepen- dently of medical imaging (Siradjuddin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Shakarami and Tarrah, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Lehmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The work of Tarjoman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2013) proposes a sim- ple feature extraction method using Support Vector Machine (SVM) (Cortes and Vapnik, 1995) and the Grey Level Co- occurrence Matrix as the main input of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' A simi- lar approximation was followed in Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2016) where 1https://purl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='com/mocae_brats SVM is used as the descriptor generator mechanism, in this case using different features of the liver images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Finally, using the weighted nearest neighbour (Cover and Hart, 1967) a classifi- cation of the query is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The work presented in (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021) proposes a CBIR scheme based on AEs to extract the most important fea- tures of brain tumour images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This work uses three different AE that generate three different image descriptors, one focused on the healthy features of the image, the other on the tumour area, and the last one uses the information from the entire im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Using these different outputs, researchers can disentangle the normal and abnormal characteristics of the query to pro- vide a controlled recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This work combines DL techniques along with traditional medical CBIR to recommend similar images of the database given a certain query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The au- thors then evaluated the accuracy of the recommendation us- ing the Sorensen-Dice coefficient (Dice, 1945;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sorensen, 1948), achieving a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='695 in normal anatomy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='201 in ab- normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Compared to previous research, the present work proposes a CBIR system for tumour segmentation using DL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The proposed MOC-AE improves the results of previous work while using less costly information to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Multimodal Brain Tumor Segmentation Challenge dataset Multimodal Brain Tumor Segmentation Challenge (BraTS) 2020 dataset (Menze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Bakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2017, 2018) was used to perform CBIR, which contains two different di- visions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' MICCAI BraTS Training contains information on 369 different cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' this subset also contains manually segmented regions of the tumoural areas of each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Additionally, MIC- CAI BraTS Validation contains 125 non-labelled scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Each scanner is available as four different Neuroimaging In- formatics Technology Initiative (NIfTI) files: Native scanner (T1), Post-contrast T1 weighted (T1Gd), T2 weighted (T2) and T2 Fluid Attenuated Inversion Recovery (T2-Flair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Segmen- tation of each scan is divided into three labels, Gd-enhancing tumour (ET), Pleritumoural edema (ED) and Non-enhancing 4 Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) tumour core (NET), manually segmented and approved by neu- roradiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Due to the fact that for the training and evaluation of the pro- posed algorithm, the labelled information for each scanner is necessary, it will only be used in the MICCAI BraTS Training partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This dataset will be divided into a training and evalu- ation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Figure 1 contains samples of the information present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' It must be noted that each image represents a full 3 dimensional scan of a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sample images from unprocessed BraTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To test the cases, we study the similarity of segmentation be- tween the query and the retrieved images will be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In the case of tumoural characteristics of each patient, similarity mea- surement is performed using the tumour segmentation informa- tion present in the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' However, to compare the similarity of the healthy areas of each patient, we have the same approximation of (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021) where the brain of each patient is divided anatomically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This process must be per- formed to compare the results of the proposed model with those of the work of (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021) and, to obtain the same information they used, each scanner must be preprocessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Dataset preprocessing Each 3 dimensional scanner consists on a scanner of 240x240x155 pixels of information, but the input of the pro- posed method is a two dimensional image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In order to obtain the images from the 3 dimensional data that are stored in the BraTS 2020 dataset the scanners must be sliced in layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Orig- inally, the data dimension was 240x240x155, which is sliced on the third axis to generate 240x240 images by taking the infor- mation about each layer separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Furthermore, each image is normalized between the range [-1, 1] to be treated with Artifi- cial Neural Networks (ANNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In addition, each healthy image is labelled with six normal anatomical labels: left and right cerebrum, left and right cere- bellum and left and right ventricle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This division is achieved us- ing BrainSuite 19a software (Shattuck and Leahy, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This program is able to obtain a voxel segmentation of the cerebrum, cerebellum and ventricle of each case, making it possible to use this information to evaluate the similarity between the query and the retrieved cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Figure 2 has a sample of the new labels generated for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' As can be seen after segmentation of the anatomical labels, each brain is divided into six different areas, as was done in Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sample images of the labelled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Figure 3 shows a brief scheme of the preprocessing process, from the NIfTI files to a 2 dimensional images of each case, obtaining in addition a segmentation of the anatomical labels of each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Multi-Output Classification Autoencoder In Figure 4 the schematic of the proposed method can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' According to the figure, the architecture presented com- bines two different approaches: an AE network that extracts the structural information of each image and a binary classifier ET ED NETET ED NET Right cerebrum Left cerebrum Right cerebellum Left cerebellum Right ventricle Left ventricleGuillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Data preparation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (*) show that the anatomical labels were obtained using BrainSuite 19a software (Shattuck and Leahy, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (**) shows that each slice corresponds to a certain depth in z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' that is responsible for extracting the tumour information from each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This dual-objective architecture enhances the fea- tures represented in the descriptor, which is obtained from the latent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' MOC-AE model definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' On the one hand, in MOC-AE an AE is used following the same approximation as (Siradjuddin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2019) where a CBIR system is designed using the latent space of an AE as image descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This simple scheme makes it possible to extract the composition features of an image by forcing the network to reduce the dimensionality of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' One of the main advantages of this method is that it does not require any label to work, as a result of the self-supervised learning scheme of AEs (Krishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Using an AE as the main base for image descriptor generation, the network can learn latent representations of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The main drawback of using the latent space of an AE as a descriptor is that it considers with the same importance every portion of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Furthermore, there is no control over the information represented in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To solve this problem, it is proposed to add an auxiliary clas- sifier that shares the descriptor with AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This addition to the network will attempt to maintain the information from the pa- tient’s tumour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Here, it is important to note that this learning scheme is focused on trying to keep certain features of the input information, in order to disentangle the healthy and tumoural information of the patient in the image descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This solution is based on the work of (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021) where three different AEs were used to disentangle tumour and normal in- formation from each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Using the classifier, the network is forced to learn the infor- mation of the tumour to be able to classify the cases where a tumour is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' At the same time, AE forces the descriptor to maintain the structural characteristics of each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This dual-objective forces the network to focus on some features that are present in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This architecture is focused on medical CBIR due to the fact that it makes possible to make the network focus on the pos- sible pathologies of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' With regard to the work of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2021), this scheme does not require a seg- mented dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In the proposed method, the binary classifier only needs information about the presence of a tumour in the image, but segmentation of the tumoural region of the image is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' As stated in Krishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2022), in the field of medicine, the process of annotating the data is particularly costly because manual annotation must be performed by specialists in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Therefore, the possibility of developing a CBIR system capable of focusing in the most relevant areas of the image while main- taining the relatively low cost of the labels used is a character- istic that is particularly desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Model training scheme The proposed architecture must combine two different learn- ing processes using the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' As stated above, this shared information forces the network to combine the normal and pathological features of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Regarding the training of the model, two different outputs will use two different losses, which must be combined to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The AE output is responsible for reconstructing the 6 Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) information in the input image, while the classifier must dif- ferentiate between healthy and regions with the presence of a tumour in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' First, the input image x must be reconstructed by the AE gen- erating ˆx, both the input and the reconstructed image will be compared using the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Lr(x, ˆx) = ||x − ˆx||2 2 (1) This reconstruction loss function Lr will force the latent space to learn the spatial features of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' At the same time, the classification head of the proposed model will be trained using the binary cross-entropy loss func- tion, formulated as follows: Lc(x) = − 1 N N � i=1 yi · log(ˆyi) + (1 − yi) · log(1 − ˆyi) (2) The head loss classification function Lc focuses on differen- tiating healthy and tumoural images, focusing on detecting the presence of tumours in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' When seeking this ob- jective, the model will be forced to maintain tumoural features in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Both losses are combined to train the model using the follow- ing equation: Lt = γ1 ∗ Lr + γ2 ∗ Lc (3) where both γ terms are normalization coefficients to balance both losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The loss function is minimised using the Adam (Kingma and Ba, 2014) optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The best values of the γ parameters were found using the grid search in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The range of values tested varied between 0 and 1, requiring that the sum of both γ must be of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The best performance was found to be achieved with γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='2 and γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Network architecture definition The proposed network composition is Convolutional Neu- ral Network (CNN) using the Residual blocks presented in the ResNet architecture (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The same principles as those used in ResNet were used to design the Encoder and Decoder networks of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' But in particular, it was de- cided to use complete pre-activation blocks as was proposed in (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2016b) that generally produce the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The AE generates the latent vector by reducing the information to a space of 500 dimensions that corresponds to the image descrip- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Regarding the classifier, it uses the latent vector information of 500 positions and generates a binary classification, using an intermediate dense layer of 64 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Figure 5 shows details of the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The input data that the Encoder receives are 4 slices of each case, cor- responding to the different scanners (T1, T1Gd, T2, T2-Flair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Each residual block contains two separable convolutions (Chol- let, 2017) along with Batch Normalization (Ioffe and Szegedy, 2015) and Dropout (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2012) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The Rectified Linear Unit (ReLU) activation function was used in the hidden layers, while the hyperbolic tangent and the sigmoid activation were used in the reconstruction and classification outputs, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' CBIR algorithm CBIR system aims to obtain, from the database of docu- mented clinical cases, the patient most similar to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In other words, each time a scan is received, the algorithm CBIR finds in the database the most similar cases in the database by comparing the healthy and tumoural structures of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To compare each query with the rest of the documented cases, the image descriptor is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This descriptor cor- responds to the latent space of MOC-AE generated using the trained Encoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To compare the query image with the rest of the database, we use the Euclidean distance, following the same approximation as (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Tarjoman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Shakarami and Tarrah, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Siradjuddin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The recommendation system will also use the classification learning of the network in order to retrieve similar cases from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Because of the inherent ability of the network of classifying a certain query as tumoural or not this information will be used to enhance the recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' If the query is classified as tumoural by the MOC-AE with certain reliability it will only search among other tumoural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This way, Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' MOC-AE architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' when the network is sure that a case contains a tumour it will only search on other tumoural cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In particular, it is decided to put a threshold of 90% of confidence from where search on tumoural cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Results The experimental results of the proposed system CBIR are shown in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The proposed algorithm performance will be analysed by its training results and by comparing it with similar works, this way evaluating its individual perfor- mance and measuring how it performs with respect state-of-the- art similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' As mentioned in Section 3 the dataset from the input of the experiment consists of 369 cases divided into 155 slices each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Thus, a total of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='195 images were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' However, 10% of the whole dataset (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='720 images) is reserved for testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Empirical evaluation of the CBIR system It is decided to empirically evaluate the results of the CBIR recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To compare the results of the model, Figure 6 shows the results of different queries used as input to the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' As can be seen, the model is capable of retrieving similar images in both anatomical and tumoural aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Comparison results Table 1 shows the results of the proposed algorithm com- pared with the results obtained in (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021) in terms of the Sorensen-Dice coefficient (Dice, 1945;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sorensen, 1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In particular, the table measures the consistency between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Two different experiments of CBIR of the trained MOC-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' the anatomy of the healthy and tumoural features of the im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This coefficient varies between 1 and 0, where the higher the value, the more similar are the query and the retrieved im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' To calculate the Sorensen-Dice coefficient of the proposed method, the same procedure will be followed as (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021), where the queries used are the slices of each case with the largest tumoural area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Three different coefficients are analysed, first, the normal Dice measures the similarity of the healthy features of each patient, using the six anatomical labels obtained with the procedure explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='1 and calcu- lating the multi-label Sorensen-Dice coefficient, the tumoural Dice uses the segmentation information of each image, measur- ing the similarity of the tumoural sections of each case and fi- nally the entire Dice combines the normal and tumoural Dice to balance the result and provide a more complete measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' OOOOO 5x15x256 30x30x128 30x30x128 57600 60x60x64 57600 60x60x64 120x120x32 120x120x32 240x240x16 240x240x16 240x240xET ED NET Right cerebrum Left cerebrum Right cerebellum Left cerebellum Right ventricle Left ventricle8 Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) Model Normal Dice Tumoural Dice Entire Dice MOC-AE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='634±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='310±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='472±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='175 Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Normal latent space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='730±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='072±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='401±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='108 Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Abnormal latent space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='505±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='289±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='397±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='137 Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Entire latent space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='695±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='201±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='448±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='123 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Sorensen-Dice coefficient values comparing MOC-AE and the work of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The presented results outperform the results of (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021), especially in terms of tumour detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Thus, it is shown that MOC-AE can achieve state-of-the-art results in tumour recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In addition, it should be remem- bered that the main strength of MOC-AE is that it does not require segmentation information of the cases to train the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' With respect to the compared work, MOC-AE shows a global improvement in terms of patient tumoural features and a performance comparable to that of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' in anatom- ical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The best tumoural performance is achieved with the proposed method, while the best anatomical similarity is ob- tained in (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Finally, the best balance be- tween normal and abnormal features of each patient is achieved with MOC-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Therefore, the proposed MOC-AE is able to balance the anatomical and tumour areas of each patient, to re- trieve more similar images from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Discussion CDSSs represent critical algorithms that could significantly improve the diagnostic tasks of doctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The proposed MOC- AE achieves state-of-the-art results in both the anatomical and tumoural features of recommended cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' When factorizing medical images using MOC-AE, the model can focus on both the normal features of the patient and the pathologies present in the case, generating a compact and significant representation of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' These image descriptors can be used to recommend similar cases from a database, helping the doctor diagnose the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The architecture presented in the current work is able to com- bine features present in the image with labels annotated by pro- fessionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' One of the main strengths that differentiates MOC- AE from similar CBIR models is that it does not require costly information on the label such as tumour segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The MOC-AE learn characteristics of each patient combining rec- ommendation and classification outputs, thus generating an en- riched image descriptor using only binary label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In an area where the cost of producing high-quality labels is espe- cially costly, it is considered crucial to develop a model that is able to produce state-of-the-art results with a low-cost associ- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The results of the model improve the previous work by obtaining better results in both retrieving cases with similar pathologies and balancing both anatomical and abnormal fea- tures of each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In view of the results, MOC-AE is con- sidered the best alternative to develop a CDSS specialized in medical imaging recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Conclusions In the present paper, a novel CBIR neural network called MOC-AE is presented to recommend the most similar cases given a query, taking particular attention to the anatomical and pathological features of the patient at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The pro- posed model combines in an unique descriptor the most relevant characteristics of the case, being able to extract from a database the most similar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This solution is capable of improving the results of state-of- the-art similar approaches, improving the results regarding the similarity of the tumoural area and balancing better healthy and abnormal features of the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Thus, the proposed system arises as the best solution to create a CDSS in tumoural brain recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' At the same time, with respect to other works, the presented MOC-AE reduces the cost of the training process, only being necessary labels for the presence of the tumour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' This cost re- Guillermo Iglesias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' / Medical Image Analysis (2023) 9 duction is particularly relevant when treating with medical im- ages, where the cost of manually annotating costly information requires the time and effort of specialised professionals in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' With MOC-AE this information is drastically reduced, while the results are improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' In conclusion, the use of MOC-AE in tumoural CBIR is a promising option to improve efficiency and accuracy in com- parative diagnostic and tumoural pathologies treatment.' metadata={'source': 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Surgery 169, 1253– 1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Declaration of interest The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' The authors alone are responsible for the content and writing of this paper Acknowledgements The authors thank the research group KNOwledge Discov- ery and Information Systems (KNODIS) for the compute in- frastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content=' Supplementary Material The source code of the project, along with trained models and results are publicly available and can be consulted in https: //purl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} +page_content='com/mocae_brats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQfKQak/content/2301.03701v1.pdf'} diff --git a/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/2301.11908v1.pdf.txt b/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/2301.11908v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bac6f3b679e9dc11fe3175620c74eec261da1991 --- /dev/null +++ b/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/2301.11908v1.pdf.txt @@ -0,0 +1,1113 @@ +A modular perspective to the jet suppression from a +small to large radius in very high transverse momentum jets +Om Shahi,1 Vaishnavi Desai,2 and Prabhakar Palni2, ∗ +1Department of Physics, BITS Pilani, Goa. +2School of Physical & Applied Sciences, Goa University, Goa. +In this work, we extend the scope of the JETSCAPE Framework to cover the jet radius(R) +dependence of the jet Nuclear Modification factor (RAA) for broader area jet cones, going all the +way up to R = 1.0 . Our primary focus has been the in-depth analysis of the high-pT inclusive jets and +the quenching effects observed in the Quark-gluon plasma formed in the Pb-Pb collisions at √sNN += 5.02 TeV for the most-central (0-10%) collisions. Nuclear modification factor (RAA) is calculated +for inclusive jets to compare with the experimental results collected at the ATLAS and the CMS +detectors in the jet-pT range from 100 GeV up to 1 TeV. The results predicted by the JETSCAPE +are consistent in the high pT range as well as for extreme jet cone sizes. We also calculate the +double ratio (RR +AA/RR=small +AA +) as a function of jet radius and jet-pT , where the observations are well +described by the JETSCAPE framework which is based on the hydrodynamic multi-stage evolution +of the parton shower. The calculations are then replicated for different low-virtuality based evolution +models like the MARTINI and the AdS/CFT, which is followed by a rigorous comparison between +the predictions from the former model combinations to the measurements at the CMS experiment. +I. +INTRODUCTION +The extremely hot and dense conditions created at the +starting of big bang which we now understand as the +soup of de-confined state of the partons, the Quark-Gluon +Plasma. The QGP is of such great interest that, to study +the properties of this state of matter, scientists have spent +decades in building up the equipment which could recre- +ate this extremely dense soupy state. +The Relativis- +tic Heavy-Ion Collider (RHIC) and the Large Hadron +Collider (LHC) conduct Heavy-ion collisions where the +QGP is created for very short instants of time, the par- +ton shower propagation and modification is greatly influ- +enced by the quark-gluon plasma. The high-pT jets pro- +duced in these heavy ion collisions undergo strong yield +suppression and medium modification which are together +referred to as jet quenching phenomena.Therefore, we +study the jet modification in nucleus-nucleus collisions +relative to proton-proton collisions to probe the prop- +erties of the QGP via constraints from model-to-data +comparison. [1–7] The jet Nuclear Modification Factor +(RAA) has been one of the most important and the prime +observable to study the properties of the Quark-gluon +Plasma. The measurement of Nuclear Modification Fac- +tor for jets as well as charged particles has revealed many +important characteristics of the Quark-gluon plasma. It +provides a strong confirmation of the interaction of par- +tons with the deconfined plasma, the respective medium +modifications and the eventual hydrodynamization with +the medium [8]. Since there are many conclusive studies +based on jet-RAA [9–13], our effort here is to push the +limits of what we hope to learn about the QGP from this +observable with the available experimental data. +Along with these measurements, we can reconstruct in- +∗ Corresponding author: prabhakar.palni@unigoa.ac.in +clusive jet spectra for p-p and Pb-Pb by varying distance +parameters R in the anti-kT algorithm, which is realized +in the FASTJET software package [14]. The inclusive jet +spectrum is of significant interest because the sensitivity +of hadronization effects is far less compared to the observ- +ables involving individual final-state hadrons. Here, the +area of the reconstructed jet cone is defined by R. Thus, +by varying R, the reconstructed jet will include differ- +ent proportions of energy from the medium response and +the quenched jet. We have also observed new sensitivity +to QGP properties and underlying jet quenching mech- +anism in a study for jet yield suppression versus R [15]. +Specifically, different dependences of the jet suppression +on R, are predicted by generators and theoretical models +based on anti-de Sitter/conformal field theory correspon- +dence [16] and perturbative QCD [17]. +We take off by calculating the Jet (RAA) for the Pb- +Pb collisions in SECTION II and compare with the ex- +perimental data from the ATLAS as well as the CMS +detector for a robust test of our configuration and the +overall JETSCAPE Framework (version 3.5.1) [18]. Our +calculations include the (2+1)D MUSIC [19] model for +hydrodynamics which is ideal for studying many aspects +of heavy ion collisions. Most notably, our calculations +cover the high-pT range of jets that is up to 1 TeV, which +enables us to probe the medium at much shorter distance +scales. +We now exploit the advantages that the JETSCAPE +framework offers in SECTION II A, that is the coupling +of several different energy loss modules such as MARTINI +and AdS/CFT with the MATTER module to explore the +quenching effects in a multi-stage manner which gives us +an insight into the interaction and energy loss mecha- +nism in the medium with respect to the virtuality of the +partonic jet. Comparing the experimental data with the +trends from these different models which handle the low +virtuality phase allows us to develop a lucid understand- +ing of the physics governing the above models. +arXiv:2301.11908v1 [hep-ph] 27 Jan 2023 + +2 +In SECTION II B, we proceed with concrete results +from the above combinations of several successful mod- +els towards the jet radius dependence studies. An impor- +tant measurement done by the CMS collaboration [20] re- +cently, was the jet-RAA for Pb-Pb collisions at the LHC +and recording the data for jet cone area covering the radii +from 0.2 up to 1. This analysis gives us an intricate un- +derstanding of the behaviour and interaction strength of +the high-pT inclusive collimated jets in proximity to the +jet axis, for R=0.2 and the energy distribution around the +cone for values of radii up to 1. The current JETSCAPE +framework is based on calculations of perturbative quan- +tum chromodynamics and evolution in a dense medium, +thus allowing us to go up to a radius of order 1. We also +plot and report the Double Ratio for jet-pT dependence +and jet radius dependence to provide a vivid picture of +the energy transactions with the Quark-gluon plasma as +the jet cone size increases and also discuss the trends that +the JETSCAPE framework predicts. +We conclude this work in SECTION III, with a concise +account of the current JETSCAPE model’s potential to +explain the jet-RAA for larger area jet cones. We also +shed light on the ability of the distinct combination of +modules to describe the jet-RAA. +A. +SIMULATION WITH MULTI-STAGE +ENERGY LOSS APPROACH IN JETSCAPE +The JETSCAPE framework provides an ideal envi- +ronment for carrying out multi-stage energy loss. The +hard scattering is generated by Pythia 8 [21] with initial +state radiation (ISR) and multiparton interaction (MPI) +turned on, and final state radiation (FSR) turned off. For +the event wise simulations, the TRENTo model [22] sets +up the initial conditions and the viscous hydrodynamic +evolution is described by the (2+1)D MUSIC [19] model +followed by Cooper-Frye prescription [23, 24]which con- +verts the fluid cells to hadrons on an isothermal hyper- +surface at TSW= 151 MeV. The jet energy loss induced +by scattering is calculated in succession of two stages: +MATTER [25, 26] takes care of the highly virtual phase +(the first stage) while the low virtuality phase (the second +stage), is handled concurrently by the LBT model [27– +29]. We have also employed the MARTINI [30–32] and +the AdS/CFT model [33] in combination with the MAT- +TER model to explore the low virtuality phase. the vir- +tuality of the parton is defined as Q. The parton un- +dergoes energy loss in two stages, when the virtuality +of the parton, Q2 > Q2 +SW, where QSW is the switching +virtuality, the MATTER model handles the energy loss +and the parton is transferred to the LBT model once +Q2 ≤ Q2 +SW. In our calculations, the jet medium interac- +tion includes inelastic medium-induced gluon radiation +and medium recoil. An extensive account on the com- +parison of the model to the existing jet-RAA is already +done [34], showing that the contribution of medium re- +coil is pretty significant in the modification of jet-RAA +for a complete set of centrality classes ranging from the +most central collisions to the peripheral collisions. This +version of the JETSCAPE framework [18] encompasses +the modifications of a hard thermal-loop (HTL) ˆq [35] +for fixed coupling, running coupling, and with a virtual- +ity dependent factor that modulates the effective value of +ˆq. It also accounts for the reduced medium-induced emis- +sion in the high virtuality phase, due to coherence effects. +The calculations based on the hard-thermal-loop (HTL) +considering weak- coupling approximation and the limits +of high temperature yield a ˆq given as [28], +ˆqHTL = Ca +42ζ(3) +π +α2 +sT 3 ln +� +2ET +6πT 2αfix +s +� +. +(1) +where Ca is the representation specific Casimir, E is the +energy of the hard parton and T is the local temperature +of the medium. The Coherence effects which reduce the +interaction strength of the parton with the medium is also +taken into consideration as the virtuality dependent mod- +ulation factor f(Q2) which regulates the effective value of +ˆq in the high virtuality MATTER event generator. The +parameterization of the virtuality-dependent modulation +factor is given as [34] +ˆq · f ≡ ˆqrun +HTLf(Q2) +(2) +f(Q2) = +� +1+10 ln2(Q2 +sw)+100 ln4(Q2 +sw) +1+10 ln2(Q2)+100 ln4(Q2) +Q2 > Q2 +sw +1 +Q2 ≤ Q2 +sw +, +(3) +here Q2 is the running virtuality of the hard parton. +Finally, the partons undergo colorless hadronization ac- +cording to the default Lund string fragmentation from +Pythia 8 [21, 36]. The major contributions in our final +jets are from hard jet shower part and the effect from +the soft medium response, the latter is calculated via the +Cooper-Frye formula. We reconstruct our jets for sev- +eral radius selections using the anti-kT algorithm which +is realised in the FASTJET Software Package [14] and +compare with experimental data. The Underlying Events +(UE) are removed by implementing a minimun track re- +quirement of ptrack,min +T +> 4 GeV. All the parameters in- +volved in the tuning of the constituent modules follow +the standard set of tunes released by the JETSCAPE +Collaboration in an elaborate recent study [34]. +II. +RESULTS AND ANALYSIS +This work covers the collision energy: √sNN = 5.02 +TeV for two centrality classes i.e.the most central(0- +10%) collisions and shows a comparison with selected +experimental data available from the ATLAS and the +CMS Collaborations. +The energy loss depicted in our +results is achieved by the coupling of MATTER with +the LBT module and the secondary module (which han- +dles the low virtuality phase) remains the same through- +out our results until and unless specified. Fig. 1 shows +the p + p simulation results for inclusive jet spectra for + +3 +100 +200 +300 +400 +500 +600 +pjet +T [GeV] +10 +8 +10 +6 +10 +4 +10 +2 +100 +d2 +jet +dpjet +T dyjet [nb/GeV] +pp, +s = 5.02 TeV +anti +kt, R = 0.4, |yjet| < 2.8 +ATLAS [PLB 790,108 (2019)] +JETSCAPE 3.5.1 +FIG. 1. +(Color online)Diffrential cross section of inclusive +jets for p+p collisions with cone size R=0.4, with a minimum +track requirement of ptrack +T +> 4 GeV. +p+p at √s = 5.02TeV for |yjet| < 2.8 which follow the +JETSCAPE PP 19 tune [37] are then compared to the +experimental data from the ATLAS [12, 13], the ratio of +inclusive jet cross-section to the ATLAS data is plotted in +fig. 2, which clearly shows that the results are in accept- +able range(≤ 20%). We report the 0-10% central Pb-Pb +jet spectra for R=0.4 in fig. 3. We also plot the ratio +of differential cross section for inclusive jets compared to +the data from ATLAS [13] shown in fig. 4. Here, RAA is +defined as +RAA = +1 +Nevent +d2Njet +dηjetdpjet +T +��� +AA +d2σjet +dηjetdpjet +T +��� +pp +, +(4) +The inclusive jet-RAA is now calculated as the ratio +of the Pb-Pb and p-p spectra plotted above, which is +shown in fig. 5 in comparison to the ATLAS data. The +JETSCAPE results agree with the data quite nicely, al- +though we see a 5% enhancement in the low pT region +which is followed by a suppression as we ascend in the +range. +A. +Exploring the MARTINI and AdS/CFT models +Both the MARTINI [30] and the AdS/CFT [33] mod- +els are designed to handle the low virtuality phase and +carry forward the energy loss once the parton is trans- +ferred from the MATTER. We now replace the successful +LBT model with the MARTINI model, to carry out the +simulations for the same settings [34] and compare the +jet-RAA with the experimental data from the ATLAS +Collaboration [13]. Similarly the calculations are done +by replacing the LBT by AdS/CFT and compared with +the recorded data. +We now present the intensive comparison between the +low virtuality parton evolution models, where the in- +100 +200 +300 +400 +500 +600 +pjet +T [GeV] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Ratio to ATLAS[ 1 +Nev +d2 +jet +dpjet +T dyjet] +pp, s = 5.02 TeV +anti +kt, R = 0.4 +(|yjet| < 2.8) +JETSCAPE 3.5.1 +FIG. 2. (Color online)Diffrential cross-section of inclusive jets +for p+p collisions with cone size R=0.4, with a minimum track +requirement of ptrack +T +> 4 GeV. +100 +200 +300 +400 +500 +600 +pjet +T [GeV] +10 +8 +10 +6 +10 +4 +10 +2 +100 +1 +Nev TAA +d2Njet +dpjet +T dyjet [nb/GeV] +PbPb (0-10%), sNN = 5.02 TeV +Qsw = 2 GeV, +fix +s = 0.35 +(|yjet| < 2.8), q = qrun +HTLf(Q2) +ATLAS[PLB 790,108 (2019)] +JETSCAPE (MATTER+LBT) +FIG. 3. +(Color online) Diffrential cross-section of inclusive +jets in Pb+Pb collisions at √sNN = 5.02TeV with cone size +R=0.4, with a minimum track requirement of ptrack +T +> 4 GeV. +100 +200 +300 +400 +500 +600 +pjet +T [GeV] +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Ratio to ATLAS[ 1 +Nev +d2 +jet +dpjet +T dyjet] +PbPb(0-10%) sNN = 5.02TeV +anti +kt, R = 0.4, (|yjet| < 2.8) +JS(MATTER + LBT), Qsw = 2GeV +q = qrun +HTLf(Q2), +fix +s = 0.35 +FIG. 4. (Color online)Ratio of differential cross section for +inclusive jets with cone size R = 0.4 in Pb+Pb collisions at +√sNN = 5.02TeV. The ratio is taken with respect to the AT- +LAS data[cite]. + +4 +102 +2 × 102 +3 × 102 4 × 102 +6 × 102 +pjet +T [GeV] +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Rjet +AA +PbPb (0-10%), sNN = 5.02 TeV +Qsw = 2 GeV, +fix +s = 0.35 +(|yjet| < 2.8), q = qrun +HTLf(Q2) +ATLAS[PLB 790,108 (2019)] +JETSCAPE (MATTER+LBT) +FIG. 5. +(Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=0.4 with +|yjet| < 2.8 and a minimum track requirement of ptrack +T +> 4 +GeV. +103 +3 × 102 +4 × 102 +6 × 102 +pjet +T [GeV] +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Rjet +AA +PbPb (0-10%), +sNN = 5.02 TeV +anti +kt, R = 0.2, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS [JHEP 05(2021)284] +MATTER+LBT +MATTER+MARTINI +MATTER+AdS/CFT +FIG. 6. +(Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=0.2 with +|yjet| < 2 and a minimum track requirement of ptrack +T +> 4 +GeV. The plot shows the comparison between the models by +red, yellow and cyan markers for (MATTER+LBT), (MAT- +TER+MARTINI) and (MATTER+AdS/CFT), respectively. +The predictions are in comparison with CMS data shown with +blue marker. +clusive jet-RAA is calculated for three models, MAT- +TER coupled with LBT (which is our default), MAT- +TER coupled with MARTINI and MATTER coupled +with AdS/CFT, the simulations are altogether staged in +contrast with the experimental data. +The above calculations for different models are ex- +tended and fig. 6 shows the jet-RAA for |yjet| < 2 and jet +cone radius R=0.2, in comparison with the experimen- +tal data from the CMS Collaboration [20]. Similarly, the +jet-RAA is plotted for R=0.4, R=0.6, R=0.8 and R=1.0 +in the fig. 7, fig. 8, fig. 9 and fig. 10 respectively. The +103 +4 × 102 +5 × 102 +6 × 102 +7 × 102 8 × 102 9 × 102 +pjet +T [GeV] +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Rjet +AA +PbPb (0-10%), +sNN = 5.02 TeV +anti +kt, R = 0.4, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS [JHEP 05(2021)284] +MATTER+LBT +MATTER+MARTINI +MATTER+AdS/CFT +FIG. 7. +(Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=0.4 with +|yjet| < 2 and a minimum track requirement of ptrack +T +> 4 +GeV. The plot shows the comparison between the models by +red, yellow and cyan markers for (MATTER+LBT), (MAT- +TER+MARTINI) and (MATTER+AdS/CFT), respectively. +The predictions are in comparison with CMS data shown with +blue marker. +103 +4 × 102 +5 × 102 +6 × 102 +7 × 102 8 × 102 9 × 102 +pjet +T [GeV] +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Rjet +AA +PbPb(0-10%) sNN = 5.02TeV +anti +kt, R = 0.6, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2GeV +CMS [JHEP 05(2021)284] +MATTER+LBT +MATTER+MARTINI +MATTER+AdS/CFT +FIG. 8. +(Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=0.6 with +|yjet| < 2 and a minimum track requirement of ptrack +T +> 4 +GeV. The plot shows the comparison between the models by +red, yellow and cyan markers for (MATTER+LBT), (MAT- +TER+MARTINI) and (MATTER+AdS/CFT), respectively. +The predictions are in comparison with CMS data shown with +blue marker. +predictions made by the JETSCAPE are consistent even +as we move to larger area jet cones. The radiative energy +loss in AdS/CFT is more dominant than the elastic jet +energy loss compared to LBT and MARTINI. Since, the +effect of the recoil partons in LBT on the total energy loss +is not very significant. It is very articulate that there is +an appreciable reduction in net elastic and radiative jet + +5 +103 +4 × 102 +5 × 102 +6 × 102 +7 × 102 8 × 102 9 × 102 +pjet +T [GeV] +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Rjet +AA +PbPb (0-10%) +sNN = 5.02 TeV +anti +kt, R = 0.8, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS [JHEP 05(2021)284] +MATTER+LBT +MATTER+MARTINI +MATTER+AdS/CFT +FIG. 9. +(Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=0.8 with +|yjet| < 2 and a minimum track requirement of ptrack +T +> 4 +GeV. The plot shows the comparison between the models by +red, yellow and cyan markers for (MATTER+LBT), (MAT- +TER+MARTINI) and (MATTER+AdS/CFT), respectively. +The predictions are in comparison with CMS data shown with +blue marker. +103 +5 × 102 +6 × 102 +7 × 102 +8 × 102 +9 × 102 +pjet +T [GeV] +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Rjet +AA +PbPb (0-10%) +sNN = 5.02 TeV +anti +kt, R = 1.0, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS [JHEP 05(2021)284] +MATTER+LBT +MATTER+MARTINI +MATTER+AdS/CFT +FIG. 10. (Color online)the jet-RAA as a function of jet-pT +for inclusive jets in the most central(0-10%) Pb+Pb colli- +sions at √sNN = 5.02 TeV for the jet cone radius R=1.0 with +|yjet| < 2 and a minimum track requirement of ptrack +T +> 4 +GeV. The plot shows the comparison between the models by +red, yellow and cyan markers for (MATTER+LBT), (MAT- +TER+MARTINI) and (MATTER+AdS/CFT), respectively. +The predictions are in comparison with CMS data shown with +blue marker. +energy loss when the jet cone includes the recoil partons. +With the above comparisons to both ATLAS and +CMS experimental data, the simulation results from the +JETSCAPE stand sturdy for credible studies using the +current model, which is carried out in the next section. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +jet +R +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +RR +AA/RR = 0.2 +AA +300 < pjet +T < 400 [GeV] +PbPb (0-10%), +sNN = 5.02 TeV +anti +kt, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS(0-10%) +JS (MATTER+LBT) +JS (MATTER+MARTINI) +JS (MATTER+AdS/CFT) +FIG. 11. +(Color online)(300 GeV ≤ pT,jet ≤ 400 GeV)the +double ratio (RR +AA/RR=0.2 +AA +) as a function of jet radius for in- +clusive jets in the most central(0-10%) Pb+Pb collisions at +√sNN = 5.02 TeV for different jet radii with |yjet| < 2 and a +minimum track requirement of ptrack +T +> 4 GeV. The plot shows +the comparison between the models by magenta, red and blue +markers for (MATTER+LBT), (MATTER+MARTINI) and +(MATTER+AdS/CFT), respectively. The predictions are in +comparison with CMS data shown with black marker. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +jet +R +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +RR +AA/RR = 0.2 +AA +400 < pjet +T < 500 [GeV] +PbPb (0-10%), +sNN = 5.02 TeV +anti +kt, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS(0-10%) +JS (MATTER+LBT) +JS (MATTER+MARTINI) +JS (MATTER+AdS/CFT) +FIG. 12. +(Color online)(400 GeV ≤ pT,jet ≤ 500 GeV)the +double ratio (RR +AA/RR=0.2 +AA +) as a function of jet radius for in- +clusive jets in the most central(0-10%) Pb+Pb collisions at +√sNN = 5.02 TeV for different jet radii with |yjet| < 2 and a +minimum track requirement of ptrack +T +> 4 GeV. The plot shows +the comparison between the models by magenta, red and blue +markers for (MATTER+LBT), (MATTER+MARTINI) and +(MATTER+AdS/CFT), respectively. The predictions are in +comparison with CMS data shown with black marker. + +6 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +jet +R +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +RR +AA/RR = 0.2 +AA +500 < pjet +T < 1000 [GeV] +PbPb (0-10%), +sNN = 5.02 TeV +anti +kt, (|yjet| < 2) +q = qrun +HTLf(Q2), +fix +s = 0.35, Qsw = 2 GeV +CMS (0-10%) +JS (MATTER+LBT) +JS (MATTER+MARTINI) +JS (MATTER+AdS/CFT) +FIG. 13. (Color online)(500 GeV ≤ pT,jet ≤ 1 TeV)the dou- +ble ratio (RR +AA/RR=0.2 +AA +) as a function of jet radius for in- +clusive jets in the most central(0-10%) Pb+Pb collisions at +√sNN = 5.02 TeV for different jet radii with |yjet| < 2 and a +minimum track requirement of ptrack +T +> 4 GeV. The plot shows +the comparison between the models by magenta, red and blue +markers for (MATTER+LBT), (MATTER+MARTINI) and +(MATTER+AdS/CFT), respectively. The predictions are in +comparison with CMS data shown with black marker. +B. +jet-pT and jet radius dependence of the RAA +In this section we emphasise on the the significant con- +tribution from the hydrodynamic medium response al- +ready highlighted in previous studies [8, 38], which is ob- +served in the jet-RAA as a function of jet radius. Through +radius dependent studies, we hope to develop a clear pic- +ture of the parts played by radiation and collisions in +energy loss. +In a recent study by the CMS Collaboration [20], where +they have compared predictions from quenched jet event +generators and theoretical models used to replicate rela- +tivistic heavy ion collisions to the experimental data for +the jet-RAA. They arrive at the conclusion that although +most state of the art models have progressed but sig- +nificant uncertainty remains for the large area jet data. +The large jet radius R also implies that the jet retains a +significant proportion of the extensively distributed mo- +mentum and energy deposited in the plasma. Since, the +JETSCAPE framework has pioneered the realisation of +multi-stage approach for modular based energy loss, the +motivation here is to challenge the JETSCAPE model +that has so far met our expectations in describing the +variety of data observed and to explore the boundaries +of the current model. +This quest is executed by calculating the double ra- +tio (RR +AA/RR=0.2 +AA +) and plotting as a function of jet ra- +dius which are in comparison with the data for the most +central (0-10%) Pb-Pb collisions over a range of jet-pT +from 200 GeV up to 1 TeV. The plots are further sub- +categorized by three jet-pT bins. +For 300 GeV ≤ pT,jet ≤ 400 GeV, fig x shows the pre- +dictions by the different combination of energy loss mod- +els and we see a consistent trend for all three models i.e. +(MATTER+LBT), (MATTER+MARTINI) and (MAT- +TER+AdS/CFT), the results are within the statistical +limits. +For 400 GeV ≤ pT,jet ≤ 500 GeV, we observe in fig. +y, that (MATTER+LBT) is in fine agreement with +the data while both (MATTER+MARTINI) and (MAT- +TER+AdS/CFT), tend to slightly over-predict the jet- +RAA(≤ 10%). +In +the +extreme +region +i.e. +for +500 GeV ≤ pT,jet ≤ 1 TeV, all the models significantly +over-predict the jet-RAA, +with the best description +provided by the (MATTER+LBT) model(≤ 10%). All +the models show saturation around jet radius R=0.8 +and R=1.0, which fits well for a realistic approach. +Altogether, the JETSCAPE framework nicely describes +the full evolution of the parton shower by adopting a +virtuality based multi-stage approach for energy loss. +We observe that as the jet is scattered in the medium, +the final state partons of the medium interacting with +the jet are also considered as a constituent of the jet. +As more and more gluons fall into the larger jet cone +and prohibit from contributing to the jet energy loss. +Therefore, energy lost by the jets is partially gained as +the area of jet cone increases. +III. +CONCLUSION +In this paper, we present the first comparisons of the +jet–RAA predictions from the JETSCAPE framework us- +ing the (2+1)D MUSIC model for viscous hydrodynamic +evolution to the ATLAS data for Pb+Pb collisions at +√sNN = 5.02 TeV in high jet transverse momentum in- +terval 100 GeV ≤ pT,jet ≤ 1 TeV for anti-kT jets of ra- +dius R=0.4. The above results put us in a strong stance +to conclude that the MUSIC model is adequate and also +successful in speculating the experimental observations +even at higher jet-pT for the most central collisions as +well as mid-central collisions. +This work also elucidates the predictions made by low +virtuality based evolution models like MARTINI and +AdS/CFT in a hydrodynamic medium generated by the +MUSIC. We observe an overall similar trend anticipa- +tions as compared to the unfolding in other hydrody- +namic models like (2+1)D VISHNU [39]. +We advance the current JETSCAPE calculations and +compare with the data of wider jet cones ranging from +R=0.2 to R=1.0 recorded at the CMS detector for +Pb+Pb collisions at √sNN = 5.02 TeV in high jet trans- +verse momentum interval 200 GeV ≤ pT,jet ≤ 1 TeV for +anti-kT jets of radii R=0.2,0.4,0.6,0.8 and 1.0. 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Commun. 199, +61 (2016), arXiv:1409.8164 [nucl-th]. + diff --git a/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/load_file.txt b/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0be0ddda215de9bd5ccfca3a2aa98edb8745e7cd --- /dev/null +++ b/pdFKT4oBgHgl3EQfyi7m/content/tmp_files/load_file.txt @@ -0,0 +1,662 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf,len=661 +page_content='A modular perspective to the jet suppression from a small to large radius in very high transverse momentum jets Om Shahi,1 Vaishnavi Desai,2 and Prabhakar Palni2, ∗ 1Department of Physics, BITS Pilani, Goa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 2School of Physical & Applied Sciences, Goa University, Goa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' In this work, we extend the scope of the JETSCAPE Framework to cover the jet radius(R) dependence of the jet Nuclear Modification factor (RAA) for broader area jet cones, going all the way up to R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Our primary focus has been the in-depth analysis of the high-pT inclusive jets and the quenching effects observed in the Quark-gluon plasma formed in the Pb-Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the most-central (0-10%) collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Nuclear modification factor (RAA) is calculated for inclusive jets to compare with the experimental results collected at the ATLAS and the CMS detectors in the jet-pT range from 100 GeV up to 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The results predicted by the JETSCAPE are consistent in the high pT range as well as for extreme jet cone sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We also calculate the double ratio (RR AA/RR=small AA ) as a function of jet radius and jet-pT , where the observations are well described by the JETSCAPE framework which is based on the hydrodynamic multi-stage evolution of the parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The calculations are then replicated for different low-virtuality based evolution models like the MARTINI and the AdS/CFT, which is followed by a rigorous comparison between the predictions from the former model combinations to the measurements at the CMS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' INTRODUCTION The extremely hot and dense conditions created at the starting of big bang which we now understand as the soup of de-confined state of the partons, the Quark-Gluon Plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The QGP is of such great interest that, to study the properties of this state of matter, scientists have spent decades in building up the equipment which could recre- ate this extremely dense soupy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The Relativis- tic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) conduct Heavy-ion collisions where the QGP is created for very short instants of time, the par- ton shower propagation and modification is greatly influ- enced by the quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The high-pT jets pro- duced in these heavy ion collisions undergo strong yield suppression and medium modification which are together referred to as jet quenching phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='Therefore, we study the jet modification in nucleus-nucleus collisions relative to proton-proton collisions to probe the prop- erties of the QGP via constraints from model-to-data comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' [1–7] The jet Nuclear Modification Factor (RAA) has been one of the most important and the prime observable to study the properties of the Quark-gluon Plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The measurement of Nuclear Modification Fac- tor for jets as well as charged particles has revealed many important characteristics of the Quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' It provides a strong confirmation of the interaction of par- tons with the deconfined plasma, the respective medium modifications and the eventual hydrodynamization with the medium [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Since there are many conclusive studies based on jet-RAA [9–13], our effort here is to push the limits of what we hope to learn about the QGP from this observable with the available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Along with these measurements, we can reconstruct in- ∗ Corresponding author: prabhakar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='palni@unigoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='in clusive jet spectra for p-p and Pb-Pb by varying distance parameters R in the anti-kT algorithm, which is realized in the FASTJET software package [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The inclusive jet spectrum is of significant interest because the sensitivity of hadronization effects is far less compared to the observ- ables involving individual final-state hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Here, the area of the reconstructed jet cone is defined by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Thus, by varying R, the reconstructed jet will include differ- ent proportions of energy from the medium response and the quenched jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We have also observed new sensitivity to QGP properties and underlying jet quenching mech- anism in a study for jet yield suppression versus R [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Specifically, different dependences of the jet suppression on R, are predicted by generators and theoretical models based on anti-de Sitter/conformal field theory correspon- dence [16] and perturbative QCD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We take off by calculating the Jet (RAA) for the Pb- Pb collisions in SECTION II and compare with the ex- perimental data from the ATLAS as well as the CMS detector for a robust test of our configuration and the overall JETSCAPE Framework (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='1) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Our calculations include the (2+1)D MUSIC [19] model for hydrodynamics which is ideal for studying many aspects of heavy ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Most notably, our calculations cover the high-pT range of jets that is up to 1 TeV, which enables us to probe the medium at much shorter distance scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We now exploit the advantages that the JETSCAPE framework offers in SECTION II A, that is the coupling of several different energy loss modules such as MARTINI and AdS/CFT with the MATTER module to explore the quenching effects in a multi-stage manner which gives us an insight into the interaction and energy loss mecha- nism in the medium with respect to the virtuality of the partonic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Comparing the experimental data with the trends from these different models which handle the low virtuality phase allows us to develop a lucid understand- ing of the physics governing the above models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='11908v1 [hep-ph] 27 Jan 2023 2 In SECTION II B, we proceed with concrete results from the above combinations of several successful mod- els towards the jet radius dependence studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' An impor- tant measurement done by the CMS collaboration [20] re- cently, was the jet-RAA for Pb-Pb collisions at the LHC and recording the data for jet cone area covering the radii from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' This analysis gives us an intricate un- derstanding of the behaviour and interaction strength of the high-pT inclusive collimated jets in proximity to the jet axis, for R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 and the energy distribution around the cone for values of radii up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The current JETSCAPE framework is based on calculations of perturbative quan- tum chromodynamics and evolution in a dense medium, thus allowing us to go up to a radius of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We also plot and report the Double Ratio for jet-pT dependence and jet radius dependence to provide a vivid picture of the energy transactions with the Quark-gluon plasma as the jet cone size increases and also discuss the trends that the JETSCAPE framework predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We conclude this work in SECTION III, with a concise account of the current JETSCAPE model’s potential to explain the jet-RAA for larger area jet cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We also shed light on the ability of the distinct combination of modules to describe the jet-RAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' SIMULATION WITH MULTI-STAGE ENERGY LOSS APPROACH IN JETSCAPE The JETSCAPE framework provides an ideal envi- ronment for carrying out multi-stage energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The hard scattering is generated by Pythia 8 [21] with initial state radiation (ISR) and multiparton interaction (MPI) turned on, and final state radiation (FSR) turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' For the event wise simulations, the TRENTo model [22] sets up the initial conditions and the viscous hydrodynamic evolution is described by the (2+1)D MUSIC [19] model followed by Cooper-Frye prescription [23, 24]which con- verts the fluid cells to hadrons on an isothermal hyper- surface at TSW= 151 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The jet energy loss induced by scattering is calculated in succession of two stages: MATTER [25, 26] takes care of the highly virtual phase (the first stage) while the low virtuality phase (the second stage), is handled concurrently by the LBT model [27– 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We have also employed the MARTINI [30–32] and the AdS/CFT model [33] in combination with the MAT- TER model to explore the low virtuality phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' the vir- tuality of the parton is defined as Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The parton un- dergoes energy loss in two stages, when the virtuality of the parton, Q2 > Q2 SW, where QSW is the switching virtuality, the MATTER model handles the energy loss and the parton is transferred to the LBT model once Q2 ≤ Q2 SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' In our calculations, the jet medium interac- tion includes inelastic medium-induced gluon radiation and medium recoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' An extensive account on the com- parison of the model to the existing jet-RAA is already done [34], showing that the contribution of medium re- coil is pretty significant in the modification of jet-RAA for a complete set of centrality classes ranging from the most central collisions to the peripheral collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' This version of the JETSCAPE framework [18] encompasses the modifications of a hard thermal-loop (HTL) ˆq [35] for fixed coupling, running coupling, and with a virtual- ity dependent factor that modulates the effective value of ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' It also accounts for the reduced medium-induced emis- sion in the high virtuality phase, due to coherence effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The calculations based on the hard-thermal-loop (HTL) considering weak- coupling approximation and the limits of high temperature yield a ˆq given as [28], ˆqHTL = Ca 42ζ(3) π α2 sT 3 ln � 2ET 6πT 2αfix s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (1) where Ca is the representation specific Casimir, E is the energy of the hard parton and T is the local temperature of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The Coherence effects which reduce the interaction strength of the parton with the medium is also taken into consideration as the virtuality dependent mod- ulation factor f(Q2) which regulates the effective value of ˆq in the high virtuality MATTER event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The parameterization of the virtuality-dependent modulation factor is given as [34] ˆq · f ≡ ˆqrun HTLf(Q2) (2) f(Q2) = � 1+10 ln2(Q2 sw)+100 ln4(Q2 sw) 1+10 ln2(Q2)+100 ln4(Q2) Q2 > Q2 sw 1 Q2 ≤ Q2 sw , (3) here Q2 is the running virtuality of the hard parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Finally, the partons undergo colorless hadronization ac- cording to the default Lund string fragmentation from Pythia 8 [21, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The major contributions in our final jets are from hard jet shower part and the effect from the soft medium response, the latter is calculated via the Cooper-Frye formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We reconstruct our jets for sev- eral radius selections using the anti-kT algorithm which is realised in the FASTJET Software Package [14] and compare with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The Underlying Events (UE) are removed by implementing a minimun track re- quirement of ptrack,min T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' All the parameters in- volved in the tuning of the constituent modules follow the standard set of tunes released by the JETSCAPE Collaboration in an elaborate recent study [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' RESULTS AND ANALYSIS This work covers the collision energy: √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for two centrality classes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='the most central(0- 10%) collisions and shows a comparison with selected experimental data available from the ATLAS and the CMS Collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The energy loss depicted in our results is achieved by the coupling of MATTER with the LBT module and the secondary module (which han- dles the low virtuality phase) remains the same through- out our results until and unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 1 shows the p + p simulation results for inclusive jet spectra for 3 100 200 300 400 500 600 pjet T [GeV] 10 8 10 6 10 4 10 2 100 d2 jet dpjet T dyjet [nb/GeV] pp, s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, |yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 ATLAS [PLB 790,108 (2019)] JETSCAPE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)Diffrential cross section of inclusive jets for p+p collisions with cone size R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, with a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' p+p at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02TeV for |yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 which follow the JETSCAPE PP 19 tune [37] are then compared to the experimental data from the ATLAS [12, 13], the ratio of inclusive jet cross-section to the ATLAS data is plotted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 2, which clearly shows that the results are in accept- able range(≤ 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We report the 0-10% central Pb-Pb jet spectra for R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We also plot the ratio of differential cross section for inclusive jets compared to the data from ATLAS [13] shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Here, RAA is defined as RAA = 1 Nevent d2Njet dηjetdpjet T ��� AA d2σjet dηjetdpjet T ��� pp , (4) The inclusive jet-RAA is now calculated as the ratio of the Pb-Pb and p-p spectra plotted above, which is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 5 in comparison to the ATLAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The JETSCAPE results agree with the data quite nicely, al- though we see a 5% enhancement in the low pT region which is followed by a suppression as we ascend in the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Exploring the MARTINI and AdS/CFT models Both the MARTINI [30] and the AdS/CFT [33] mod- els are designed to handle the low virtuality phase and carry forward the energy loss once the parton is trans- ferred from the MATTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We now replace the successful LBT model with the MARTINI model, to carry out the simulations for the same settings [34] and compare the jet-RAA with the experimental data from the ATLAS Collaboration [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Similarly the calculations are done by replacing the LBT by AdS/CFT and compared with the recorded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We now present the intensive comparison between the low virtuality parton evolution models, where the in- 100 200 300 400 500 600 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 Ratio to ATLAS[ 1 Nev d2 jet dpjet T dyjet] pp, s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 (|yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8) JETSCAPE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)Diffrential cross-section of inclusive jets for p+p collisions with cone size R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, with a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 100 200 300 400 500 600 pjet T [GeV] 10 8 10 6 10 4 10 2 100 1 Nev TAA d2Njet dpjet T dyjet [nb/GeV] PbPb (0-10%), sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV Qsw = 2 GeV, fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35 (|yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8), q = qrun HTLf(Q2) ATLAS[PLB 790,108 (2019)] JETSCAPE (MATTER+LBT) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online) Diffrential cross-section of inclusive jets in Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02TeV with cone size R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, with a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 100 200 300 400 500 600 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 Ratio to ATLAS[ 1 Nev d2 jet dpjet T dyjet] PbPb(0-10%) sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, (|yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8) JS(MATTER + LBT), Qsw = 2GeV q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)Ratio of differential cross section for inclusive jets with cone size R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 in Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The ratio is taken with respect to the AT- LAS data[cite].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 4 102 2 × 102 3 × 102 4 × 102 6 × 102 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 Rjet AA PbPb (0-10%), sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV Qsw = 2 GeV, fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35 (|yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8), q = qrun HTLf(Q2) ATLAS[PLB 790,108 (2019)] JETSCAPE (MATTER+LBT) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)the jet-RAA as a function of jet-pT for inclusive jets in the most central(0-10%) Pb+Pb colli- sions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 with |yjet| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 103 3 × 102 4 × 102 6 × 102 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 Rjet AA PbPb (0-10%), sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2 GeV CMS [JHEP 05(2021)284] MATTER+LBT MATTER+MARTINI MATTER+AdS/CFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)the jet-RAA as a function of jet-pT for inclusive jets in the most central(0-10%) Pb+Pb colli- sions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 with |yjet| < 2 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plot shows the comparison between the models by red, yellow and cyan markers for (MATTER+LBT), (MAT- TER+MARTINI) and (MATTER+AdS/CFT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The predictions are in comparison with CMS data shown with blue marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' clusive jet-RAA is calculated for three models, MAT- TER coupled with LBT (which is our default), MAT- TER coupled with MARTINI and MATTER coupled with AdS/CFT, the simulations are altogether staged in contrast with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The above calculations for different models are ex- tended and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 6 shows the jet-RAA for |yjet| < 2 and jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2, in comparison with the experimen- tal data from the CMS Collaboration [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Similarly, the jet-RAA is plotted for R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 and R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 in the fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 7, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 8, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 9 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The 103 4 × 102 5 × 102 6 × 102 7 × 102 8 × 102 9 × 102 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 Rjet AA PbPb (0-10%), sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2 GeV CMS [JHEP 05(2021)284] MATTER+LBT MATTER+MARTINI MATTER+AdS/CFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)the jet-RAA as a function of jet-pT for inclusive jets in the most central(0-10%) Pb+Pb colli- sions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 with |yjet| < 2 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plot shows the comparison between the models by red, yellow and cyan markers for (MATTER+LBT), (MAT- TER+MARTINI) and (MATTER+AdS/CFT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The predictions are in comparison with CMS data shown with blue marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 103 4 × 102 5 × 102 6 × 102 7 × 102 8 × 102 9 × 102 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='00 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2GeV CMS [JHEP 05(2021)284] MATTER+LBT MATTER+MARTINI MATTER+AdS/CFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)the jet-RAA as a function of jet-pT for inclusive jets in the most central(0-10%) Pb+Pb colli- sions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 with |yjet| < 2 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plot shows the comparison between the models by red, yellow and cyan markers for (MATTER+LBT), (MAT- TER+MARTINI) and (MATTER+AdS/CFT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The predictions are in comparison with CMS data shown with blue marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' predictions made by the JETSCAPE are consistent even as we move to larger area jet cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The radiative energy loss in AdS/CFT is more dominant than the elastic jet energy loss compared to LBT and MARTINI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Since, the effect of the recoil partons in LBT on the total energy loss is not very significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' It is very articulate that there is an appreciable reduction in net elastic and radiative jet 5 103 4 × 102 5 × 102 6 × 102 7 × 102 8 × 102 9 × 102 pjet T [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='25 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2 GeV CMS [JHEP 05(2021)284] MATTER+LBT MATTER+MARTINI MATTER+AdS/CFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)the jet-RAA as a function of jet-pT for inclusive jets in the most central(0-10%) Pb+Pb colli- sions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for the jet cone radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 with |yjet| < 2 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plot shows the comparison between the models by red, yellow and cyan markers for (MATTER+LBT), (MAT- TER+MARTINI) and (MATTER+AdS/CFT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The predictions are in comparison with CMS data shown with blue marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 103 5 × 102 6 × 102 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+page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='50 Rjet AA PbPb (0-10%) sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2 GeV CMS [JHEP 05(2021)284] MATTER+LBT MATTER+MARTINI MATTER+AdS/CFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} 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blue marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' energy loss when the jet cone includes the recoil partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' With the above comparisons to both ATLAS and CMS experimental data, the simulation results from the JETSCAPE stand sturdy for credible studies using the current model, which is carried out in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 RR AA/RR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 AA 500 < pjet T < 1000 [GeV] PbPb (0-10%), sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV anti kt, (|yjet| < 2) q = qrun HTLf(Q2), fix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='35, Qsw = 2 GeV CMS (0-10%) JS (MATTER+LBT) JS (MATTER+MARTINI) JS (MATTER+AdS/CFT) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (Color online)(500 GeV ≤ pT,jet ≤ 1 TeV)the dou- ble ratio (RR AA/RR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 AA ) as a function of jet radius for in- clusive jets in the most central(0-10%) Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV for different jet radii with |yjet| < 2 and a minimum track requirement of ptrack T > 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plot shows the comparison between the models by magenta, red and blue markers for (MATTER+LBT), (MATTER+MARTINI) and (MATTER+AdS/CFT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The predictions are in comparison with CMS data shown with black marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' jet-pT and jet radius dependence of the RAA In this section we emphasise on the the significant con- tribution from the hydrodynamic medium response al- ready highlighted in previous studies [8, 38], which is ob- served in the jet-RAA as a function of jet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Through radius dependent studies, we hope to develop a clear pic- ture of the parts played by radiation and collisions in energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' In a recent study by the CMS Collaboration [20], where they have compared predictions from quenched jet event generators and theoretical models used to replicate rela- tivistic heavy ion collisions to the experimental data for the jet-RAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' They arrive at the conclusion that although most state of the art models have progressed but sig- nificant uncertainty remains for the large area jet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The large jet radius R also implies that the jet retains a significant proportion of the extensively distributed mo- mentum and energy deposited in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Since, the JETSCAPE framework has pioneered the realisation of multi-stage approach for modular based energy loss, the motivation here is to challenge the JETSCAPE model that has so far met our expectations in describing the variety of data observed and to explore the boundaries of the current model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' This quest is executed by calculating the double ra- tio (RR AA/RR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 AA ) and plotting as a function of jet ra- dius which are in comparison with the data for the most central (0-10%) Pb-Pb collisions over a range of jet-pT from 200 GeV up to 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The plots are further sub- categorized by three jet-pT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' For 300 GeV ≤ pT,jet ≤ 400 GeV, fig x shows the pre- dictions by the different combination of energy loss mod- els and we see a consistent trend for all three models i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' (MATTER+LBT), (MATTER+MARTINI) and (MAT- TER+AdS/CFT), the results are within the statistical limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' For 400 GeV ≤ pT,jet ≤ 500 GeV, we observe in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' y, that (MATTER+LBT) is in fine agreement with the data while both (MATTER+MARTINI) and (MAT- TER+AdS/CFT), tend to slightly over-predict the jet- RAA(≤ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' In the extreme region i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' for 500 GeV ≤ pT,jet ≤ 1 TeV, all the models significantly over-predict the jet-RAA, with the best description provided by the (MATTER+LBT) model(≤ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' All the models show saturation around jet radius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 and R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0, which fits well for a realistic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Altogether, the JETSCAPE framework nicely describes the full evolution of the parton shower by adopting a virtuality based multi-stage approach for energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We observe that as the jet is scattered in the medium, the final state partons of the medium interacting with the jet are also considered as a constituent of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' As more and more gluons fall into the larger jet cone and prohibit from contributing to the jet energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Therefore, energy lost by the jets is partially gained as the area of jet cone increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' CONCLUSION In this paper, we present the first comparisons of the jet–RAA predictions from the JETSCAPE framework us- ing the (2+1)D MUSIC model for viscous hydrodynamic evolution to the ATLAS data for Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV in high jet transverse momentum in- terval 100 GeV ≤ pT,jet ≤ 1 TeV for anti-kT jets of ra- dius R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The above results put us in a strong stance to conclude that the MUSIC model is adequate and also successful in speculating the experimental observations even at higher jet-pT for the most central collisions as well as mid-central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' This work also elucidates the predictions made by low virtuality based evolution models like MARTINI and AdS/CFT in a hydrodynamic medium generated by the MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We observe an overall similar trend anticipa- tions as compared to the unfolding in other hydrody- namic models like (2+1)D VISHNU [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' We advance the current JETSCAPE calculations and compare with the data of wider jet cones ranging from R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2 to R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0 recorded at the CMS detector for Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='02 TeV in high jet trans- verse momentum interval 200 GeV ≤ pT,jet ≤ 1 TeV for anti-kT jets of radii R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' Although the JETSCAPE framework is still under improvisation to define the jet medium interactions at wide angles, this 7 work highlights the current standing of the model to de- scribe the energy loss and medium response phenomenon for broad area jet cones and the JETSCAPE predictions are well within the statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to acknowledge Yasuki Tachibana and Chun Shen for their useful discussion and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfyi7m/content/2301.11908v1.pdf'} +page_content=' The authors would also 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Wulff1,2, Dominik S. Meier1, and and Rui Mata1 +1University of Basel +2Max Planck Institute for Human Development +A number of labeling systems based on text have been +proposed to help monitor work on the United Nations +(UN) Sustainable Development Goals (SDGs). Here, we +present a systematic comparison of systems using a va- +riety of text sources and show that systems differ con- +siderably in their specificity (i.e., true-positive rate) and +sensitivity (i.e., true-negative rate), have systematic bi- +ases (e.g., are more sensitive to specific SDGs relative to +others), and are susceptible to the type and amount of +text analyzed. +We then show that an ensemble model +that pools labeling systems alleviates some of these lim- +itations, exceeding the labeling performance of all cur- +rently available systems. We conclude that researchers +and policymakers should care about the choice of label- +ing system and that ensemble methods should be favored +when drawing conclusions about the absolute and rela- +tive prevalence of work on the SDGs based on automated +methods. +Monitoring work on the SDGs is crucial for their advance- +ment and one promising approach is to screen the increasing +amount of digitally available text using automated, natural +language processing methods. This approach has taken hold, +for example, in scientometric efforts that monitor the SDGs +in academic publications (e.g., Aurora Universities Network +(AUR), 2020; Bautista, 2019; Duran-Silva et al., 2019; Jaya- +balasingham et al., 2021; Sustainable Development Solu- +tions Network (SDSN), 2021). One such system has iden- +tified millions of SDG-related academic publications and is +already used by the Times Higher Education Impact Rank- +ings to rank over 1,000 universities worldwide according to +their SDG-related outputs. This example illustrates the im- +portant role that SDG labeling from text can play in academic +research and funding policy (Smith et al., 2021). +The increased and widespread use of SDG labeling sys- +Dirk U. Wulff +https://orcid.org/0000-0002-4008-8022 Do- +minik S. Meier +https://orcid.org/0000-0002-3999-1388 Rui Mata +https://orcid.org/0000-0002-1679-906X +Correspondence concerning this article should be addressed +to Dirk U. Wulff, Department of Psychology, University of +Basel, Missionsstrasse 60-62, 4055 Basel, Switzerland. +E-mail: +dirk.wulff@gmail.com +tems should be accompanied by efforts to validate the differ- +ent approaches and ensure that their predictions accurately +reflect the SDGs. However, systematic and comprehensive +evaluations of the accuracy of these labeling systems are +largely lacking. Crucially, the few existing results indicate +some striking differences in the predictions of the different +labeling systems (e.g., Armitage et al., 2020; Pukelis et al., +2020; Schmidt & Vanderfeesten, 2021). +Three key reasons prevent rigorous evaluation of the dif- +ferent SDG labeling systems. First, most labeling systems +have been developed for and tested with specific proprietary +citation databases thus limiting their portability to other text +sources. Second, there has been a paucity of publicly avail- +able data that can be used to validate the predictions of differ- +ent approaches. Third, systems have often been analyzed in +isolation without systematic comparisons involving various +performance metrics or assessment of biases. +In this paper, we aimed to help overcome these limitations +by relying on our recently developed open-source R package, +text2sdg ((text2sdg.io) Meier et al., 2021), which can be used +with any text source, and collating various labeled and non- +labeled data sources to provide a comparison of seven exist- +ing SDG labeling systems using a comprehensive set of per- +formance metrics. We provide a comparative evaluation of +seven labeling systems; namely, the Aurora (Aurora Univer- +sities Network (AUR), 2020), Elsevier (Jayabalasingham et +al., 2021), SIRIS (Duran-Silva et al., 2019), Auckland (Mu & +Wang, 2022), SDGO (Bautista, 2019), and SDSN (Sustain- +able Development Solutions Network; Sustainable Devel- +opment Solutions Network (SDSN), 2021) systems imple- +mented in the text2sdg R package and, additionally, OSDG.ai +(Pukelis et al., 2022), a publicly available tool. These sys- +tems account for a majority of, albeit not all (cf. Fane et al., +2022), systems currently available for automated detection of +SDGs from text. Finally, we also aimed to assess the poten- +tial of using an ensemble approach that integrates several of +the existing systems. Ensemble models can potentially im- +prove accuracy and generalizability by considering the pre- +dictions of multiple models, each of which may have differ- +ent biases, resulting in a more balanced and representative +prediction. +Our contribution is structured as follows. We first intro- +duce and use three labeled data sets containing hand-coded +arXiv:2301.11353v1 [cs.CL] 25 Jan 2023 + +2 +WULFF, MEIER, MATA +labels to evaluate the categorizations of the seven automated +labeling systems against those of human experts. The data +sets cover different text sources, including titles and abstracts +of academic publications, as well as news articles, thus sig- +nificantly increasing the scope of sources considered in past +work (e.g., Armitage et al., 2020; Pukelis et al., 2020). Cru- +cially, we use a number of different metrics to compare the +different labeling systems (e.g., sensitivity, specificity, over- +all accuracy) thus providing a comprehensive assessment of +the systems’ strengths and weaknesses. Second, we com- +pare the predictions of the SDG labeling systems across data +sets to reveal SDG-specific biases; that is, whether differ- +ent systems tend to make differential categorizations for spe- +cific SDGs. The potential for bias is particularly important +concerning the application of automated labeling systems +to make relative statements about the presence of specific +SDGs. For example, the use of biased systems in the detec- +tion of some SDGs could lead to an incorrect assessment of +investment in some domains (e.g., health) relative to others +(e.g., education). Third, we introduce and leverage several +novel unlabeled data sets (e.g., Disneyland reviews, cook- +ing recipes, math lectures, random text) to better understand +the labeling systems’ susceptibility to the type and amount +of text analyzed. Fourth, and finally, we explore whether +ensemble-modeling approaches can address some of the po- +tential limitations of existing SDG labeling systems. +Results +In the following subsections, we first compare the seven +labeling systems on a variety of metrics on three labeled data +sets covering texts from academic publications and news ar- +ticles. Second, we assess to what extent the different label- +ing systems show biases concerning different SDGs. Third, +we assess the susceptibility of the different labeling systems +to produce false positives as a function of the length of the +text source in novel unlabeled data sets. Fourth, and finally, +we assess the potential of ensemble models that integrate the +different labeling systems to address potential limitations of +individual labeling systems. +SDG labeling systems differ in their sensitivity–specificity +trade-offs +Our first set of analyses consisted of comparing seven la- +beling systems to generate predicted labels for documents +from three labeled data sets. All three data sets provide hu- +man judges’ ratings of SDGs for different document types: +the first data set consists of over 10,000 titles of academic +publications (titles; Vanderfeesten et al., 2020), the second +data set consists of over 30,000 abstracts of academic pub- +lications (abstracts; OSDG et al., 2021), and the third data +set consists of over 9,000 news articles scraped, with permis- +sion, from the SDG knowledge hub website (news articles; +https://sdg.iisd.org Wulff & Meier, 2023). The data sets +differ in a number of respects, most notably in the number +of words per document, with titles, abstracts, and news arti- +cles being composed of 18, 90, and 673 words per document +on average, respectively, and in the number SDGs they were +evaluated for, with titles and abstracts having each been eval- +uated for only one of the SDGs and news articles having been +evaluated for all 17 SDGs. +Figure 1 presents the main classification results separately +for the three data sets (titles, abstracts, news articles), with +panel A showing a breakdown of SDG labels per document. +To better assess the differences between each data set, it is +helpful to look initially at the first column of each subpanel +in Figure 1A, which contains information on the expert rat- +ings for each data set. For the titles data set, a large majority +of documents were judged to contain one SDG (63%). In the +abstracts data set, a majority of documents were assigned +an SDG (80.2%). In turn, in the news articles data set, all +of the documents (100%) were assigned one or more SDGs. +As a whole, across data sets, only a minority of documents +have not been assigned any SDG by the experts, which, as +we discuss below, can limit the validation of SDG labeling +systems. +The remaining columns in Figure 1A depict the labels as- +signed to each document by the labeling systems and, there- +fore, allow a first comparison between systems and the hu- +man experts. +Visual inspection suggests that all systems +showed reasonable accuracy in terms of recovering the "true" +SDG labels assigned by experts. This is illustrated by the +similar pattern of colored stripes across experts and the seven +labeling systems. Nevertheless, the systems showed consid- +erable differences in their ability to detect SDGs, as illus- +trated by the relative intensity of the colored stripes and gray +stripes, respectively. The results below quantify these differ- +ences using common performance metrics. +We compared the seven labeling systems quantitatively +using a number of common metrics typically used in catego- +rization problems (i.e., sensitivity, specificity, accuracy, F1 +score). Sensitivity measures a system’s ability to correctly +identify true SDG text sources as positive whereas specificity +measures a system’s ability to identify non-SDG sources as +negative. Ideally, SDG labeling systems should be both sen- +sitive and specific but there is often a trade-off between the +two. Considering both metrics can help compare the systems +and their criteria for dealing with trade-offs between the two. +In practice, we do so by visually comparing such metrics in +an ROC (receiver operating characteristic) space as well as +considering general measures of categorization performance, +such as overall accuracy and a composite measure of sensi- +tivity and specificity (i.e., F1 score) in the different data sets. +As can be seen in Figure 1B, the systems differ substan- +tially in their trade-off between sensitivity and specificity or, +in other words, in how conservatively they assign SDGs. +This is shown by the fact that the systems’ performance— + +SDG LABELING SYSTEMS +3 +Figure 1 +Performance of SDG labeling systems. +Panel A shows the SDG labels assigned by experts and seven labeling systems +for three data sets consisting of titles of research articles (Titles), abstracts of research articles (Abstracts), and online +news articles (News Articles), respectively. The colored lines represent the assigned SDG by either experts or SDG labeling +system and gray lines represent missing labels (for experts) or false negatives (for SDG labeling systems). Panel B shows the +performance of the seven systems across the three data sets by plotting the systems’ sensitivity against 1-specificity forming +an ROC (receiver operating characteristic) plot. Lines in the background show several levels of balanced accuracy. Panel C +and D show the performance of the seven systems across the three data sets in terms of accuracy and the F1 score. +in terms of sensitivity and specificity—varies mostly along +the diagonal of the ROC plot. We also observe differences in +overall accuracy but these are not stable across data sets (Fig- +ure 1C), which is due to the different levels of conservatism +of the different labeling systems: More conservative systems +(e.g., Elsevier) outperform liberal systems for news articles, +in which many labels are negative, and vice versa for ab- +stracts and titles, where almost all labels are positive. These +patterns hold for the alternative measure of performance, the +F1 score (Figure 1D). +All in all, these results point to labeling systems being +differently conservative; that is, they solve the sensitivity– +specificity trade-off differently. +As a consequence, label- +ing systems do best in different data sets and it is difficult +to identify a single best-performing model. The results fur- +ther point to a limitation of currently available data sets that +include only a small proportion of non-SDG-related docu- +ments, which introduces difficulties in assessing systems’ +susceptibility to producing false positives. We return to this +point below when we introduce novel synthetic data sets. + +Expert +Aurora +Elsevier +A +SIRIS +Auckland +OSDG +SDSN +SDGO +B +C +1 +0.8 +Abstracts +0.9 +·SDSN +SDGO +Accurac +Titles +(news articles) +0.6 +News articles +(news articles +0.8 +SDSN +0.4 +SDGO +(abstracts) +(abstracts +0.7 +0.2 +Auckland +SIRIS +(news articles) +itivity +0.6 +0 +(news articles) +SDSN +Elsevier +SIRIS +S Auckland OSDG +Aurora +SDSN +SDGO +C +Auckland +(titles) +SDGO +Alpora) +ensi +0.5 +OSDG (ites) +D +(abstracts) +Elsevier +S +(news articles) +0.4 +OSDG +Abstracts +Auckland +SIRIS +0.8 +(abstracts) OSDG +(titles) +0.3 +Titles +score +Elsevier +0.6 +SIRIS +News articles +Elsevier +0.4 +(abstracts) +Aurora +0.1 +(titles) +0.2 +0 +0 +0 +0.1 +0.2 +0.3 +0.5 +0.6 +0.7 +0.4 +AuroraElsevier + SIRIS Auckland OSDG +SDSN +SDGO +1 - Specificity4 +WULFF, MEIER, MATA +Biases in SDG labeling systems distort SDG profiles +Biases in SDG labeling systems can lead to inaccurate +representation of the prevalence and importance of different +SDGs. If a system is more sensitive to certain SDGs than +others, it will overestimate the prevalence of those SDGs +and underestimate the prevalence of the others. +This re- +sults in a misleading picture of the work being done to ad- +dress the SDGs. +Biased systems may also create an un- +fair advantage or disadvantage for certain organizations or +groups. For example, if a method is more sensitive to certain +SDGs than others, organizations may aim to portray them- +selves as focusing on those SDGs by relying on such sys- +tems. Finally, biased systems may create confusion or mis- +trust among stakeholders. If different methods produce sig- +nificantly different results, it may be difficult for stakeholders +to know which results to trust. This can lead to confusion or +mistrust in the results, which could ultimately undermine the +credibility of the work being done to address the SDGs. +We estimated SDG-specific biases by comparing the rela- +tive frequency of SDGs between predicted and observed la- +bels. Specifically, we calculate bias = +predicted−observed +observed +. We +then evaluate the profile bias of a given system by correlating +the biases across data sets. We also compare the SDG profiles +obtained from the seven labeling systems to the SDG profile +of experts. This analysis amounts to assessing the similarity +between the profile across SDGs identified by experts and the +profile identified by each of the labeling systems. +Figure 2A shows the biases of the different labeling sys- +tems. Visual inspection suggests that some biases appear sys- +tematically across data sets. For example, considering the +Aurora labeling system, the pattern suggests consistent un- +derestimation of SDGs 2, 6, 7, 8, 9, and 10 across data sets +but overestimation of SDG 13. For Elsevier, one observes +consistent underestimation of SDGs 4, 6, 9, 12, 14, and 15 +but overestimation of SDGs 3 and 16. To quantify the sys- +tems’ profile biases we correlated the SDG biases between +titles and news articles and between abstracts and news arti- +cles. We did not consider the correlation between titles and +abstracts because of a moderate correlation in expert profiles +for these two sources. +The strongest average profile bias +was observed for Elsevier (¯r = 0.72), followed by OSDG +(¯r = 0.41), Aurora (¯r = 0.38), SDGO (¯r = 0.31), SDSN +(¯r = 0.2), Auckland (¯r = 0.08), and SIRIS (¯r = −.05). +Figure 2B shows that these biases imply substantial differ- +ences in SDG profiles derived from the different labeling sys- +tems relative to the one derived from experts. This difference +is most pronounced for SDG 3 (Good health and well-being). +A number of systems overestimate the relative frequency of +SDG 3, with Elsevier giving SDG 3 twice as much weight +(.26) as experts (.13). In turn, SDGs 9 (Industry, Innovation, +and Infrastructure) and 10 (Reduced inequalities) tend to re- +ceive less weight than assigned by experts. To further evalu- +ate profile fidelity, we calculated the Spearman’s rank corre- +lation coefficient between expert and system relative frequen- +cies separately for the three data sets. We found Auckland to +have the highest average correlation (¯r = .76), followed by +SDGO (¯r = .72), Elsevier (¯r = .70), SIRIS (¯r = .67), Aurora +(¯r = .57), and, finally, OSDG (¯r = .54). +To summarize, we find that systems appear to have differ- +ent biases, with a number of these underestimating or over- +estimating the presence of specific SDGs relative to experts. +These biases can be important for the relative assessment of +work on the SDGs and emphasize that the labeling systems +should not be used interchangeably. +SDG labeling systems can produce many false positives +when applied to large text sources +Existing validation data sets—such as those used above +to test the accuracy of SDG labeling systems—may not ac- +curately reflect the performance of SDG labeling systems in +many real-world applications that involve larger samples of +text. As mentioned above, existing data sets only include a +small proportion of documents unrelated to the SDGs, which +can lead to an overestimation of the accuracy of the labeling +systems relative to applications in which labeling systems +are tested on a diverse range of documents that can poten- +tially contain many false positives. To address this issue, we +conducted evaluations of the labeling systems using data sets +that are ostensibly unrelated to SDGs in order to better un- +derstand the systems’ tendency to produce false positives; in +particular, as a function of text length. +In this analysis, we used existing and synthetic data sets; +specifically, three natural language data sets consisting of +Disneyland reviews (N = 42, 656), cooking recipes (N = +82, 245), and math lectures (N = 860). In addition, we gen- +erated synthetic texts by sampling from a word frequency list +derived from Wikipedia and generating documents contain- +ing 10, 100, 1,000, and 10,000 randomly sampled words. +Figure 3 presents the results of the SDG labeling systems +for the various data sets. The plot shows the number of SDGs +identified in each document by each system, plotted against +the average length of the documents in the respective data +set. The plot includes data from the novel data sets (those +likely not related to the SDGs), as well as the three expert- +labeled data sets. We use the number of SDGs per document +as a proxy for the false-positive rate. Consequently, by com- +paring the number of SDGs identified by the systems across +different data sets and document lengths, we hope to under- +stand the systems’ susceptibility to producing false positives +and identify any patterns or trends in their accuracy as a func- +tion of document length. +Several noteworthy results emerge. First, all systems pro- +duce false positives for all types of data sets. Second, the +tendency of systems to produce false positives is in line with +their level of conservatism, which we discussed above. For +example, Aurora and Elsevier appear to be very conserva- + +SDG LABELING SYSTEMS +5 +Figure 2 +Biases in SDG classifications. +Panel A illustrates the biases of SDG labeling systems in classifying each of the 17 +SDGs. Biases are calculated as the difference between the observed and predicted SDG proportion for a given data set +divided by the observed proportion. A positive bias means that a given SDG was assigned more often than its relative +frequency in expert judgments and vice versa for a negative bias. +The three vertical stripes for each SDG and system +correspond to the titles, abstracts, and news articles data sets. Panel B shows the profile of expert (red) and system labels +generated by averaging the relative frequencies of SDGs across data sets. The circles and numbers highlight the three most +prevalent SDGs per system. +tive in identifying SDGs, leading to an overall low false- +positive rate. Third, across all systems the tendency to de- +tect SDGs increases considerably when the length of the +texts increases. Specifically, whereas systems produced be- +tween .0003 (Aurora) and .144 (SDGO) for synthetic texts of +length 10, systems produced between 1.71 (OSDG) and 9.03 +(SDSN) SDGs per document for synthetic texts of length +10,000. +The word clouds illustrate the keywords that triggered the +assignment of SDGs for the three natural language data sets +(Disneyland reviews, cooking recipes, math lectures). As can +be seen, the keywords relate to many frequent topics that fit +the respective SDGs but also the mundane contexts of the +three data sets. These word clouds can be seen to highlight +the limitations of query-based approaches that do not control +for the frequency of keywords in natural language. +Trained ensemble models alleviate the shortcomings of +existing labeling systems +Previous sections illustrated serious shortcomings of ex- +isting SDG labeling systems. Labeling systems differ more +in conservatism than in accuracy, they have considerable +SDG-specific biases, and they commit many false positives +when the length of documents increases. +However, these +shortcomings did not affect systems uniformly—some sys- +tems were more prone to bias whereas others were more +prone to false alarms. We leverage this fact to train ensemble +models of SDG labeling systems. +We consider the six publicly available labeling systems +implemented in text2sdg and document length as features +and combine them in a random forest model. We train the +model on the three expert-labeled data sets and, to control the +false-positive rate of the model, on synthetic data sets gener- + +A +Aurora +Elsevier +SIRIS +Auckland +OSDG +SDSN +SDGO +Bias +Bias +Bias +Bias +Bias +Bias +Bias +SDG-01 +SDG-02 +SDG-03 +SDG-04 +SDG-05 +SDG-06 +SDG-07 +SDG-08 +SDG-09 +SDG-10 +SDG-11 +SDG-12 +SDG-13 +SDG-14 +SDG-15 +SDG-16 +SDG-17 +B +0.25 +Relative frequency +Aurora +0.2 +Elsevier +SIRIS +Auckland +OSDG +0.15 +SDSN +SDGO +Expert +0.1 +0.05 +0 +01 +02 +03 +04 +05 +06 +07 +08 +09 +10 +12 +13 +14 +15 +16 +17 +11 +SDG6 +WULFF, MEIER, MATA +Figure 3 +False positive predictions of SDG labeling systems. +The figure shows the number of SDGs assigned by the seven +systems to three natural language data sets unrelated to SDGs (Disneyland reviews, cooking recipes, and math lectures), four +synthetic data sets of different lengths created from Wikipedia word frequencies (Wikipedia random), and three expert-labeled +data sets (Titles, Abstracts, News Articles), as a function of the number of words per document. The word clouds at the bottom +consists of the keywords that triggered the assignment of SDGs for the three natural language data sets, with size coding the +frequency of keyword hits and the color coding the SDG labeling system. +ated from Wikipedia word frequency, matching in length the +documents of the three labeled data sets. We train the model +by assigning equal weight to the three labeled data sets and +vary the weight of the synthetic data to vary the focus on +reducing false positives. +Figure 4 shows the performance of two ensemble mod- +els including (black) and excluding (gray) document length +as a predictor compared to the six labeling systems used to +train the ensemble model and the OSDG labeling system. We +find that for a wide band of moderate synthetic data weights +the ensemble model achieves a higher average out-of-sample +accuracy than all individual labeling systems, while commit- +ting only as many false alarms as the most conservative label- +ing systems. Furthermore, we find that including document +length as a predictor substantially improves the performance +of the ensemble model. +We analyzed whether the ensemble model suffers from the +same sensitivity–specificity trade-off and found that this was +not the case. Specifically, for a synthetic data weight of 1, +we find that the ensemble model is on a par with the best- +performing conservative model for the news article data (En- +semble: Accuracy = .83, F1 = .54; Elsevier: Accuracy.82, +F1 = .46) and also with the best-performing liberal model +for the titles and abstract data (Ensemble: Accuracy = .69, +F1 = .71; SDSN: Accuracy = .69, F1 = .73), implying +an even performance across data sets. +We also evaluated +the profile bias and fidelity of the ensemble model. We ob- +served a profile bias at the lower end of the individual sys- +tems (¯r = .14) and a expert profile fidelity far outperforming +the individual systems (¯r = .92). +The ensemble model can further be used to understand +the usefulness of the individual labeling systems through +an analysis of feature importance. +Figure 4C shows the +permutation-based feature importance separately for the dif- +ferent SDGs. It can be seen that importance varies consid- +erably across SDGs, which highlights not only differences + +News +10 +Abstracts +articles +per document +■ +Wikipedia +1 +random +■ +Wikipedia +Titles +■ +random +Wikipedia +■ +Math +Aurora +random +lectures +Elsevier +.01 +SDG + SIRIS +Disneyland + Auckland +S +.001 +OSDG +reviews +# +SDSN +Wikipedia +Cooking +SDGO +.0001 +random +recipes +10 +30 +100 +300 +1000 +3000 +10000 +# Words +contaminate +rainforest transportation +language water +integration +reduction +transportation +child +expectancy learning +school +smoking potable wood +energy +food child +charcoal +clean acidic +stem food +stock +adults travel +rice mountain +.age state +water +cancer +disaster +land +prices children prices +waste +waste +landfood +coast +.waste +water + torture decent +law +milklevel +lawwho pay +sea +age +watersea water +Torigin forest +Imoney +aidsworktrade +safety +waterland food +waste human +town +tree +wastemountain. +distribution + plastic +poaching ocean, +wate +water +diet +safety +publicqualityocean city forest +temperature. +overfishing +rainforest drinking work +public +enough +school +préss charcoal +produce number +cars + violence +school + status educational +increase +student +discriminationSDG LABELING SYSTEMS +7 +in the quality of labeling systems across the SDGs but also +the lack of comparable data on the different SDGs. Despite +the considerable variance across SDGs, the ensemble model +preferred to rely on some systems more than others. Specif- +ically, Auckland, SDGO, and SDSN received higher feature +importance relative to the Aurora, Elsevier, and SIRIS sys- +tems. +Discussion +We aimed to compare a number of existing automated la- +beling systems that have been proposed to identify work on +the Sustainable Development Goals (SDGs) from text. Sys- +tematic comparison of these systems, including their relative +performance and potential biases, is a crucial step towards +identifying the most reliable and accurate tools and, further- +more, building confidence in the use of automated methods +for monitoring work on the SDGs. +We compared seven systems using a variety of text +sources—including research papers, news articles, and non- +SDG-related texts—and a variety of metrics. +Our results +suggest that the existing labeling systems differ in accuracy +and that their performance varies considerably across text +sources. These differences are due to the systems’ differ- +ences in their specificity (true-positive rate) and sensitivity +(true negative rate). Additionally, the systems have differ- +ent biases that can have an impact on the overall profile of +the SDGs identified, with some systems emphasizing spe- +cific SDGs (e.g., health) relative to experts. Our finding of +biases is important because it reveals a potential for mis- +leading representation of work on the SDGs that could cre- +ate confusion about the relative investment in different SDGs +and even undermine trust in the use of automated methods. +Some researchers have pointed out how institutional rank- +ings using several, often non-transparent criteria can lead to +a lack of convergent validity between rankings and associ- +ated confusion (Berg et al., 2022). The increasing reliance +on automated systems for ranking institutions’ contributions +to the SDGs requires reliable systems that aim to reduce bias. +Finally, and more broadly, our results suggest that labeling +systems should not be used interchangeably. +One alternative to the use of single systems is the use of +ensemble models that pool multiple labeling systems. Our +results suggest that ensemble models can overcome some +of the limitations of individual labeling systems. In partic- +ular, our results suggest that an ensemble approach is able +to achieve higher performance compared to existing systems, +and is less susceptible to biases and variations in the type and +amount of text analyzed. All in all, these results suggest that +ensemble models may be a good alternative to existing sys- +tems to detect SDGs in an automated fashion. Our ensemble +model is freely available in our text2sdg R package. +There are several limitations and opportunities for future +research in the field of automated SDG labeling. First, none +of the expert-labeled data sets used in our work allow for a +good estimation of false positives, as these documents were +not randomly or representatively sampled from the respective +class of documents. We alleviated this problem by relying on +synthetic data sets that were unrelated to SDGs but this ap- +proach does not account for the word co-occurrence patterns +of actual negative SDG texts. Future work should focus on +producing better validation data sets, ideally also from other +domains beyond academic papers or news articles, if these +methods are to be used beyond these domains, for example, +concerning policy documents. +Second, although we compared seven labeling systems +that represent the majority of the existing systems for auto- +mated detection of SDGs from text, there are additional sys- +tems that we could not consider. Prominent among these are +proprietary systems from Dimension.ai (Fane et al., 2022) +and novel versions from Elsevier. Overall, we believe that +making such automated tools publicly and easily available +would be an important step towards assessing and improving +automated labeling of SDGs. +Third, although the ensemble approach used in this study +was effective in addressing the shortcomings of existing sys- +tems, it is still limited by the keywords used from the seven +included systems. These keywords were selected using pro- +cedures specific to certain data sets, and it is likely that the +current keyword set is incomplete for some or all of the +SDGs. Once larger, more diverse, and representative expert- +labeled data sets become available, it would be beneficial to +learn relevant keywords from these expert labels using natu- +ral language processing techniques. One particularly promis- +ing approach is to fine-tune large-scale language models, +which could not only provide high accuracy but also allow +for the identification of effective keyword sets using explain- +able machine learning methods. Large language models can +analyze the content and context of text to identify relevant +keywords and phrases that are indicative of the SDGs, and +can also incorporate information about the structure and or- +ganization of text, leading to a better understanding of the re- +lationships between different concepts. Crucially, such mod- +els can be trained and fine-tuned in an iterative fashion to +specific domains or languages, which can help to improve +their performance and accuracy in specific contexts. +Al- +though such attempts exist, these have been built based on +queries rather than the expert-labeled data (Vanderfeesten & +Jaworek, 2022) thus developing well-labeled data sources re- +mains a priority for moving such efforts forward. +In conclusion, we presented a comparison of existing la- +beling systems to identify work on the Sustainable Devel- +opment Goals (SDGs) in text sources. Our approach was +based on the use of novel data sources and several perfor- +mance metrics. We found that current systems suffer from +several shortcomings but that ensemble modeling techniques +allow us to overcome some of the limitations of existing la- + +8 +WULFF, MEIER, MATA +Figure 4 +Ensemble model performance. Panel A and B illustrate the performance of ensemble models drawing on the classifications of +the six labeling systems relative to the labeling systems on their own. Panel A shows the accuracy for the three expert-labeled +data sets as a function of the amount of training weight given to synthetic non-SDG documents generated from Wikipedia +word frequencies. Panel B shows the false-positive rate as a function of the training weight. In both panels, the black line +shows the performance of an ensemble model that includes document length as a feature, whereas the gray line corresponds +to a model that does not include document length. Panel C shows the SDG-specific feature importance for the ensemble model +that includes document length and has been trained with a non-SDG data weight of 1. +beling systems. We demonstrate that an ensemble approach +is able to achieve higher specificity and sensitivity compared +to existing systems, and is less susceptible to biases and vari- +ations in the type and amount of text analyzed. Our find- +ings have important implications for researchers and policy- +makers seeking to accurately monitor progress on the SDGs, +and we recommend the use of ensemble approaches as best +practice when drawing conclusions about the absolute and +relative prevalence of work on the SDGs based on automated +methods. +Methods +SDG labeling systems +The systems, which all are based on Lucene-style queries, +vary considerably in complexity. The SDSN and OSDG sys- +tems are least complex because they only make use of OR- +operations, implying that they assign an SDG as soon as a +single keyword is matched. The SIRIS and, in particular, +the Elsevier systems are more complex as they additionally +include AND-operations, meaning that multiple keywords +must be present to trigger a match. The Aurora system is +most complex because it further includes NEAR-operations, +meaning that keywords must co-occur within a maximum +distance to result in match. +Labeled data +The Aurora data set (Vanderfeesten et al., 2020) was cre- +ated to validate the Aurora classification system. The part +of the data we use consists of a survey where people had to +indicate whether a research paper was relevant for a given +SDG. Each of the 244 respondents did this for 100 papers +randomly selected from a pool of research papers detected +by the Aurora system as SDG relevant. +The OSDG Community Dataset (OSDG et al., 2021) con- + +A +c +Labeled data +1 +SDG-16 +0.08 +0.9 +Accuracy +0.8 +0.07 +0.7 +01010010101010101010 +SDGO +0.06 +Auckland +SDSN +0.6 +O +SDG-04 +0.5 +Feature importance +0.05 +0 +0.2 +0.5 +1.3 +3.6 +10 +Synthetic data weight +SDG-03 +0.04 +B +SDG-05 +Synthetic data +SDG-11 +0.4 + rate +0.03 +SDG-06 +O Ensemble incl. # words +0.3 +SDG-07 +O Ensemble excl. # words +0.02 +0.2 +SDGO +SDSN +SDG-01 +SDG-02 +SBG-88 +0.1 +0.01 +OSDG +SDG-13 +Auckland +G-09 +mooroioioiolod +0 +Aurora +0.2 +0.5 +1.3 +3.6 +0 +10 +0 +Synthetic data weight +words +RSDG LABELING SYSTEMS +9 +tains tens of thousands of text excerpts which were labeled by +Community volunteers. To make this labeling more efficient, +the volunteers only had to indicate whether or not a suggested +label suited the text excerpt. Thus, the volunteers simply had +to accept or reject a given SDG but were not asked to select +one or more SDGs that might relate to the given text excerpt. +Each text was rated by multiple volunteers +The SDG Knowledge Hub data (Wulff & Meier, 2023) +consists of news articles posted on the SDG Knowledge Hub +website (sdg.iisd.org). This website was launched in October +2016 and is managed by the International Institute for Sus- +tainable Development (IISD). It hosts news and commentary +regarding the implementation of the SDGs. The news articles +contain labels that show which SDGs they cover. These la- +bels are assigned by the subject experts who write these news +articles and confirmed by SDG Knowledge Hub editors. We +downloaded 9,172 news articles that have been published on +the website along with the assigned SDG labels. +Unlabeled data +The Disneyland reviews, cooking recipes, and math lec- +tures were obtained from Kaggle. The Disneyland data set +contains 42,656 reviews of Disneyland locations in Paris, +California, or Hong Kong posted by visitors on Trip Advi- +sor (see kaggle.com). The cooking recipe data set contains +82,245 recipes scraped from food-related websites, such as +skinnytaste.com (see kaggle.com). The math lecture data set +contains 860 lectures posted on YouTube by institutions or +creators covering 11 subjects, ranging from algebra to natu- +ral language processing, that are related to computer science +and mathematics (see kaggle.com). +The synthetic data sets were generated by concatenating +words sampled at random based on the words’ frequencies +in the (Wikipedia corpus. The synthetic texts, thereby, re- +flect the natural word frequency distribution found in natural +language. +text2sdg +To detect SDGs in these different texts, we used the +text2sdg R package (text2sdg.io Meier et al., 2021). text2sdg +provides a common framework for implementing the differ- +ent systems to detect SDGs in text and makes it easy to quan- +titatively compare and visualize their results. The text2sdg +also makes available the ensemble model presented in this +article. +Ensemble modeling +We recruited two types of algorithms, random forest (R +package ranger Wright & Ziegler, 2015) and extreme gra- +dient boosting (Chen et al., 2015, R package xgboost), to +train ensembles of SDG labeling systems. We trained the +algorithms separately for each SDG to predict the presence +or absence of the SDG based on the predictions of six dif- +ferent SDG labeling systems implemented in text2sdg.io and +the number of words in the documents. Training and evalua- +tion were performed using a repeated k-fold cross validation +procedure. The models were trained using all three expert- +labeled and synthetic data sets, with the latter matching the +former in numbers and word lengths. The cases in the data +sets were initially weighted by 1/N with N being the num- +ber of cases in a data set to give each data set equal weight. +Furthermore, we multiplied the weight of the synthetic data +sets by a factor k ∈ [0, 10] to vary the weight of synthetic +data relative to the expert-labeled data. A factor of k = 0 +means that the synthetic data receives no weight whereas a +factor of k = 10 means that the synthetic data receive ten +times as much weight as the expert-labeled data. Overall, we +found the random forest to perform slightly better than the +gradient-boosting algorithm; hence, we report the results of +the random forest in the main text. +References +Armitage, C. S., Lorenz, M., & Mikki, S. (2020). Mapping +scholarly publications related to the sustainable de- +velopment goals: Do independent bibliometric ap- +proaches get the same results? Quantitative Science +Studies, 1(3), 1092–1108. +Aurora Universities Network (AUR). Search Queries for +"Mapping Research Output to the Sustainable De- +velopment Goals (SDGs)" v5.0. 2020. https://doi. +org/10.5281/zenodo.3817445. +Bautista, N. (2019). Sdg ontology. https://doi.org/10.6084/ +m9.figshare.11106113.v1 +Berg, F., Kölbel, J. F., & Rigobon, R. (2022). Aggregate Con- +fusion: The Divergence of ESG Ratings. Review of +Finance, 26(6), 1315–1344. https : / / doi . org / 10 . +1093/rof/rfac033 +Chen, T., He, T., Benesty, M., Khotilovich, V., Tang, Y., Cho, +H., Chen, K., et al. (2015). Xgboost: Extreme gra- +dient boosting. R package version 0.4-2, 1(4), 1–4. +Duran-Silva, N., Fuster, E., Massucci, F. A., & Quinquillà, A. +(2019). A controlled vocabulary defining the seman- +tic perimeter of Sustainable Development Goals +(Version 1.2). 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(2022). +Osdg 2.0: A multilingual tool for classifying text +data by un sustainable development goals (sdgs). +https://doi.org/10.48550/arXiv.2211.11252 +Pukelis, L., Puig, N. B., Skrynik, M., & Stanciauskas, V. +(2020). Osdg–open-source approach to classify text +data by un sustainable development goals (sdgs). +arXiv preprint arXiv:2005.14569. +Schmidt, F., & Vanderfeesten, M. (2021). Evaluation on ac- +curacy of mapping science to the United Nations’ +Sustainable Development Goals (SDGs) of the Au- +rora SDG queries. https://doi.org/10.5281/zenodo. +4964606 +Smith, T. B., Vacca, R., Mantegazza, L., & Capua, I. (2021). +Natural language processing and network analysis +provide novel insights on policy and scientific dis- +course around sustainable development goals. Sci- +entific reports, 11(1), 22427. https://doi.org/10. +1038/s41598-021-01801-6 +Sustainable +Development +Solutions +Network +(SDSN). +(2021). Compiled list of sdg keywords. Retrieved +January 4, 2023, from https : / / ap - unsdsn . org / +regional-initiatives/universities-sdgs/ +Vanderfeesten, M., & Jaworek, R. (2022). AI for map- +ping multi-lingual academic papers to the United +Nations’ Sustainable Development Goals (SDGs). +https://doi.org/10.5281/zenodo.5939866 +Vanderfeesten, M., Spielberg, E., & Gunes, Y. (2020). Survey +data of "Mapping Research Output to the Sustain- +able Development Goals (SDGs)" (Version 1.0.1). +Zenodo. https://doi.org/10.5281/zenodo.3813230 +Wright, M. N., & Ziegler, A. (2015). Ranger: A fast imple- +mentation of random forests for high dimensional +data in c++ and r. arXiv preprint arXiv:1508.04409. +Wulff, D. U., & Meier, D. S. (2023). SDG Knowledge Hub +Dataset of SDG-labeled News Articles. Zenodo. +https://doi.org/10.5281/zenodo.7523032 +Data availability +The text2sdg R package is available on the Compre- +hensive R Network (https://CRAN.R-project.org/package= +text2sdg). All data sets used in this analysis are publicly +available. Links are included in the main text. +Acknowledgements +We are grateful to Laura Wiles for editing the manuscript. +This work was supported by a grant from the Swiss Science +Foundation (100015_197315) to Dirk U. Wulff. + diff --git a/qNFIT4oBgHgl3EQfwytZ/content/tmp_files/load_file.txt b/qNFIT4oBgHgl3EQfwytZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc81bc5ab39d05be346653fcaed61c38da1a08c0 --- /dev/null +++ b/qNFIT4oBgHgl3EQfwytZ/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf,len=587 +page_content='Using novel data and ensemble models to improve automated labeling of Sustainable Development Goals Dirk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Wulff1,2, Dominik S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Meier1, and and Rui Mata1 1University of Basel 2Max Planck Institute for Human Development A number of labeling systems based on text have been proposed to help monitor work on the United Nations (UN) Sustainable Development Goals (SDGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Here, we present a systematic comparison of systems using a va- riety of text sources and show that systems differ con- siderably in their specificity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', true-positive rate) and sensitivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', true-negative rate), have systematic bi- ases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', are more sensitive to specific SDGs relative to others), and are susceptible to the type and amount of text analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We then show that an ensemble model that pools labeling systems alleviates some of these lim- itations, exceeding the labeling performance of all cur- rently available systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We conclude that researchers and policymakers should care about the choice of label- ing system and that ensemble methods should be favored when drawing conclusions about the absolute and rela- tive prevalence of work on the SDGs based on automated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Monitoring work on the SDGs is crucial for their advance- ment and one promising approach is to screen the increasing amount of digitally available text using automated, natural language processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This approach has taken hold, for example, in scientometric efforts that monitor the SDGs in academic publications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', Aurora Universities Network (AUR), 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Bautista, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Duran-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Jaya- balasingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Sustainable Development Solu- tions Network (SDSN), 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' One such system has iden- tified millions of SDG-related academic publications and is already used by the Times Higher Education Impact Rank- ings to rank over 1,000 universities worldwide according to their SDG-related outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This example illustrates the im- portant role that SDG labeling from text can play in academic research and funding policy (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The increased and widespread use of SDG labeling sys- Dirk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Wulff https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org/0000-0002-4008-8022 Do- minik S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Meier https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org/0000-0002-3999-1388 Rui Mata https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org/0000-0002-1679-906X Correspondence concerning this article should be addressed to Dirk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Wulff, Department of Psychology, University of Basel, Missionsstrasse 60-62, 4055 Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' E-mail: dirk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='wulff@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='com tems should be accompanied by efforts to validate the differ- ent approaches and ensure that their predictions accurately reflect the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' However, systematic and comprehensive evaluations of the accuracy of these labeling systems are largely lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Crucially, the few existing results indicate some striking differences in the predictions of the different labeling systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', Armitage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Pukelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Schmidt & Vanderfeesten, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Three key reasons prevent rigorous evaluation of the dif- ferent SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' First, most labeling systems have been developed for and tested with specific proprietary citation databases thus limiting their portability to other text sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Second, there has been a paucity of publicly avail- able data that can be used to validate the predictions of differ- ent approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Third, systems have often been analyzed in isolation without systematic comparisons involving various performance metrics or assessment of biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In this paper, we aimed to help overcome these limitations by relying on our recently developed open-source R package, text2sdg ((text2sdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='io) Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021), which can be used with any text source, and collating various labeled and non- labeled data sources to provide a comparison of seven exist- ing SDG labeling systems using a comprehensive set of per- formance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We provide a comparative evaluation of seven labeling systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' namely, the Aurora (Aurora Univer- sities Network (AUR), 2020), Elsevier (Jayabalasingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021), SIRIS (Duran-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2019), Auckland (Mu & Wang, 2022), SDGO (Bautista, 2019), and SDSN (Sustain- able Development Solutions Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Sustainable Devel- opment Solutions Network (SDSN), 2021) systems imple- mented in the text2sdg R package and, additionally, OSDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='ai (Pukelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2022), a publicly available tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These sys- tems account for a majority of, albeit not all (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Fane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2022), systems currently available for automated detection of SDGs from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Finally, we also aimed to assess the poten- tial of using an ensemble approach that integrates several of the existing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Ensemble models can potentially im- prove accuracy and generalizability by considering the pre- dictions of multiple models, each of which may have differ- ent biases, resulting in a more balanced and representative prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our contribution is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We first intro- duce and use three labeled data sets containing hand-coded arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='11353v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='CL] 25 Jan 2023 2 WULFF, MEIER, MATA labels to evaluate the categorizations of the seven automated labeling systems against those of human experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The data sets cover different text sources, including titles and abstracts of academic publications, as well as news articles, thus sig- nificantly increasing the scope of sources considered in past work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', Armitage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Pukelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Cru- cially, we use a number of different metrics to compare the different labeling systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', sensitivity, specificity, over- all accuracy) thus providing a comprehensive assessment of the systems’ strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Second, we com- pare the predictions of the SDG labeling systems across data sets to reveal SDG-specific biases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' that is, whether differ- ent systems tend to make differential categorizations for spe- cific SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The potential for bias is particularly important concerning the application of automated labeling systems to make relative statements about the presence of specific SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For example, the use of biased systems in the detec- tion of some SDGs could lead to an incorrect assessment of investment in some domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', health) relative to others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', education).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Third, we introduce and leverage several novel unlabeled data sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', Disneyland reviews, cook- ing recipes, math lectures, random text) to better understand the labeling systems’ susceptibility to the type and amount of text analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Fourth, and finally, we explore whether ensemble-modeling approaches can address some of the po- tential limitations of existing SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Results In the following subsections, we first compare the seven labeling systems on a variety of metrics on three labeled data sets covering texts from academic publications and news ar- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Second, we assess to what extent the different label- ing systems show biases concerning different SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Third, we assess the susceptibility of the different labeling systems to produce false positives as a function of the length of the text source in novel unlabeled data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Fourth, and finally, we assess the potential of ensemble models that integrate the different labeling systems to address potential limitations of individual labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' SDG labeling systems differ in their sensitivity–specificity trade-offs Our first set of analyses consisted of comparing seven la- beling systems to generate predicted labels for documents from three labeled data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' All three data sets provide hu- man judges’ ratings of SDGs for different document types: the first data set consists of over 10,000 titles of academic publications (titles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Vanderfeesten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020), the second data set consists of over 30,000 abstracts of academic pub- lications (abstracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' OSDG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021), and the third data set consists of over 9,000 news articles scraped, with permis- sion, from the SDG knowledge hub website (news articles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' https://sdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='iisd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org Wulff & Meier, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The data sets differ in a number of respects, most notably in the number of words per document, with titles, abstracts, and news arti- cles being composed of 18, 90, and 673 words per document on average, respectively, and in the number SDGs they were evaluated for, with titles and abstracts having each been eval- uated for only one of the SDGs and news articles having been evaluated for all 17 SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 1 presents the main classification results separately for the three data sets (titles, abstracts, news articles), with panel A showing a breakdown of SDG labels per document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To better assess the differences between each data set, it is helpful to look initially at the first column of each subpanel in Figure 1A, which contains information on the expert rat- ings for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For the titles data set, a large majority of documents were judged to contain one SDG (63%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In the abstracts data set, a majority of documents were assigned an SDG (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In turn, in the news articles data set, all of the documents (100%) were assigned one or more SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' As a whole, across data sets, only a minority of documents have not been assigned any SDG by the experts, which, as we discuss below, can limit the validation of SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The remaining columns in Figure 1A depict the labels as- signed to each document by the labeling systems and, there- fore, allow a first comparison between systems and the hu- man experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Visual inspection suggests that all systems showed reasonable accuracy in terms of recovering the "true" SDG labels assigned by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This is illustrated by the similar pattern of colored stripes across experts and the seven labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Nevertheless, the systems showed consid- erable differences in their ability to detect SDGs, as illus- trated by the relative intensity of the colored stripes and gray stripes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The results below quantify these differ- ences using common performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We compared the seven labeling systems quantitatively using a number of common metrics typically used in catego- rization problems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', sensitivity, specificity, accuracy, F1 score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Sensitivity measures a system’s ability to correctly identify true SDG text sources as positive whereas specificity measures a system’s ability to identify non-SDG sources as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Ideally, SDG labeling systems should be both sen- sitive and specific but there is often a trade-off between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Considering both metrics can help compare the systems and their criteria for dealing with trade-offs between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In practice, we do so by visually comparing such metrics in an ROC (receiver operating characteristic) space as well as considering general measures of categorization performance, such as overall accuracy and a composite measure of sensi- tivity and specificity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', F1 score) in the different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' As can be seen in Figure 1B, the systems differ substan- tially in their trade-off between sensitivity and specificity or, in other words, in how conservatively they assign SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This is shown by the fact that the systems’ performance— SDG LABELING SYSTEMS 3 Figure 1 Performance of SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel A shows the SDG labels assigned by experts and seven labeling systems for three data sets consisting of titles of research articles (Titles), abstracts of research articles (Abstracts), and online news articles (News Articles), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The colored lines represent the assigned SDG by either experts or SDG labeling system and gray lines represent missing labels (for experts) or false negatives (for SDG labeling systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel B shows the performance of the seven systems across the three data sets by plotting the systems’ sensitivity against 1-specificity forming an ROC (receiver operating characteristic) plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Lines in the background show several levels of balanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel C and D show the performance of the seven systems across the three data sets in terms of accuracy and the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' in terms of sensitivity and specificity—varies mostly along the diagonal of the ROC plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We also observe differences in overall accuracy but these are not stable across data sets (Fig- ure 1C), which is due to the different levels of conservatism of the different labeling systems: More conservative systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', Elsevier) outperform liberal systems for news articles, in which many labels are negative, and vice versa for ab- stracts and titles, where almost all labels are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These patterns hold for the alternative measure of performance, the F1 score (Figure 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' All in all, these results point to labeling systems being differently conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' that is, they solve the sensitivity– specificity trade-off differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' As a consequence, label- ing systems do best in different data sets and it is difficult to identify a single best-performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The results fur- ther point to a limitation of currently available data sets that include only a small proportion of non-SDG-related docu- ments, which introduces difficulties in assessing systems’ susceptibility to producing false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We return to this point below when we introduce novel synthetic data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Expert Aurora Elsevier A SIRIS Auckland OSDG SDSN SDGO B C 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='8 Abstracts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='9 SDSN SDGO Accurac Titles (news articles) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 News articles (news articles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='8 SDSN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='4 SDGO (abstracts) (abstracts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 Auckland SIRIS (news articles) itivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 0 (news articles) SDSN Elsevier SIRIS S Auckland OSDG Aurora SDSN SDGO C Auckland (titles) SDGO Alpora) ensi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5 OSDG (ites) D (abstracts) Elsevier S (news articles) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='4 OSDG Abstracts Auckland SIRIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='8 (abstracts) OSDG (titles) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='3 Titles score Elsevier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 SIRIS News articles Elsevier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='4 (abstracts) Aurora 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='1 (titles) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='4 AuroraElsevier SIRIS Auckland OSDG SDSN SDGO 1 - Specificity4 WULFF, MEIER, MATA Biases in SDG labeling systems distort SDG profiles Biases in SDG labeling systems can lead to inaccurate representation of the prevalence and importance of different SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' If a system is more sensitive to certain SDGs than others, it will overestimate the prevalence of those SDGs and underestimate the prevalence of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This re- sults in a misleading picture of the work being done to ad- dress the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Biased systems may also create an un- fair advantage or disadvantage for certain organizations or groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For example, if a method is more sensitive to certain SDGs than others, organizations may aim to portray them- selves as focusing on those SDGs by relying on such sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Finally, biased systems may create confusion or mis- trust among stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' If different methods produce sig- nificantly different results, it may be difficult for stakeholders to know which results to trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This can lead to confusion or mistrust in the results, which could ultimately undermine the credibility of the work being done to address the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We estimated SDG-specific biases by comparing the rela- tive frequency of SDGs between predicted and observed la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Specifically, we calculate bias = predicted−observed observed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We then evaluate the profile bias of a given system by correlating the biases across data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We also compare the SDG profiles obtained from the seven labeling systems to the SDG profile of experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This analysis amounts to assessing the similarity between the profile across SDGs identified by experts and the profile identified by each of the labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 2A shows the biases of the different labeling sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Visual inspection suggests that some biases appear sys- tematically across data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For example, considering the Aurora labeling system, the pattern suggests consistent un- derestimation of SDGs 2, 6, 7, 8, 9, and 10 across data sets but overestimation of SDG 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For Elsevier, one observes consistent underestimation of SDGs 4, 6, 9, 12, 14, and 15 but overestimation of SDGs 3 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To quantify the sys- tems’ profile biases we correlated the SDG biases between titles and news articles and between abstracts and news arti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We did not consider the correlation between titles and abstracts because of a moderate correlation in expert profiles for these two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The strongest average profile bias was observed for Elsevier (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='72), followed by OSDG (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='41), Aurora (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='38), SDGO (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='31), SDSN (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2), Auckland (¯r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='08), and SIRIS (¯r = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 2B shows that these biases imply substantial differ- ences in SDG profiles derived from the different labeling sys- tems relative to the one derived from experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This difference is most pronounced for SDG 3 (Good health and well-being).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' A number of systems overestimate the relative frequency of SDG 3, with Elsevier giving SDG 3 twice as much weight (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='26) as experts (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In turn, SDGs 9 (Industry, Innovation, and Infrastructure) and 10 (Reduced inequalities) tend to re- ceive less weight than assigned by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To further evalu- ate profile fidelity, we calculated the Spearman’s rank corre- lation coefficient between expert and system relative frequen- cies separately for the three data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We found Auckland to have the highest average correlation (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='76), followed by SDGO (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='72), Elsevier (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='70), SIRIS (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='67), Aurora (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='57), and, finally, OSDG (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To summarize, we find that systems appear to have differ- ent biases, with a number of these underestimating or over- estimating the presence of specific SDGs relative to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These biases can be important for the relative assessment of work on the SDGs and emphasize that the labeling systems should not be used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' SDG labeling systems can produce many false positives when applied to large text sources Existing validation data sets—such as those used above to test the accuracy of SDG labeling systems—may not ac- curately reflect the performance of SDG labeling systems in many real-world applications that involve larger samples of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' As mentioned above, existing data sets only include a small proportion of documents unrelated to the SDGs, which can lead to an overestimation of the accuracy of the labeling systems relative to applications in which labeling systems are tested on a diverse range of documents that can poten- tially contain many false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To address this issue, we conducted evaluations of the labeling systems using data sets that are ostensibly unrelated to SDGs in order to better un- derstand the systems’ tendency to produce false positives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' in particular, as a function of text length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In this analysis, we used existing and synthetic data sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' specifically, three natural language data sets consisting of Disneyland reviews (N = 42, 656), cooking recipes (N = 82, 245), and math lectures (N = 860).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In addition, we gen- erated synthetic texts by sampling from a word frequency list derived from Wikipedia and generating documents contain- ing 10, 100, 1,000, and 10,000 randomly sampled words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 3 presents the results of the SDG labeling systems for the various data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The plot shows the number of SDGs identified in each document by each system, plotted against the average length of the documents in the respective data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The plot includes data from the novel data sets (those likely not related to the SDGs), as well as the three expert- labeled data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We use the number of SDGs per document as a proxy for the false-positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Consequently, by com- paring the number of SDGs identified by the systems across different data sets and document lengths, we hope to under- stand the systems’ susceptibility to producing false positives and identify any patterns or trends in their accuracy as a func- tion of document length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Several noteworthy results emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' First, all systems pro- duce false positives for all types of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Second, the tendency of systems to produce false positives is in line with their level of conservatism, which we discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' For example, Aurora and Elsevier appear to be very conserva- SDG LABELING SYSTEMS 5 Figure 2 Biases in SDG classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel A illustrates the biases of SDG labeling systems in classifying each of the 17 SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Biases are calculated as the difference between the observed and predicted SDG proportion for a given data set divided by the observed proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' A positive bias means that a given SDG was assigned more often than its relative frequency in expert judgments and vice versa for a negative bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The three vertical stripes for each SDG and system correspond to the titles, abstracts, and news articles data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel B shows the profile of expert (red) and system labels generated by averaging the relative frequencies of SDGs across data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The circles and numbers highlight the three most prevalent SDGs per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' tive in identifying SDGs, leading to an overall low false- positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Third, across all systems the tendency to de- tect SDGs increases considerably when the length of the texts increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Specifically, whereas systems produced be- tween .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='0003 (Aurora) and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='144 (SDGO) for synthetic texts of length 10, systems produced between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='71 (OSDG) and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='03 (SDSN) SDGs per document for synthetic texts of length 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The word clouds illustrate the keywords that triggered the assignment of SDGs for the three natural language data sets (Disneyland reviews, cooking recipes, math lectures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' As can be seen, the keywords relate to many frequent topics that fit the respective SDGs but also the mundane contexts of the three data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These word clouds can be seen to highlight the limitations of query-based approaches that do not control for the frequency of keywords in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Trained ensemble models alleviate the shortcomings of existing labeling systems Previous sections illustrated serious shortcomings of ex- isting SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Labeling systems differ more in conservatism than in accuracy, they have considerable SDG-specific biases, and they commit many false positives when the length of documents increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' However, these shortcomings did not affect systems uniformly—some sys- tems were more prone to bias whereas others were more prone to false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We leverage this fact to train ensemble models of SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We consider the six publicly available labeling systems implemented in text2sdg and document length as features and combine them in a random forest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We train the model on the three expert-labeled data sets and, to control the false-positive rate of the model, on synthetic data sets gener- A Aurora Elsevier SIRIS Auckland OSDG SDSN SDGO Bias Bias Bias Bias Bias Bias Bias SDG-01 SDG-02 SDG-03 SDG-04 SDG-05 SDG-06 SDG-07 SDG-08 SDG-09 SDG-10 SDG-11 SDG-12 SDG-13 SDG-14 SDG-15 SDG-16 SDG-17 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='25 Relative frequency Aurora 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 Elsevier SIRIS Auckland OSDG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='15 SDSN SDGO Expert 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='05 0 01 02 03 04 05 06 07 08 09 10 12 13 14 15 16 17 11 SDG6 WULFF, MEIER, MATA Figure 3 False positive predictions of SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The figure shows the number of SDGs assigned by the seven systems to three natural language data sets unrelated to SDGs (Disneyland reviews, cooking recipes, and math lectures), four synthetic data sets of different lengths created from Wikipedia word frequencies (Wikipedia random), and three expert-labeled data sets (Titles, Abstracts, News Articles), as a function of the number of words per document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The word clouds at the bottom consists of the keywords that triggered the assignment of SDGs for the three natural language data sets, with size coding the frequency of keyword hits and the color coding the SDG labeling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' ated from Wikipedia word frequency, matching in length the documents of the three labeled data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We train the model by assigning equal weight to the three labeled data sets and vary the weight of the synthetic data to vary the focus on reducing false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 4 shows the performance of two ensemble mod- els including (black) and excluding (gray) document length as a predictor compared to the six labeling systems used to train the ensemble model and the OSDG labeling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We find that for a wide band of moderate synthetic data weights the ensemble model achieves a higher average out-of-sample accuracy than all individual labeling systems, while commit- ting only as many false alarms as the most conservative label- ing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Furthermore, we find that including document length as a predictor substantially improves the performance of the ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We analyzed whether the ensemble model suffers from the same sensitivity–specificity trade-off and found that this was not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Specifically, for a synthetic data weight of 1, we find that the ensemble model is on a par with the best- performing conservative model for the news article data (En- semble: Accuracy = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='83, F1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='54;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Elsevier: Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='82, F1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='46) and also with the best-performing liberal model for the titles and abstract data (Ensemble: Accuracy = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='69, F1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='71;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' SDSN: Accuracy = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='69, F1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='73), implying an even performance across data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We also evaluated the profile bias and fidelity of the ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We ob- served a profile bias at the lower end of the individual sys- tems (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='14) and a expert profile fidelity far outperforming the individual systems (¯r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The ensemble model can further be used to understand the usefulness of the individual labeling systems through an analysis of feature importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Figure 4C shows the permutation-based feature importance separately for the dif- ferent SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' It can be seen that importance varies consid- erably across SDGs, which highlights not only differences News 10 Abstracts articles per document ■ Wikipedia 1 random ■ Wikipedia Titles ■ random Wikipedia ■ Math Aurora random lectures Elsevier .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='01 SDG SIRIS Disneyland Auckland S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='001 OSDG reviews # SDSN Wikipedia Cooking SDGO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='0001 random recipes 10 30 100 300 1000 3000 10000 # Words contaminate rainforest transportation language water integration reduction transportation child expectancy learning school smoking potable wood energy food child charcoal clean acidic stem food stock adults travel rice mountain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='age state water cancer disaster land prices children prices waste waste landfood coast .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='waste water torture decent law milklevel lawwho pay sea age watersea water Torigin forest Imoney aidsworktrade safety waterland food waste human town tree wastemountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' distribution plastic poaching ocean, wate water diet safety publicqualityocean city forest temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' overfishing rainforest drinking work public enough school préss charcoal produce number cars violence school status educational increase student discriminationSDG LABELING SYSTEMS 7 in the quality of labeling systems across the SDGs but also the lack of comparable data on the different SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Despite the considerable variance across SDGs, the ensemble model preferred to rely on some systems more than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Specif- ically, Auckland, SDGO, and SDSN received higher feature importance relative to the Aurora, Elsevier, and SIRIS sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Discussion We aimed to compare a number of existing automated la- beling systems that have been proposed to identify work on the Sustainable Development Goals (SDGs) from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Sys- tematic comparison of these systems, including their relative performance and potential biases, is a crucial step towards identifying the most reliable and accurate tools and, further- more, building confidence in the use of automated methods for monitoring work on the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We compared seven systems using a variety of text sources—including research papers, news articles, and non- SDG-related texts—and a variety of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our results suggest that the existing labeling systems differ in accuracy and that their performance varies considerably across text sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These differences are due to the systems’ differ- ences in their specificity (true-positive rate) and sensitivity (true negative rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Additionally, the systems have differ- ent biases that can have an impact on the overall profile of the SDGs identified, with some systems emphasizing spe- cific SDGs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', health) relative to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our finding of biases is important because it reveals a potential for mis- leading representation of work on the SDGs that could cre- ate confusion about the relative investment in different SDGs and even undermine trust in the use of automated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Some researchers have pointed out how institutional rank- ings using several, often non-transparent criteria can lead to a lack of convergent validity between rankings and associ- ated confusion (Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The increasing reliance on automated systems for ranking institutions’ contributions to the SDGs requires reliable systems that aim to reduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Finally, and more broadly, our results suggest that labeling systems should not be used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' One alternative to the use of single systems is the use of ensemble models that pool multiple labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our results suggest that ensemble models can overcome some of the limitations of individual labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In partic- ular, our results suggest that an ensemble approach is able to achieve higher performance compared to existing systems, and is less susceptible to biases and variations in the type and amount of text analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' All in all, these results suggest that ensemble models may be a good alternative to existing sys- tems to detect SDGs in an automated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our ensemble model is freely available in our text2sdg R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' There are several limitations and opportunities for future research in the field of automated SDG labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' First, none of the expert-labeled data sets used in our work allow for a good estimation of false positives, as these documents were not randomly or representatively sampled from the respective class of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We alleviated this problem by relying on synthetic data sets that were unrelated to SDGs but this ap- proach does not account for the word co-occurrence patterns of actual negative SDG texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Future work should focus on producing better validation data sets, ideally also from other domains beyond academic papers or news articles, if these methods are to be used beyond these domains, for example, concerning policy documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Second, although we compared seven labeling systems that represent the majority of the existing systems for auto- mated detection of SDGs from text, there are additional sys- tems that we could not consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Prominent among these are proprietary systems from Dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='ai (Fane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2022) and novel versions from Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Overall, we believe that making such automated tools publicly and easily available would be an important step towards assessing and improving automated labeling of SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Third, although the ensemble approach used in this study was effective in addressing the shortcomings of existing sys- tems, it is still limited by the keywords used from the seven included systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These keywords were selected using pro- cedures specific to certain data sets, and it is likely that the current keyword set is incomplete for some or all of the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Once larger, more diverse, and representative expert- labeled data sets become available, it would be beneficial to learn relevant keywords from these expert labels using natu- ral language processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' One particularly promis- ing approach is to fine-tune large-scale language models, which could not only provide high accuracy but also allow for the identification of effective keyword sets using explain- able machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Large language models can analyze the content and context of text to identify relevant keywords and phrases that are indicative of the SDGs, and can also incorporate information about the structure and or- ganization of text, leading to a better understanding of the re- lationships between different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Crucially, such mod- els can be trained and fine-tuned in an iterative fashion to specific domains or languages, which can help to improve their performance and accuracy in specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Al- though such attempts exist, these have been built based on queries rather than the expert-labeled data (Vanderfeesten & Jaworek, 2022) thus developing well-labeled data sources re- mains a priority for moving such efforts forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In conclusion, we presented a comparison of existing la- beling systems to identify work on the Sustainable Devel- opment Goals (SDGs) in text sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our approach was based on the use of novel data sources and several perfor- mance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We found that current systems suffer from several shortcomings but that ensemble modeling techniques allow us to overcome some of the limitations of existing la- 8 WULFF, MEIER, MATA Figure 4 Ensemble model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel A and B illustrate the performance of ensemble models drawing on the classifications of the six labeling systems relative to the labeling systems on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel A shows the accuracy for the three expert-labeled data sets as a function of the amount of training weight given to synthetic non-SDG documents generated from Wikipedia word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel B shows the false-positive rate as a function of the training weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' In both panels, the black line shows the performance of an ensemble model that includes document length as a feature, whereas the gray line corresponds to a model that does not include document length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Panel C shows the SDG-specific feature importance for the ensemble model that includes document length and has been trained with a non-SDG data weight of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' beling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We demonstrate that an ensemble approach is able to achieve higher specificity and sensitivity compared to existing systems, and is less susceptible to biases and vari- ations in the type and amount of text analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Our find- ings have important implications for researchers and policy- makers seeking to accurately monitor progress on the SDGs, and we recommend the use of ensemble approaches as best practice when drawing conclusions about the absolute and relative prevalence of work on the SDGs based on automated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Methods SDG labeling systems The systems, which all are based on Lucene-style queries, vary considerably in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The SDSN and OSDG sys- tems are least complex because they only make use of OR- operations, implying that they assign an SDG as soon as a single keyword is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The SIRIS and, in particular, the Elsevier systems are more complex as they additionally include AND-operations, meaning that multiple keywords must be present to trigger a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The Aurora system is most complex because it further includes NEAR-operations, meaning that keywords must co-occur within a maximum distance to result in match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Labeled data The Aurora data set (Vanderfeesten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2020) was cre- ated to validate the Aurora classification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The part of the data we use consists of a survey where people had to indicate whether a research paper was relevant for a given SDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Each of the 244 respondents did this for 100 papers randomly selected from a pool of research papers detected by the Aurora system as SDG relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The OSDG Community Dataset (OSDG et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021) con- A c Labeled data 1 SDG-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='9 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='7 01010010101010101010 SDGO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='06 Auckland SDSN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 O SDG-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5 Feature importance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 10 Synthetic data weight SDG-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='04 B SDG-05 Synthetic data SDG-11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='4 rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='03 SDG-06 O Ensemble incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' # words 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='3 SDG-07 O Ensemble excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' # words 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 SDGO SDSN SDG-01 SDG-02 SBG-88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='01 OSDG SDG-13 Auckland G-09 mooroioioiolod 0 Aurora 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='6 0 10 0 Synthetic data weight words RSDG LABELING SYSTEMS 9 tains tens of thousands of text excerpts which were labeled by Community volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' To make this labeling more efficient, the volunteers only had to indicate whether or not a suggested label suited the text excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Thus, the volunteers simply had to accept or reject a given SDG but were not asked to select one or more SDGs that might relate to the given text excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Each text was rated by multiple volunteers The SDG Knowledge Hub data (Wulff & Meier, 2023) consists of news articles posted on the SDG Knowledge Hub website (sdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='iisd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This website was launched in October 2016 and is managed by the International Institute for Sus- tainable Development (IISD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' It hosts news and commentary regarding the implementation of the SDGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The news articles contain labels that show which SDGs they cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' These la- bels are assigned by the subject experts who write these news articles and confirmed by SDG Knowledge Hub editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We downloaded 9,172 news articles that have been published on the website along with the assigned SDG labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Unlabeled data The Disneyland reviews, cooking recipes, and math lec- tures were obtained from Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The Disneyland data set contains 42,656 reviews of Disneyland locations in Paris, California, or Hong Kong posted by visitors on Trip Advi- sor (see kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The cooking recipe data set contains 82,245 recipes scraped from food-related websites, such as skinnytaste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='com (see kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The math lecture data set contains 860 lectures posted on YouTube by institutions or creators covering 11 subjects, ranging from algebra to natu- ral language processing, that are related to computer science and mathematics (see kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The synthetic data sets were generated by concatenating words sampled at random based on the words’ frequencies in the (Wikipedia corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The synthetic texts, thereby, re- flect the natural word frequency distribution found in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' text2sdg To detect SDGs in these different texts, we used the text2sdg R package (text2sdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='io Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' text2sdg provides a common framework for implementing the differ- ent systems to detect SDGs in text and makes it easy to quan- titatively compare and visualize their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The text2sdg also makes available the ensemble model presented in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Ensemble modeling We recruited two types of algorithms, random forest (R package ranger Wright & Ziegler, 2015) and extreme gra- dient boosting (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', 2015, R package xgboost), to train ensembles of SDG labeling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' We trained the algorithms separately for each SDG to predict the presence or absence of the SDG based on the predictions of six dif- ferent SDG labeling systems implemented in text2sdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='io and the number of words in the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Training and evalua- tion were performed using a repeated k-fold cross validation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The models were trained using all three expert- labeled and synthetic data sets, with the latter matching the former in numbers and word lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' The cases in the data sets were initially weighted by 1/N with N being the num- ber of cases in a data set to give each data set equal weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Furthermore, we multiplied the weight of the synthetic data sets by a factor k ∈ [0, 10] to vary the weight of synthetic data relative to the expert-labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' A factor of k = 0 means that the synthetic data receives no weight whereas a factor of k = 10 means that the synthetic data receive ten times as much weight as the expert-labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Overall, we found the random forest to perform slightly better than the gradient-boosting algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' hence, we report the results of the random forest in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' References Armitage, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Ranger: A fast imple- mentation of random forests for high dimensional data in c++ and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' arXiv preprint arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='04409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Wulff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=', & Meier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' SDG Knowledge Hub Dataset of SDG-labeled News Articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='7523032 Data availability The text2sdg R package is available on the Compre- hensive R Network (https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content='org/package= text2sdg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' All data sets used in this analysis are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Links are included in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Acknowledgements We are grateful to Laura Wiles for editing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' This work was supported by a grant from the Swiss Science Foundation (100015_197315) to Dirk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} +page_content=' Wulff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFIT4oBgHgl3EQfwytZ/content/2301.11353v1.pdf'} diff --git a/rNAzT4oBgHgl3EQfO_uf/vector_store/index.faiss b/rNAzT4oBgHgl3EQfO_uf/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..413ce7f7dc29cef00219d5a08d818fd0f6babce0 --- /dev/null +++ b/rNAzT4oBgHgl3EQfO_uf/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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b/utE0T4oBgHgl3EQf9gLX/content/tmp_files/2301.02803v1.pdf.txt @@ -0,0 +1,538 @@ +From graph theory and geometric probabilities to +a representative width for three-dimensional detonation cells +Vianney Monnier, Vincent Rodriguez, Pierre Vidal, Ratiba Zitoun +Institut Pprime, UPR 3346 CNRS, ENSMA, BP 40109, 86961 Futuroscope-Chasseneuil, France +January 10, 2023 +We present a model for predicting a representative width λ for the three-dimensional cells observed on +the detonation fronts in reactive gases. The physical premise is that the 3D unsteady cellular process +for irregular cells is stochastic and produce the same burnt mass as the average planar steady ZND +process per unit of time. Graph theory defines an ideal cell whose grouping is equivalent to that of the +actual 3D cellular front. Geometric probabilities determine the mean burnt fraction that parameterizes +the model. The ZND model closes the problem with the relation time-position of a fluid element in the +ZND steady reaction zone. The comparison of measured and calculated λ shows agreement to better +than or within the accepted experimental uncertainties, depending on the reactive mixture. The quality +of this estimate is dependent solely on that of the detailed kinetic scheme used for the ZND calculations, +the modelling assumptions aside. The model is easily implementable as a post-process of ZND profiles +that produces instantaneously the estimates of the cell width, length and reaction time, and the ZND +reaction length and time. +1 +Introduction +First identified experimentally in the late 1950s [1], the cellular structure of the detonation reaction zone in +gases is viewed today as an example of non-linear instability of combustion waves in compressible reactive +fluids, e.g. [2, 3, 4]. It is now recognized that physical representations of this unsteady structure can only +be three-dimensional. Experimental and numerical analyses of front views of detonation waves, e.g. [5, 6], +evidence that the cellular structure is made up of irregular patterns if the number of cells on the front +surface is sufficiently large. That is typically observed in the case of detonation propagation in tubes with +cross sections sufficiently large because the usual cell descriptor, namely its mean width λ, decreases when +increasing the initial pressure p0 of the gas. The accepted modelling framework then involves hydrodynamics +and chemical kinetics solely and no participation of viscosity as, for example, in boundary layers and +turbulence. A topic of debate is whether a unique characteristic length is relevant to characterizing a 3D +cell. The model below assumes that such is the mean width λ and, therefore, elaborates on the analysis +outlined from our recordings of three-dimensional detonation cells [5] and presented to the 28th ICDERS +(Naples 2022). +The three ingredients are graph theory, geometric probabilities and the Zel’dovich-Von Neuman-D¨oring +(ZND) model of planar detonation. First (Sect.2), we express the physical premise that the 3D unsteady +cellular process for irregular cells is stochastic and should produce the same burnt mass as the average +planar steady ZND process per unit of time. Then (Sect.3), we use graph theory to define an ideal cell +whose grouping is equivalent to the actual 3D cellular front [5] and geometric probabilities to determine +the mean burnt fraction that parameterizes the model. Finally (Sect.4), we implement the ZND model +with detailed schemes of chemical kinetics to calculate the relation time-position of a fluid element in the +ZND steady reaction zone, respective to its leading shock, which closes the problem of determining λ. The +comparison of measured and calculated λ shows agreement to better than or within the accepted experi- +mental uncertainties, depending on the reactive mixture, its initial pressure p0 and equivalence ratio. Thus, +the quality of this estimate is dependent solely on that of the kinetic scheme, the modelling assumptions +aside. The model is easily implementable as a post-process of ZND profiles that produces instantaneously +the estimates of the cell width, length and reaction time, and ZND reaction length and time. +Correspondence to: pierre.vidal@ensma.fr +1 +arXiv:2301.02803v1 [physics.flu-dyn] 7 Jan 2023 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +2 +Model +The basic assumption is that the cellular and ZND processes burn the fresh mixture at the same mass +rate for sufficiently large periods and the same projected front area. Let t and z denote the time and the +position in the ZND reaction zone, respective to its leading shock, and ∆tC the period during which the +ZND front travels the distance LC representing the length of the ideal cell. For the self-sustained detonation +propagating at the Chapman-Jouguet (CJ) velocity DCJ, +LC = DCJ∆tC. +(1) +A reaction time below refers to the period necessary to fully burn all fluid elements captured by a front at +the initial instant t = 0 and through the same reference surface area. Thus, in the ZND process, denoting +by tZ its reaction time, the fluid elements entered in the reaction zone during the period 0 < t ⩽ tZ can +only be partially burnt at tZ. That results in the mean ZND burnt fraction ¯yZ and reaction rate ¯yZ/tZ. In +the cellular process, the front is a grouping of forward-convex waves whose forefront velocities for irregular +cells randomly vary about the ZND mean velocity, such as DCJ. Their boundaries are the intersections with +transverse waves that sweep the surfaces of the forward waves with lower velocities. High-speed recordings, +e.g. [7, 8, 9] indicate that combustion is ensured at an instantaneous rate much rapid in the domains behind +the transverse waves and the forward waves with higher velocities, that is, much faster than the mean +cellular rate. A symmetry argument then suggests that the reaction time – as defined above – of the ideal +cell should be half the cell time ∆tC/2. Indeed, the period [0, ∆tC/2] is that necessary, on average, for the +transverse waves to sweep a projected front area equivalent to the maximum area of the ideal cell, which, by +symmetry, occurs every cell half length LC/2. Thus, during this period, they cover the surface of the ideal +cell, and they can capture and burn all the fluid elements that have crossed the lower-velocity front surfaces +since t = 0. That results in the mean cell burnt fraction ¯yC and reaction rate 2/∆tC – not ¯yC × 2/∆tC. +These means of the mass fractions yZ and yC are relative to periods elapsed since t = 0, +¯yZ = 1 +tZ +� tZ +0 +yZ(t′)dt′, +¯yC = +2 +∆tC +� ∆tC/2 +0 +yC(t′)dt′. +(2) +where the subscripts Z and C denote the ZND and the cellular processes. The first definition above also +applies to any variable, for example, the material speed UZ (t) = dz (t) /dt at the time t - or the position +z (t) of a fluid element - in the ZND reaction zone. This defines the ZND reaction length ℓZ by +ℓZ = +� tZ +0 +UZ(t′)dt′ = ¯UZ × tZ, +¯UZ = ℓZ +tZ +, +(3) +where ¯UZ denotes the mean of UZ (t). With v denoting the specific volume, and v0 its initial value, the +relation of mass conservation written as vZ(t)DCJ = v0UZ(t) at the position z (t) can also be averaged, so +(3) rewrites +ℓZ += +¯vZ +v0 +DCJ × tZ, +(4) +¯vZ (¯yZ) += +(1 − ¯yZ) vH + ¯yZvCJ. +(5) +Relation (5) results from the averaging of the volume additivity constraint v = � yivi, where vi and yi +denote the specific volume and the mass fraction of the chemical species i, and vH and vCJ the specific +volumes at the ZND shock and reaction end positions. +The equality of the mean reaction rates of the cellular and ZND processes implies that of their mean +reaction progress variables ¯yC and ¯yZ respective to their reaction times ∆tC/2 and tZ, so +2 +∆tC += ¯y +tZ +, +(6) +where ¯y denotes ¯yC = ¯yZ. The combination of (1) with (6) gives the relation (7) between the cell length +LC and the ZND reaction time tZ, which, with (4) and (5), gives the relation (8) between LC and the ZND +Institut Pprime UPR3346 CNRS +January 10, 2023 +2 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +reaction length lZ, +LC1 (¯y, tZ) += +k1 × tZ, +k1 (¯y) = 2 +¯y DCJ, +(7) +LC2 (¯y, ℓZ) += +k2 × ℓZ, +k2 (¯y) = 2 +¯y +v0 +¯vZ (¯y). +(8) +In the next section, graph theory is used to define a cellular pattern statistically equivalent to the 3D +cellular front, hence the ideal cell to which geometric probabilities are applied to obtain ¯y. The cell length +LC is defined by the intersection of the curves LC1 (t) = k1 × t and LC2 (t) = k2 × z (t), with the relation +time-position z (t) of a fluid element preliminary determined through classical ZND numerical calculations +with a detailed scheme of chemical kinetics. The ZND reaction time tZ and length ℓZ are then obtained +from (7) and (8), and, finally, the cell width ¯λ is geometrically related to LC. +3 +Graph theory and geometric probabilities +In our preliminary analysis [5], we considered the front-view distribution of the same pattern, for example, +rectangle, pentagon, hexagon, etc., to be statistically independent on the front position if the cell number +F, that is the initial pressure p0, is sufficiently large. Indeed, several experiments carried out in the same +conditions should return the same distribution. We used elements from planar graph theory to show that +these irregular front views are equivalent to tessellations of hexagons. This was obtained by combining the +physical condition that only three transverse waves can intersect with the mathematical limit at large F of +the Descartes-Euler-Poincar´e relation F − E + V = 2 that connects the number of faces F (the cells), edges +E (the transverse waves) and vertices V (the edge intersections) in a tessellation. For three-edge vertices, +2E = 3V , so the limit at large F of the edge number per face 2E/F is 6. One consequence is that a cell +counting on an experimental recording gives an estimate of the cell mean width through +¯λ = 3 ln 3 +π +√ +3 +2 dx, +dx = +� +8 +3 +√ +3AC, +AC = AT +F , +(9) +where dx and AC are the outer diameter and the area of the hexagon, and AT the cross-section area of +the tube. Another consequence, detailed below, is that the mean reaction progress variable ¯y and hence +the cell mean width ¯λ (Sect.2) can be predicted by combining geometric probabilities with properties of +this representative tessellation. The premise is that the motion of the transverse waves for irregular cells is +stochastic. +First, we define a control volume with the surface area AC and the half length LC/2 of the ideal cell. We +denote by MC the mass contained in this volume, M (t) the mass having crossed the area AC since t = 0 at +the intermediate instant 0 < t ⩽ ∆tC/2 (or the front position 0 < x (t) ⩽ LC/2 in the laboratory frame), +and MB (t) the mass burnt at this instant t. They write +MC = ρ0AC × LC +2 , +M (t) = ρ0AC × x (t) , +MB (t) = ρ0AB (t) × x (t) , +(10) +where ρ0 denotes the initial specific mass and AB (t) the surface area swept by the transverse waves at the +instant t since t = 0. Thus, the burnt mass fraction yC at the instant t is +yC (t) = MB (t) +M (t) = AB (t) +AC +, +(11) +so its mean ¯yC (2) is the mean combustion area respective to the cell area AC. +Next, we express the stochasticity of the transverse-wave motion. The successive positions of the trans- +verse waves in a same experiment, projected onto the surface of the ideal cell, should be statistically +equivalent to those obtained from one experiment to another at the same front position, that is, to those of +line segments randomly dropped onto the surface. This ensures that combustion efficiency is, on average, +Institut Pprime UPR3346 CNRS +January 10, 2023 +3 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +independent of experiment. Thus, ¯yC is the probability that the segments are completely contained in the +cell surface, that is, the non-intersect probability, for the propagation period ∆tC/2. The calculation is a +classical problem of geometric probabilities, namely the Buffon’s needle problem extended to a surface with +a hexagonal tiling and needle lengths varying between 0 and the hexagon outer diameter dx. In the many +accounts of such problems, the non-intersection probabilities are expressed as a ratio µC/µ, where µ is the +measure of the space of the independent variables representing all the random orientations and positions +of a segment, and µC the measure of the subspace in which these variables should vary so the segments +do not intersect the boundaries of the typical tessellation element. For the hexagon, we extend below to a +variable-length segment the solution by Vassallo [10] for a constant-length segment. We do not reproduce +his calculations for conciseness and because of the clarity of his account. The independent variables are the +segment length s varying in [0, dx], its angle θ varying in [0, 2π], and the coordinates of its center varying +in the cell surface of area AC. The non-intersection constraint demands this center to be contained into +a smaller surface than that of the cell, and whose shape and area depend on the segment orientation and +length. Nondimensionalizing the lengths by the side length dx/2 of the typical hexagon, and denoting by +r = 2s/dx the non-dimensional segment lengths, we have +µ += +6π +√ +3, +µC = m1 + m2 + m3, +mi = +� ri2 +ri1 +µi (r) dr, +(12) +r +∈ +[r11 = 0, r12 = 1] , +µ1 (r) = 3π +√ +3 − 12r + r2 � +3 − π/ +√ +3 +� +, +(13) +r +∈ +� +r21 = 1, r22 = +√ +3 +� +, +µ2 (r) = π +√ +3 +� +r2 + 5 +� +− 9 +� +4r2 − 3 ... +(14) +... − 2 +√ +3 +� +3 + 2r2� +arcsin +�√ +3/2r +� +, +(15) +r +∈ +� +r31 = +√ +3, r32 = 2 +� +, +µ3 (r) = 2 +√ +3 +� +r2 + 12 +� +arcsin +�√ +3/r +� +... +(16) +... + 30 +� +r2 − 3 − +� +8π +√ +3 + 18 +� +− r2 � +3 + 2π/ +√ +3 +� +, +(17) +m1 += +10.720, +m2 = 1. 837 4, +m3 = 1.154 7 × 10−2, +(18) +where the µis are Vassallo’s non-intersection measures for constant segment lengths [10] and the mis ours +for segment lengths varying in the intervals [0, dx/2], +� +dx/2, +√ +3dx/2 +� +and +�√ +3dx/2, dx +� +. +This gives the +non-intersection probability, that is, the mean burnt fraction ¯yC (≡ ¯y, Sect.2), by +µ = 32.648, +µC = 12.569, +¯y = µC +µ ≈ 0.38498, +(19) +so, from (6), the cellular process takes, on average, ∆tC/tZ ≈ 5.2 times longer than the ZND process to +achieve combustion. +Finally, we obtain the aspect ratio ¯λ/LC by combining stochasticity and, inspired by [11], geometry. +Since the transverse waves have a stochastic motion, their positions can be considered to be the same every +period ∆tC, so the longitudinal overdriven front waves of the model cellular front should superimpose on +each other every distance LC. Equivalently, these waves can be viewed as the upper surface elements of +spheres arranged in the hexagonal closest packing, that is, with alternate layers in the ABAB ... sequence. +The sphere diameter is also the distance between the centers of adjacent spheres and the inner diameter of +a hexagon, di, so the ratio LC/di comes out as twice the height of the tetrahedral pyramid whose base is +the triangle with vertices the centers of the three closest spheres in the same layer. Simple geometry then +gives LC/di = +� +8/3 and di/dx = +√ +3/2. With the first relation (9), that yields the mean cell aspect ratio +¯λ +LC += 3 ln 3 +π +� +3 +8 ≈ 0.64244 +(20) +and opening angle 65.4o well representing the measurements on longitudinal recordings and, with (6), (19) +and the basic relation between mass and volume fractions, the ratio of lengths LC/ℓZ and ¯λ/ℓZ which, +depending on vH and vCJ, take the accepted large values of ≈ 20 − 40. +Institut Pprime UPR3346 CNRS +January 10, 2023 +4 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +4 +Results and discussion +For conciseness, we do not detail the classical system of 1st order ordinary differential equations that +governs the ZND model. The quality of the chemical-kinetics schemes implemented in the ZND calculations +determines that of the cell mean width estimates by the model above, its assumptions aside. We selected +mixtures with H2, C3H8 or CH4 as the fuel and O2 or Air as the oxidizer because of their practical importance +and the attention they have received from kineticists, and we thus used the Konnov [12] and the FFCM-1 +[13] detailed schemes. +Figure 1 compares the calculation results (full and open symbols +, +, +) to measurements in tubes +(crosses) as presented in the detonation database [14]. Denoting by d the transverse dimensions of the +tubes, the black (+) and grey (×) crosses denote values of the ratio ¯λ/d smaller and larger, respectively, +than the arbitrary magnitude O(10). That gives O(100) cells on the front surface, which we considered +to be a number sufficiently large to meet the model’s basic assumption. We collected the tube dimensions +from the original references. +The comparison shows a very good agreement by the trends and the magnitudes and, in most cases, +to better than the ≈ 50 − 100% typical standard deviation for the mean width of irregular cells. Overall, +the smaller the cell mean width compared to the tube dimension (i.e., the black crosses +), the better the +agreement, consistent with the assumption of many cells on the front surface. The agreement is also better +with mixture compositions close to the stoichiometric balance, perhaps because this is the usual calibration +range of kinetic schemes. +The cell mean width is still considered to be a useful characteristic length in detonation dynamics, +although the complex 3D patterns observed on detonation front views question the sufficiency of its repre- +sentativeness and are the likely reason for the large standard deviations. In view of this intrinsic limitation, +we felt that simple global modelling could yield a representative cell width sufficiently accurate for practical +purposes without detailing the complex wave interactions that form the cellular structure. +Our approach does not pretend to explain how the detonation reaction zones are unstable and only +assumes a global equivalence of the ZND and cellular processes expressed by basic conservation and action +principles. Since its implementation is an easy post-process of ZND profiles, a complementary application +is its use as an inverse method to assess the representative capacity of kinetic schemes from cell measure- +ments obtained in conditions that eliminate the confinement effects. Current work includes other mixtures +and schemes – for example, we obtained the same good agreement with the San-Diego mechanism, where +applicable – and a comparative discussion with regular cells. +Institut Pprime UPR3346 CNRS +January 10, 2023 +5 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +References +[1] Y. Denisov, Y. Troshin, Pulsating and spinning detonation of gaseous mixtures in tubes, dokl. akad. +nauk sssr, 125(1) (1959) 110–113, , and Structure of gaseous detonation in tubes, Sov. Phys. Tech. +Phys., 5(4) (1960) 419-431 (1959). +[2] M. Short, G. Sharpe, Pulsating instability of detonations with a two-step chain-branching reaction +model: theory and numerics, Combust. Theor. Model. 7 (2) (2003) 401–416. +URL https://www.tandfonline.com/doi/abs/10.1088/1364-7830/7/2/311 +[3] M. Radulescu, G. Sharpe, D. Bradley, A universal parameter quantifying explosion hazards, deton- +ability and hot spot formation: χ number, in: Proc. 7th Int. Seminar on Fire and Explosion Hazards, +Research Publishing, 2013, pp. 617–626. doi:10.3850/978-981-07-5936-0_10-01. +[4] P. Clavin, G. Searby, Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation +Fronts and Explosion of Stars, Cambridge University Press, 2016. doi:10.1017/CBO9781316162453. +[5] V. Monnier, V. Rodriguez, P. Vidal, R. Zitoun, An analysis of three-dimensional patterns of experi- +mental detonation cells, Combustion and Flame 245 (2022) 112310. doi:10.1016/j.combustflame. +2022.112310. +[6] J. Crane, J. Lipkowicz, X. Shi, I. Wlokas, A. Kempf, H. Wang, Three-dimensional detonation structure +and its response to confinement, Proceedings of the Combustion Institute (2022). doi:10.1016/j. +proci.2022.10.019. +[7] F. Pintgen, J. Shepherd, Simultaneous soot foil and plif imaging of propagating detonations, Proc. 19th +ICDERS Paper 119 (2003). +[8] F. Pintgen, C. Eckett, J. Austin, J. Shepherd, Direct observations of reaction zone structure in prop- +agating detonations, Combustion and Flame 133 (3) (2003) 211–229. doi:10.1016/S0010-2180(02) +00458-3. +[9] J. Austin, F. Pintgen, J. Shepherd, Reaction zones in highly unstable detonations, Proc. Combust. +Inst. 30 (2) (2005) 1849–1857. doi:10.1016/j.proci.2004.08.157. +[10] S. Vassallo, Buffon’s coin and needle problems for the snub hexagonal tiling, Advances in Mathematics: +Scientific Journal 10 (2021) 2223–2233. doi:https://doi.org/10.37418/amsj.10.4.36. +[11] R. Takai, K. Yoneda, T. Hikita, Study of detonation wave structure, Symposium (International) on +Combustion 15 (1) (1975) 69–78, fifteenth Symposium (International) on Combustion. doi:10.1016/ +S0082-0784(75)80285-2. +[12] F. Coppens, J. De Ruyck, A. Konnov, The effects of composition on burning velocity and nitric oxide +formation in laminar premixed flames of ch4 + h2 + o2 + n2, Combustion and Flame 149 (4) (2007) +409–417. doi:10.1016/j.combustflame.2007.02.004. +[13] G. Smith, Y. Tao, H. Wang, Foundational fuel chemistry model version 1.0 (ffcm-1) 2016 (2016). +URL https://web.stanford.edu/group/haiwanglab/FFCM1/pages/download.html +[14] M. Kaneshige, J. Shepherd, Detonation database, California Institute of Technology; Online available; +accessed march 2022 (2002). +URL https://shepherd.caltech.edu/detn_db/html/db.html +Institut Pprime UPR3346 CNRS +January 10, 2023 +6 + +Monnier et al. +From graph theory and geometric probabilities to detonation cell widths +Figures +Figure 1: +Comparison of calculated and measured cell mean widths ¯λ. Full and open symbols: calculations using +the Konnov [12] ( +and +) and the FFCM-1 [13] ( ) schemes of chemical kinetics. Crosses: measurements [14] with +small (⩽ O(10), black +) and large (⩾ O(10), grey ×) ratios ¯λ/d, with d the transverse dimensions of the tubes. +Institut Pprime UPR3346 CNRS +January 10, 2023 +7 + +102 +H2-O2 +ER=1 +Cell mean width (mm) +101 +XXX ++ ++ ++ +100 ++ ++ ++# +10-1 +10 +100 +1000 +Initial pressure po (kPa)101 +C,Hg-O2 +ER= 1 +100 +10 +100 +Initial pressure po (kPa)103 +Hz-Air +Po = 101.3 kPa +102 +X +X +X +X +101 ++ +100 +0.5 +2 +Equivalence Ratio103 +H2-Air +ER=1.0: ++ +ER=0.5: X^ +X +X +102 +X ++ ++ +101 +100 +10 +100 +1000 +Initial pressure po (kPa)15 +CH4-O2 +Po = 120.0 kPa +10 ++ +0.7 +1.4 +Equivalence Ratio102 +CH,-O2 +ER= 1 +101 +X +# +100 ++ ++ +# +10-1 +10 +100 +1000 +Initial pressure po (kPa) \ No newline at end of file diff --git a/utE0T4oBgHgl3EQf9gLX/content/tmp_files/load_file.txt b/utE0T4oBgHgl3EQf9gLX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cae847c0e054ee6d9889fc75972d9867a6604e0e --- /dev/null +++ b/utE0T4oBgHgl3EQf9gLX/content/tmp_files/load_file.txt @@ -0,0 +1,265 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf,len=264 +page_content='From graph theory and geometric probabilities to a representative width for three-dimensional detonation cells Vianney Monnier, Vincent Rodriguez, Pierre Vidal, Ratiba Zitoun Institut Pprime, UPR 3346 CNRS, ENSMA, BP 40109, 86961 Futuroscope-Chasseneuil, France January 10, 2023 We present a model for predicting a representative width λ for the three-dimensional cells observed on the detonation fronts in reactive gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The physical premise is that the 3D unsteady cellular process for irregular cells is stochastic and produce the same burnt mass as the average planar steady ZND process per unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Graph theory defines an ideal cell whose grouping is equivalent to that of the actual 3D cellular front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Geometric probabilities determine the mean burnt fraction that parameterizes the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The ZND model closes the problem with the relation time-position of a fluid element in the ZND steady reaction zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The comparison of measured and calculated λ shows agreement to better than or within the accepted experimental uncertainties, depending on the reactive mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The quality of this estimate is dependent solely on that of the detailed kinetic scheme used for the ZND calculations, the modelling assumptions aside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The model is easily implementable as a post-process of ZND profiles that produces instantaneously the estimates of the cell width, length and reaction time, and the ZND reaction length and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' 1 Introduction First identified experimentally in the late 1950s [1], the cellular structure of the detonation reaction zone in gases is viewed today as an example of non-linear instability of combustion waves in compressible reactive fluids, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' It is now recognized that physical representations of this unsteady structure can only be three-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Experimental and numerical analyses of front views of detonation waves, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' [5, 6], evidence that the cellular structure is made up of irregular patterns if the number of cells on the front surface is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' That is typically observed in the case of detonation propagation in tubes with cross sections sufficiently large because the usual cell descriptor, namely its mean width λ, decreases when increasing the initial pressure p0 of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The accepted modelling framework then involves hydrodynamics and chemical kinetics solely and no participation of viscosity as, for example, in boundary layers and turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' A topic of debate is whether a unique characteristic length is relevant to characterizing a 3D cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The model below assumes that such is the mean width λ and, therefore, elaborates on the analysis outlined from our recordings of three-dimensional detonation cells [5] and presented to the 28th ICDERS (Naples 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The three ingredients are graph theory, geometric probabilities and the Zel’dovich-Von Neuman-D¨oring (ZND) model of planar detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' First (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='2), we express the physical premise that the 3D unsteady cellular process for irregular cells is stochastic and should produce the same burnt mass as the average planar steady ZND process per unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Then (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='3), we use graph theory to define an ideal cell whose grouping is equivalent to the actual 3D cellular front [5] and geometric probabilities to determine the mean burnt fraction that parameterizes the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Finally (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='4), we implement the ZND model with detailed schemes of chemical kinetics to calculate the relation time-position of a fluid element in the ZND steady reaction zone, respective to its leading shock, which closes the problem of determining λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The comparison of measured and calculated λ shows agreement to better than or within the accepted experi- mental uncertainties, depending on the reactive mixture, its initial pressure p0 and equivalence ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Thus, the quality of this estimate is dependent solely on that of the kinetic scheme, the modelling assumptions aside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The model is easily implementable as a post-process of ZND profiles that produces instantaneously the estimates of the cell width, length and reaction time, and ZND reaction length and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Correspondence to: pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='vidal@ensma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='fr 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='02803v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='flu-dyn] 7 Jan 2023 Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' From graph theory and geometric probabilities to detonation cell widths 2 Model The basic assumption is that the cellular and ZND processes burn the fresh mixture at the same mass rate for sufficiently large periods and the same projected front area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Let t and z denote the time and the position in the ZND reaction zone, respective to its leading shock, and ∆tC the period during which the ZND front travels the distance LC representing the length of the ideal cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' For the self-sustained detonation propagating at the Chapman-Jouguet (CJ) velocity DCJ, LC = DCJ∆tC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (1) A reaction time below refers to the period necessary to fully burn all fluid elements captured by a front at the initial instant t = 0 and through the same reference surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Thus, in the ZND process, denoting by tZ its reaction time, the fluid elements entered in the reaction zone during the period 0 < t ⩽ tZ can only be partially burnt at tZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' That results in the mean ZND burnt fraction ¯yZ and reaction rate ¯yZ/tZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' In the cellular process, the front is a grouping of forward-convex waves whose forefront velocities for irregular cells randomly vary about the ZND mean velocity, such as DCJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Their boundaries are the intersections with transverse waves that sweep the surfaces of the forward waves with lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' High-speed recordings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' [7, 8, 9] indicate that combustion is ensured at an instantaneous rate much rapid in the domains behind the transverse waves and the forward waves with higher velocities, that is, much faster than the mean cellular rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' A symmetry argument then suggests that the reaction time – as defined above – of the ideal cell should be half the cell time ∆tC/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Indeed, the period [0, ∆tC/2] is that necessary, on average, for the transverse waves to sweep a projected front area equivalent to the maximum area of the ideal cell, which, by symmetry, occurs every cell half length LC/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Thus, during this period, they cover the surface of the ideal cell, and they can capture and burn all the fluid elements that have crossed the lower-velocity front surfaces since t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' That results in the mean cell burnt fraction ¯yC and reaction rate 2/∆tC – not ¯yC × 2/∆tC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' These means of the mass fractions yZ and yC are relative to periods elapsed since t = 0, ¯yZ = 1 tZ � tZ 0 yZ(t′)dt′, ¯yC = 2 ∆tC � ∆tC/2 0 yC(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (2) where the subscripts Z and C denote the ZND and the cellular processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The first definition above also applies to any variable, for example, the material speed UZ (t) = dz (t) /dt at the time t - or the position z (t) of a fluid element - in the ZND reaction zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' This defines the ZND reaction length ℓZ by ℓZ = � tZ 0 UZ(t′)dt′ = ¯UZ × tZ, ¯UZ = ℓZ tZ , (3) where ¯UZ denotes the mean of UZ (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' With v denoting the specific volume, and v0 its initial value, the relation of mass conservation written as vZ(t)DCJ = v0UZ(t) at the position z (t) can also be averaged, so (3) rewrites ℓZ = ¯vZ v0 DCJ × tZ, (4) ¯vZ (¯yZ) = (1 − ¯yZ) vH + ¯yZvCJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (5) Relation (5) results from the averaging of the volume additivity constraint v = � yivi, where vi and yi denote the specific volume and the mass fraction of the chemical species i, and vH and vCJ the specific volumes at the ZND shock and reaction end positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The equality of the mean reaction rates of the cellular and ZND processes implies that of their mean reaction progress variables ¯yC and ¯yZ respective to their reaction times ∆tC/2 and tZ, so 2 ∆tC = ¯y tZ , (6) where ¯y denotes ¯yC = ¯yZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The combination of (1) with (6) gives the relation (7) between the cell length LC and the ZND reaction time tZ, which, with (4) and (5), gives the relation (8) between LC and the ZND Institut Pprime UPR3346 CNRS January 10, 2023 2 Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' From graph theory and geometric probabilities to detonation cell widths reaction length lZ, LC1 (¯y, tZ) = k1 × tZ, k1 (¯y) = 2 ¯y DCJ, (7) LC2 (¯y, ℓZ) = k2 × ℓZ, k2 (¯y) = 2 ¯y v0 ¯vZ (¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (8) In the next section, graph theory is used to define a cellular pattern statistically equivalent to the 3D cellular front, hence the ideal cell to which geometric probabilities are applied to obtain ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The cell length LC is defined by the intersection of the curves LC1 (t) = k1 × t and LC2 (t) = k2 × z (t), with the relation time-position z (t) of a fluid element preliminary determined through classical ZND numerical calculations with a detailed scheme of chemical kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The ZND reaction time tZ and length ℓZ are then obtained from (7) and (8), and, finally, the cell width ¯λ is geometrically related to LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' 3 Graph theory and geometric probabilities In our preliminary analysis [5], we considered the front-view distribution of the same pattern, for example, rectangle, pentagon, hexagon, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=', to be statistically independent on the front position if the cell number F, that is the initial pressure p0, is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Indeed, several experiments carried out in the same conditions should return the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' We used elements from planar graph theory to show that these irregular front views are equivalent to tessellations of hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' This was obtained by combining the physical condition that only three transverse waves can intersect with the mathematical limit at large F of the Descartes-Euler-Poincar´e relation F − E + V = 2 that connects the number of faces F (the cells), edges E (the transverse waves) and vertices V (the edge intersections) in a tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' For three-edge vertices, 2E = 3V , so the limit at large F of the edge number per face 2E/F is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' One consequence is that a cell counting on an experimental recording gives an estimate of the cell mean width through ¯λ = 3 ln 3 π √ 3 2 dx, dx = � 8 3 √ 3AC, AC = AT F , (9) where dx and AC are the outer diameter and the area of the hexagon, and AT the cross-section area of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Another consequence, detailed below, is that the mean reaction progress variable ¯y and hence the cell mean width ¯λ (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='2) can be predicted by combining geometric probabilities with properties of this representative tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The premise is that the motion of the transverse waves for irregular cells is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' First, we define a control volume with the surface area AC and the half length LC/2 of the ideal cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' We denote by MC the mass contained in this volume, M (t) the mass having crossed the area AC since t = 0 at the intermediate instant 0 < t ⩽ ∆tC/2 (or the front position 0 < x (t) ⩽ LC/2 in the laboratory frame), and MB (t) the mass burnt at this instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' They write MC = ρ0AC × LC 2 , M (t) = ρ0AC × x (t) , MB (t) = ρ0AB (t) × x (t) , (10) where ρ0 denotes the initial specific mass and AB (t) the surface area swept by the transverse waves at the instant t since t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Thus, the burnt mass fraction yC at the instant t is yC (t) = MB (t) M (t) = AB (t) AC , (11) so its mean ¯yC (2) is the mean combustion area respective to the cell area AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Next, we express the stochasticity of the transverse-wave motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The successive positions of the trans- verse waves in a same experiment, projected onto the surface of the ideal cell, should be statistically equivalent to those obtained from one experiment to another at the same front position, that is, to those of line segments randomly dropped onto the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' This ensures that combustion efficiency is, on average, Institut Pprime UPR3346 CNRS January 10, 2023 3 Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' From graph theory and geometric probabilities to detonation cell widths independent of experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Thus, ¯yC is the probability that the segments are completely contained in the cell surface, that is, the non-intersect probability, for the propagation period ∆tC/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The calculation is a classical problem of geometric probabilities, namely the Buffon’s needle problem extended to a surface with a hexagonal tiling and needle lengths varying between 0 and the hexagon outer diameter dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' In the many accounts of such problems, the non-intersection probabilities are expressed as a ratio µC/µ, where µ is the measure of the space of the independent variables representing all the random orientations and positions of a segment, and µC the measure of the subspace in which these variables should vary so the segments do not intersect the boundaries of the typical tessellation element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' For the hexagon, we extend below to a variable-length segment the solution by Vassallo [10] for a constant-length segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' We do not reproduce his calculations for conciseness and because of the clarity of his account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The independent variables are the segment length s varying in [0, dx], its angle θ varying in [0, 2π], and the coordinates of its center varying in the cell surface of area AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The non-intersection constraint demands this center to be contained into a smaller surface than that of the cell, and whose shape and area depend on the segment orientation and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Nondimensionalizing the lengths by the side length dx/2 of the typical hexagon, and denoting by r = 2s/dx the non-dimensional segment lengths, we have µ = 6π √ 3, µC = m1 + m2 + m3, mi = � ri2 ri1 µi (r) dr, (12) r ∈ [r11 = 0, r12 = 1] , µ1 (r) = 3π √ 3 − 12r + r2 � 3 − π/ √ 3 � , (13) r ∈ � r21 = 1, r22 = √ 3 � , µ2 (r) = π √ 3 � r2 + 5 � − 9 � 4r2 − 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (14) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' − 2 √ 3 � 3 + 2r2� arcsin �√ 3/2r � , (15) r ∈ � r31 = √ 3, r32 = 2 � , µ3 (r) = 2 √ 3 � r2 + 12 � arcsin �√ 3/r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' (16) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' + 30 � r2 − 3 − � 8π √ 3 + 18 � − r2 � 3 + 2π/ √ 3 � , (17) m1 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='720, m2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' 837 4, m3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='154 7 × 10−2, (18) where the µis are Vassallo’s non-intersection measures for constant segment lengths [10] and the mis ours for segment lengths varying in the intervals [0, dx/2], � dx/2, √ 3dx/2 � and �√ 3dx/2, dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' This gives the non-intersection probability, that is, the mean burnt fraction ¯yC (≡ ¯y, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='2), by µ = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='648, µC = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='569, ¯y = µC µ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='38498, (19) so, from (6), the cellular process takes, on average, ∆tC/tZ ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='2 times longer than the ZND process to achieve combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Finally, we obtain the aspect ratio ¯λ/LC by combining stochasticity and, inspired by [11], geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Since the transverse waves have a stochastic motion, their positions can be considered to be the same every period ∆tC, so the longitudinal overdriven front waves of the model cellular front should superimpose on each other every distance LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Equivalently, these waves can be viewed as the upper surface elements of spheres arranged in the hexagonal closest packing, that is, with alternate layers in the ABAB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The sphere diameter is also the distance between the centers of adjacent spheres and the inner diameter of a hexagon, di, so the ratio LC/di comes out as twice the height of the tetrahedral pyramid whose base is the triangle with vertices the centers of the three closest spheres in the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Simple geometry then gives LC/di = � 8/3 and di/dx = √ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' With the first relation (9), that yields the mean cell aspect ratio ¯λ LC = 3 ln 3 π � 3 8 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='64244 (20) and opening angle 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='4o well representing the measurements on longitudinal recordings and, with (6), (19) and the basic relation between mass and volume fractions, the ratio of lengths LC/ℓZ and ¯λ/ℓZ which, depending on vH and vCJ, take the accepted large values of ≈ 20 − 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Institut Pprime UPR3346 CNRS January 10, 2023 4 Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' From graph theory and geometric probabilities to detonation cell widths 4 Results and discussion For conciseness, we do not detail the classical system of 1st order ordinary differential equations that governs the ZND model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The quality of the chemical-kinetics schemes implemented in the ZND calculations determines that of the cell mean width estimates by the model above, its assumptions aside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' We selected mixtures with H2, C3H8 or CH4 as the fuel and O2 or Air as the oxidizer because of their practical importance and the attention they have received from kineticists, and we thus used the Konnov [12] and the FFCM-1 [13] detailed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Figure 1 compares the calculation results (full and open symbols , , ) to measurements in tubes (crosses) as presented in the detonation database [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Denoting by d the transverse dimensions of the tubes, the black (+) and grey (×) crosses denote values of the ratio ¯λ/d smaller and larger, respectively, than the arbitrary magnitude O(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' That gives O(100) cells on the front surface, which we considered to be a number sufficiently large to meet the model’s basic assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' We collected the tube dimensions from the original references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The comparison shows a very good agreement by the trends and the magnitudes and, in most cases, to better than the ≈ 50 − 100% typical standard deviation for the mean width of irregular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Overall, the smaller the cell mean width compared to the tube dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=', the black crosses +), the better the agreement, consistent with the assumption of many cells on the front surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The agreement is also better with mixture compositions close to the stoichiometric balance, perhaps because this is the usual calibration range of kinetic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' The cell mean width is still considered to be a useful characteristic length in detonation dynamics, although the complex 3D patterns observed on detonation front views question the sufficiency of its repre- sentativeness and are the likely reason for the large standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' In view of this intrinsic limitation, we felt that simple global modelling could yield a representative cell width sufficiently accurate for practical purposes without detailing the complex wave interactions that form the cellular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Our approach does not pretend to explain how the detonation reaction zones are unstable and only assumes a global equivalence of the ZND and cellular processes expressed by basic conservation and action principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Since its implementation is an easy post-process of ZND profiles, a complementary application is its use as an inverse method to assess the representative capacity of kinetic schemes from cell measure- ments obtained in conditions that eliminate the confinement effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Current work includes other mixtures and schemes – for example, we obtained the same good agreement with the San-Diego mechanism, where applicable – and a comparative discussion with regular cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Institut Pprime UPR3346 CNRS January 10, 2023 5 Monnier et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Smith, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Tao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Wang, Foundational fuel chemistry model version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='0 (ffcm-1) 2016 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' URL https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='edu/group/haiwanglab/FFCM1/pages/download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='html [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Kaneshige, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Shepherd, Detonation database, California Institute of Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Online available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' accessed march 2022 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' URL https://shepherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='edu/detn_db/html/db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='html Institut Pprime UPR3346 CNRS January 10, 2023 6 Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' From graph theory and geometric probabilities to detonation cell widths Figures Figure 1: Comparison of calculated and measured cell mean widths ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Full and open symbols: calculations using the Konnov [12] ( and ) and the FFCM-1 [13] ( ) schemes of chemical kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Crosses: measurements [14] with small (⩽ O(10), black +) and large (⩾ O(10), grey ×) ratios ¯λ/d, with d the transverse dimensions of the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content=' Institut Pprime UPR3346 CNRS January 10, 2023 7 102 H2-O2 ER=1 Cell mean width (mm) 101 XXX + + + 100 + + +# 10-1 10 100 1000 Initial pressure po (kPa)101 C,Hg-O2 ER= 1 100 10 100 Initial pressure po (kPa)103 Hz-Air Po = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='3 kPa 102 X X X X 101 + 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='5 2 Equivalence Ratio103 H2-Air ER=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='0: ++ ER=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='5: X^ X X 102 X + + 101 100 10 100 1000 Initial pressure po (kPa)15 CH4-O2 Po = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='0 kPa 10 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} +page_content='4 Equivalence Ratio102 CH,-O2 ER= 1 101 X # 100 + + # 10-1 10 100 1000 Initial pressure po (kPa)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE0T4oBgHgl3EQf9gLX/content/2301.02803v1.pdf'} diff --git a/utFKT4oBgHgl3EQf3i7U/content/2301.11929v1.pdf b/utFKT4oBgHgl3EQf3i7U/content/2301.11929v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aebf7a23b275a53a0736c9d09f76fcc23595d6a2 --- /dev/null +++ b/utFKT4oBgHgl3EQf3i7U/content/2301.11929v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a9b75465d19f32d723867a7c628e5ea4a9afcad74afa6c74175681f0474bd69d +size 1774632 diff --git a/utFKT4oBgHgl3EQf3i7U/vector_store/index.pkl b/utFKT4oBgHgl3EQf3i7U/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..35b5a5858f26b818a9dbc06f0716979faae06bfb --- /dev/null +++ b/utFKT4oBgHgl3EQf3i7U/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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University of Gothenburg, 41296 Gothenburg, Sweden +‡Department of Chemistry, University College London, 20 Gordon Street, WC1H 0AJ +London, United Kingdom +E-mail: marcel.rey@physics.gu.se; giovanni.volpe@physics.gu.se; g.volpe@ucl.ac.uk +Abstract +Light carries energy and momentum. It can therefore alter the motion of objects +from atomic to astronomical scales. Being widely available, readily controllable and +broadly biocompatible, light is also an ideal tool to propel microscopic particles, drive +them out of thermodynamic equilibrium and make them active. +Thus, light-driven +particles have become a recent focus of research in the field of soft active matter. In +this perspective, we discuss recent advances in the control of soft active matter with +light, which has mainly been achieved using light intensity. We also highlight some +first attempts to utilize light’s additional degrees of freedom, such as its wavelength, +polarization, and momentum. We then argue that fully exploiting light with all of its +properties will play a critical role to increase the level of control over the actuation +of active matter as well as the flow of light itself through it. This enabling step will +advance the design of soft active matter systems, their functionalities and their transfer +towards technological applications. +1 +arXiv:2301.13034v1 [cond-mat.soft] 30 Jan 2023 + +Introduction +In the last half century, the possibility of transporting and actuating objects with light has +left the realm of science fiction to impact several fields of science and technology. Nowadays, +a tremendous number of disciplines and applications benefit from actuating objects with +light. These include optical manipulation,1 microfluidics,2 nanomedicine,3 manufacturing4 +and even space exploration.5,6 +Light carries energy and momentum that can be transferred to materials via different +types of light–matter interactions.7 While these effects are usually too small to be appreciated +in our everyday life, their magnitude is big enough to influence the motion of microscopic +objects,8 whose energy fluctuations are comparable to the characteristic thermal energy kBT +with kB the Boltzmann constant and T the absolute temperature (kBT ≈ 4.14 · 10−21 J at +room temperature). This is the realm of soft matter, i.e., the branch of science that studies +systems and materials that can be deformed by relatively low energies on the order of thermal +fluctuations. Since the characteristic energy of a visible light photon is comparable to kBT, +light is particularly well suited to interact with soft materials (e.g., a green photon has energy +Ephoton = hc/λ ≈ 3.8 · 10−19 J ≈ 90 kBT, where h is the Planck constant, c is the speed of +light, and λ = 532 nm is the wavelength). +Active matter is a term used to include all living and artificial systems that can au- +tonomously perform work for different tasks (e.g., move, transport cargoes, energy conver- +sion) by utilizing energy available to them in their environment. These systems can develop +rich forms of self-organization and collective dynamics,9 leading to the emergence of com- +plex properties, such as the possibility of interacting and evolving autonomously.10,11 At +the macroscopic scale, examples of living active matter include animal groups and human +crowds,9 while active granular matter10 and robotic swarms12 represent their artificial coun- +terparts. At the microscale (the focus of this perspective), examples of living active matter +include, e.g., bacterial cells13 and sperm cells,14 while self-propelling colloids and micro- +robots are their man-made analogues.15 Systems at this scale are characterized by two main +2 + +features: (1) Brownian fluctuations can influence these systems’ motion and (2) inertia can +be neglected.16 +Figure 1: Actuation of active matter by different properties of light. +Different +properties of light (intensity, wavelength, polarization and momentum) can be employed to +control active matter. This schematics represents the main properties of light (inner circle) +and some prominent examples of actuation (outer circle). (Top) The intensity of light typ- +ically correlates with the magnitude of the respective particle’s propulsion mechanism,17 so +that the speed of the active particles can be adjusted by intensity. (Right) Different wave- +lengths can address different parts of heterogeneous active Janus particles, enabling control +over propulsion direction and magnitude.18 (Bottom) Nanomotors consisting of nanowires +with a high dichroic ratio preferentially absorb polarized light, enabling polarotactic active +movement controlled by the polarization state of the incident light.19 (Left) Light also carries +momentum, which can propel matter: for example, microvehicles bearing metasurfaces that +scatter light directionally can be accelerated via transfer of light momentum.20 +Being widely available, readily controllable and broadly biocompatible, light is an ideal +tool to control microscopic particles and drive them out of thermodynamic equilibrium, +thus making them active (Fig. 1). While the active matter community has enthusiastically +adopted this tool to control microscopic active particles (e.g., bacteria, active colloids and +3 + +OF +'S +SPEED +NSI +E +IN +L +E +O +7 +7 +W +Z +G +米 +TH +NOI +MO' +E +Eactive droplets), most studies have focused on exploiting the intensity of light, while neglect- +ing the other degrees of freedom offered by light, such as its wavelength, its polarization +and its linear and angular momenta (Fig. 1). Higher control will enable more fundamental +scientific discoveries about far-from-equilibrium phenomena,15 while being also useful for +applications, e.g., in sensing, nanomedicine and materials science.21 Nowadays the prospects +for light actuation and advanced particle tracking are ever brighter thanks to the develop- +ment of several new technologies, such as cheaper lasers at all wavelengths, more versatile +spatial light modulators, and higher-speed cameras. +In this perspective, we first discuss recent advances in the control of soft active matter +actuation with light using its simplest degree of freedom (light intensity), with a focus +on microscopic active systems such as microorganisms, active colloids and droplets. +We +then highlight first attempts and potential future mechanisms to utilize light’s additional +degrees of freedom, such as its wavelength, polarization and momentum. Finally, we propose +potential avenues to increase the level of control over the actuation of soft active matter based +on fully exploiting light’s degrees of freedom. +Active matter actuation by light intensity +Thanks to the technological developments in the last decades, highly-controllable lasers and +other light sources are nowadays easily available to most research labs. Light intensity is the +most obvious light property that can be exploited to enable control over the actuation of +microscopic active matter. This has been done at various scales, from molecular motors22,23 +(Fig. 2a), to microscopic colloids, bacteria and droplets15 (Fig. 2b-d), to microrobots24,25 +(Fig. 2e) and macroscopic robots26 (Fig. 2f). In the following, we analyze how light intensity +has been used to control the behavior of these systems, with an emphasis on systems at the +micrometric scale (Fig. 2b-e). +4 + +Figure 2: Active matter systems controlled by light intensity. Light intensity has +been employed to control active matter systems at all length scales. a) On the molecular +scale, light-responsive nanomotors can undergo a photochemical isomerization around the +central double bond upon irradiation with UV light that results in helicity inversion (from +right-handed to left-handed).27 This motor is very effective at inducing helical organization +in a liquid-crystal film, which can be harnessed to move microparticles placed on top of +it ((i)-(iv)). Adapted with permission from ref.27 Copyright [2006] [Springer Nature]. b- +e) Microscopic systems. b) light-activated Janus colloids self-organize into clusters under +blue light but dissolve when the light source is turned off. Adapted with permission from +ref.28 Copyright [2013] [American Association for the Advancement of Science]. c) Bacte- +ria, genetically modified to swim smoothly with a light-controllable speed, can be arranged +into complex and reconfigurable density patterns such as a portrait of Mona Lisa using a +simple digital light projector. Adapted with permission from ref.29 Copyright [2018] [eLife +Sciences Publications Ltd]. d) Self-propelling droplets of chiral nematic liquid crystals in +surfactant-rich water propel in a screw-like motion. Photo-invertible chiral dopants allow +converting between right-handed and left-handed trajectories upon UV irradiation. Adapted +with permission from ref.30 Copyright [2019] [Springer Nature]. e) Electronically-integrated +micromotors consisting of a body containing standard silicon electronics and surface elec- +trochemical actuator legs are able to walk by directing laser light to its photovoltaics that +alternately bias the front and back legs. Adapted with permission from ref.31 Copyright +[2020] [Springer Nature]. f) On the macroscale, phototactic robots can respond to light gra- +dients, e.g., by adjusting their speed in response to the measured light intensity. Adapted +with permission from ref.26 Copyright [2016] [American Physical Society]. +Micro-organisms +Several micro-organisms, including archea, bacteria and protists, have evolved to sense and +respond to light. Phototaxis, whether towards (positive) or away from (negative) a light +source, can be advantageous to optimize biological and physiological functions, such as pho- +5 + +Molecular motors +Soft active matter +Macroscopic active matter +nm' +μm +mm +m +a +Rotor +&t, Yt +Axle +Stator +200μm +C +50 μm +iv +200 μm +20 μmtosynthesis, growth and the uptake of resources in competitive ecological contexts. For exam- +ple, positive phototaxis can be beneficial for phototropic microorganisms, helping them to po- +sition and orient themselves to efficiently perform photosynthesis.32 In these microorganisms, +the response to light is usually mediated by photoreceptor proteins (e.g., proteorhodopsin +and rhodopsin pigments) that are sensitive to light.32 Most of these micro-organisms can +measure light intensity gradients and move accordingly performing a biased random walk +towards higher or lower light intensities. This is either achieved by probing changes in the +signal over time33 or, in more complex micro-organisms, by directly measuring the gradient +direction.34,35 A beautiful example of the latter is represented by the unicellular cyanobac- +terium Synechocystis that can accurately and directly sense the position of a light source as +the cell itself acts as a spherical microlens, allowing it to see the source and move towards +it.35 The interaction among multiple phototactic micro-organisms can lead to the emergence +of collective phenomena. +For example, bioconvective flows form in systems of phototac- +tic algae due to the formation of uneven mass distributions of the cells moving towards a +light source.36,37 Time and space variations of the source lead to the dynamic triggering and +reconfiguration of these bioconvective plumes.36,37 +Beyond naturally photoresponsive microorganisms, the advent of optogenetics has en- +abled researchers to introduce exogenous DNA into non-photosensitive cells to express the +production of light-sensitive proteins.38 For example, scientists have engineered E. coli bac- +terial cells to respond to red, green and blue light with the production of different pigments +creating color photographs.39 Light-sensitive proteins have been also expressed in E. coli to +modulate their motility and consequently their population density by light,40 permitting the +generation of dynamic bacterial patterns and images (Fig. 2c).29,41 +Micromotors +Inspired by these phototactic micro-organisms, various man-made self-propelling microscopic +particles that can move in response to light have been developed (see, e.g., these recent +6 + +reviews21,42–48). In a homogeneous light-field, their motion can be described as a persis- +tent random walk, similar to motile micro-organisms in homogeneous environments. In the +presence of a light intensity gradient, their motion can become biased.15 A paradigmatic +example of micromotors is constituted by Janus particles (named after the two-faced Roman +god).49 These are colloidal particles whose surface presents two different physico-chemical +properties.50,51 This asymmetry induces a local gradient in some thermodynamic properties +(e.g., concentration, interfacial energy or temperature) across the particle that leads to its +self-propulsion.15,49 Light can be used to create such an asymmetry across an illuminated +particle (e.g., in its temperature profile or surface chemistry). One approach relies on coat- +ing one side of the particle with a photocatalytic material (such as platinum, palladium, +hematite or titania) to locally decompose a chemical fuel (usually hydrogen peroxide) in +water and create a local concentration to drive the particle’s self-diffusiophoresis28,52–54 or +self-electrophoresis.55–57 Alternatively, light absorption in Janus particles half-coated with +a light-absorbing material (e.g., gold or carbon) can also lead to self-propulsion directly or +indirectly due to the formation of a local temperature gradient across the particle because of +selective heating at the absorbing side.17,51,58–62 Differently from their biological counterparts +which can move by body deformation, most of these synthetic micromotors have rigid shapes +and only recently light-responsive reconfigurable microswimmers have been proposed.63–67 +As in the case of micro-organisms, artificial Janus particles can also orient in the light-field +and feature a biased directional phototactic behavior in a light intensity gradient.17,68 +While at the individual particle level these artificial micromotors have found interest +as a promising route to develop novel applications in nanomedicine and environmental re- +mediation,44,46 complex collective behaviors have also been reported when these individual +units self-organize into larger light-activated clusters, including the formation of living crys- +tals (Fig. 2b),28 inverse crystallization,69 crystal annealing,70 different lattice structures,71 +active colloidal molecules,72–74 dynamic pattern formation,75–77 metamachines78 and even +functional photonic materials.79 These emerging behaviors are interesting as model systems +7 + +to study self-organization in living matter but also as a novel route to develop next-generation +materials. Mostly, these complex collective behaviors are governed by physical forces such as +steric, phoretic and hydrodynamic interactions. Recently, however, light has been used to en- +code more complex behaviors which also include feedback interactions among the units.80–82 +Active Droplets +Droplets are small volumes of liquid separated from their surroundings by at least one in- +terface.83 Because of their small size, they can be used as versatile transport vessels and +reactors in microfluidics for applications in chemistry, biology, and nanomedicine.83 Active +droplets are a particular class of droplets that can either self-propel in isolation or do so +in response to other neighboring droplets in an emulsion.84,85 The main physical effect that +induces these droplets’ motion is the Marangoni effect, where a gradient of surface tension +drives mass transport towards areas of higher surface tension. In the case of active droplets, +the Marangoni effect is (self-)induced by the droplet itself or by surrounding droplets. Such +gradients of surface energy can be produced optically, e.g., by illuminating the droplet sur- +face and harnessing the thermal or photochemical effects of the light absorbed within the +droplet.2,85–87 For example, lipophilic droplets stabilized by photoresponsive surfactants can +move in light gradients. Light irradiation induces the dissociation of photoresponsive sur- +factants combined with a rapid pH change in the surrounding aqueous phase, which results +in fast movement of the droplet away from the light source due to a change in surface ten- +sion.88 In self-propelling liquid crystal droplets containing photo-invertible chiral dopants, +light irradiation allows converting between right-handed or left-handed screw-like trajecto- +ries (Fig. 2d).30 Recently, similar light-induced motility effects away or towards a light source +were also reported in droplets of various materials with the addition of different photore- +sponsive molecules, such as surfactants89,90 and inorganic particles.91 +8 + +Microrobots +The possibility of fabricating active particles with complex functionalities has driven the +development of novel microrobots, i.e., robots with characteristc sizes below 1 mm.24,92 +While these systems are at the very edge of what is typically considered soft active mat- +ter, they are a powerful reminder of what can be achieved by light actuation. Due to their +relatively larger size, inertial effects, rather than viscous, are more prominent in determin- +ing their motion, and Brownian fluctuations less relevant.93 In particular, liquid crystalline +elastomers (LCE) are a common material employed to realize biomimetic micromotors ca- +pable of autonomous locomotion in response to light.94 Beyond the realization of devices +at the millimeter scale,95–99 these materials have been successfully employed to realize mi- +crorobots in the submillimeter range, such as walkers on solid surfaces100 and biomimetic +swimmers in fluids.101 Similar shape-changing walkers and grippers were realized based on +photo-sensitive spiropyran-based hydrogels.102,103 Recently, a new class of microrobots has +been demonstrated that integrated electronic components with light actuation, thus pro- +viding a stepping stone towards mass-manufacturing silicon-based functional robots at the +microscopic scale (Fig. 2e).31 +Actuation by other properties of light +Differently from intensity, other light properties have not been extensively exploited for the +actuation of soft active matter yet. In this section, we will briefly review how they have +been used so far in this context (with a focus on wavelength, polarization and transfer of +momentum) and what further possibilities they offer. +Wavelength +The wavelength determines the color of the light as well as the energy carried by each +photon. +Optimal light conditions crucial for microorganism, e.g. +cyanobacteria,104 that +9 + +grow by capturing energy from sunlight. +They demonstrate positive phototaxis towards +green light as it is their preferred energy source for oxygenic photosynthesis, while they show +negative phototaxis away from strong light or UV light as it causes cell damage.104 Similarly, +marine zooplankton Platynereis larvae exposed to UV light swim downwards, away from the +light source, while cyan light makes the larvae swim in the reverse, upwards direction.105 In +the ocean, UV light is most intense near the surface, while cyan light reaches greater depths. +Platynereis larvae may thus use the ratio between UV and cyan light as a “depth gauge” +during vertical migration.105 +In artificial active matter, wavelength is the second most important and exploited prop- +erty of light after intensity. The wavelength is often a boundary condition imposed by the +materials employed in the experiment. For example, Janus particles with metallic caps can +be heated by light of a specific wavelength depending on the cap’s material, e.g., green +matches the plasmon resonance of gold (λ ≈ 530 nm), blue that of silver (λ ≈ 400 nm), and +UV that of platinum (λ ≈ 260 nm). +Multiple wavelengths have been combined in a single experiment to control the propulsion +of different types of active particles108 (or different parts of an active particle) to achieve +more complex particle’s behaviors.18,109–111 For example, TiO2 Janus particles with either +cobalt oxide caps111 or metal caps18,109,110 combine a complex interplay between adsorption +of light at different wavelengths and the respective catalytic and photochemical processes +occurring on each side of the particle. Adjusting the wavelength of light enables control +over the propulsion direction (including its on-demand reversal)18,109 and magnitude18,109–111 +(Fig. 3a). Similarly, hybrid active particles made from two different photocatalysts, e.g. TiO2 +and CuO2, catalyze hydrogen peroxide over differing ranges of wavelength,112 which can lead +to a wavelength-dependent translational and rotational swimming behavior.112 Alternatively, +photoelectrochemically driven nanotree microswimmer loaded with photosensitizer dyes can +be driven and steered by visible light using different wavelengths.108 +Strategies to manipulate liquid interfaces and droplets typically employ azobenzene- +10 + +Figure 3: Active matter systems controlled by wavelength. a) Forward trajectory +of Au-coated anatase TiO2 Janus particles in a H2O2 solution upon illumination with UV +light (top, magenta trajectory) and reverse direction upon illumination with green light +(bottom, green trajectory). Adapted with permission from ref.18 Copyright [2020] [Springer +Nature]. +b) AzoTAB-stabilized Janus emulsions under bright-field blue light irradiation +self-assemble towards a localized UV light spot (filled circle). +Adapted with permission +from ref.106 Copyright [2020] [Springer Nature]. c-e) Potential future uses of wavelength +as control strategy for active particles. +c) Elongated particles (e.g., ellipsoids) with two +different metal patches can be steered using two wavelengths of light. d) U-shaped particle +with three metal patches can work as cargo carriers, with full control over their planar +movement. e) In stimuli-responsive polymer brush coated Janus particles decorated with +plasmonic nanoparticles, external stimuli such as temperature, pH or salt concentration can +collapse the polymer and shift the absorbance spectrum,107 affecting the particle mobility. +derived surfactants such as AzoTAB.113 These molecules can be reversibly switched between +two conformations of different polarity by subsequent illumination with UV (365 nm) and +blue (475 nm) light, which affects the interfacial tension of interfaces stabilized by the surfac- +tant. For example, this principle enabled the omnidirectional manipulation of oil droplets114 +and liquid marbles115 floating on the surface of an aqueous solution of AzoTAB surfactants. +By changing the wavelength of the light used to illuminate the edge of the droplets or liquid +marbles, it was possible to reversibly repel them from the incident beam (UV illumination) +11 + +Os +35 s +110 s +a +5 μm +100μm +入~260 nm +Ag +入~400nm +入400 nm +Ag +D: +Au +入530 nm +Au +入~530 nm +Absorbance [a.u.] +[a.u.] +e +Temperature +Hd +Absorbance I +Salt +400 +500 +600 +700 +800 +400 +500 +600 +700 +800 +Wavelength [nm] +Wavelength [nm]or attract them towards it (blue illumination).114,115 Similarly, oil droplets can be propelled +in the proximity of azobenzene-stabilized micelles.116 Changing the wavelength of the light +induces a change in micelle geometry, which impacts the movement pattern of the droplets, +resulting in a run-and-halt behavior.116 Further, Janus emulsions stabilized with AzoTAB +under blue light irradiation move towards a UV light spot around which they self-assemble +in an ordered fashion (Fig. 3b).106 Finally, fatty acid droplets containing photo-sensitive +spiropyran can move towards visible light sources and away from UV light sources, thus +enabling their manipulation in three dimensions.117 +Potential future uses of the wavelength as a control mechanism of an active system may +include: +• Multiple resonant shapes. Complex active particle shapes can be designed to re- +spond to different wavelengths, e.g., by exploiting characteristic plasmonic resonances +(hence wavelength-selective enhanced light absorption) of different metals. +For ex- +ample, ellipsoidal or rod-shaped anisotropic Janus particles with two different metal +patches instead of one118 would enable addressing each individual patch or both si- +multaneously by adjusting the wavelength of light. +The anisotropic nature of the +particle should allow control over the direction and magnitude of propulsion, where +particles can either move straight or rotate clockwise or counterclockwis depending on +the light wavelength and intensity (Fig. 3c). The same underlying principle can be +used to design even more complex units, e.g., U-shaped particles with multiple plas- +monic patches, which could additionally allow the reversal of the direction of motion +for loading/unloading cargoes (Fig. 3d). +• Stimuli-responsive plasmonic or photonic resonance shifts via elastic defor- +mations. The elastic deformation of stimuli-responsive polymer brushes decorated +with plasmonic nanoparticles has been employed to tune the wavelength of the light +absorbed by this composite structure: external stimuli such as changes in pH,119 tem- +perature120 or salt concentration107 can collapse the polymer brush and bring the +12 + +plasmonic nanoparticles in closer contact, thus red-shifting the absorption spectrum. +We suggest to employ the same concept for active particles, which will then be able +to adapt their activity to their local environment (Fig. 3e). A further, though slightly +more futuristic, approach is inspired by the skin of chameleons and cephalopods: these +animals can camouflage by actively tuning the photonic response of their skin through +elastic deformation, thus changing the wavelength of the reflected and absorbed light. +Similarly, photonic active particles could be realized with stimuli-responsive hydrogel +opal films,121 where the photonic band gap can be adjusted via external stimuli such +as temperature. +Polarization +Polarization is the property of light waves which describes the oscillation of the electric field +in the direction perpendicular to the wave propagation. For example, light from the sun is +unpolarized, i.e., there is no preferred orientation for this oscillation. Unpolarized light can +become polarized when it is scattered or passes through polarizing filters that select only +certain orientations of the electric field. In linearly polarized light, the electric field oscillates +in a single direction perpendicular to the propagation direction. In circularly polarized light, +the field rotates at a constant rate around the direction of propagation as the wave travels. +Sensitivity to polarization is not uncommon in nature. For example, many insect species +bear photoreceptors in a small dorsal rim area of the eye that detect polarized skylight to +improve their navigation skills.122 Furthermore, polarization sensitivity helps squids detect +transparent, yet polarization-active zooplankton under partly polarized light.123 The vision +of the mantis shrimp holds the world record for the most complex visual system: these +marine crustaceans have up to 16 photoreceptors and can see UV, visible and polarized +light; they are also the only animal known to detect circularly polarized light, which may +serve as a secret communication system.124 At the microscale, Euglena gracilis cells (an alga) +exhibit polarotaxis behavior, which aligns their motion direction perpendicular to the light +13 + +polarization.125 Their polarotaxis can also be used to guide the collective movement of these +algae (Fig. 4a).126 +In the manmade world, polarized light has been widely exploited in optical manipulation +to control the orientation and rotation of particles via transfer of linear and angular mo- +mentum.7 However, only recently, nanomotors with a high dichroic ratio have been shown +to respond to polarized light. These nanomotors are constituted of nanowires with a ZnO +shell and a Sb2Se3 core, whose anisotropic crystal structure preferentially absorbs light po- +larized along the wire, hence enhancing their self-propulsion speed.19 By connecting two +cross-aligned dichroic nanowires, the authors of this work were able to realize artificial po- +larotactic active particles, whose navigation can be controlled by the polarization state of +the incident light.19 +However, when it comes to actuating artificial active matter, polarization is still under- +explored. Examples of possible future uses of polarization to control the motion of active +particles are: +• Polarization-dependent absorbers. +The scope for active particles whose self- +phoretic forces depend on the absorption of specific polarizations of light is broad. +This can be achieved employing dielectric structures (as in ref.19) or metallic struc- +tures, e.g., the plasmonic nanocrescents in Fig. 4c, which feature multiple polarization- +dependent resonances combined with near-field enhancement at their tips.128,129 The +propulsion speed of active particles featuring similar nanostructures would therefore +depend on their orientation relative to the polarization of light. For example, in Fig. 4c, +this would lead to an enhanced propulsion of the particles in the direction parallel to +the polarization of light, thus leadign to a polarotactic behavior as the algae cells in +Fig. 4a.126 +• Polarized photovoltaics. The integration of electronic components in microrobots31 +and metavehicles20 could be exploited to generate polarization-dependent motion. The +14 + +Figure 4: Active matter systems controlled by polarization. a) Polarotaxis in pho- +toresponsive algae (Euglena gracilis) leads to movement perpendicular to the polarization of +light. Adapted with permission from ref.126 Copyright [2021] [American Physical Society]. b) +Nanomotors consisting of nanowires with a high dichroic ratio preferentially absorb polarized +light, enabling polarotactic active movement and steering controlled by the polarization state +of the incident light. Adapted with permission from ref.19 Copyright [2019] [John Wiley and +Sons]. c,d) Potential future uses of polarization as control mechanism in active particles. c) +Plasmonic nanocrescents feature polarization-dependent resonances and near-field enhance- +ment at their tips.127 For a fixed wavelength, the propulsion strength of active particles +driven by similar nanostructures therefore would depend on their orientation relative to the +polarization of light, leading to predominant motion in the direction where the absorption of +a given polarization is stronger. d) Electronically-integrated micromotors31 equipped with +polarizing filters in front of photovoltaic components could allow the polarization-dependent +control of specific actuators to steer the particle’s self-propulsion with the light polarization. +electronic components could be simple circuits made from standard inorganic or or- +ganic photovoltaics and metal interconnects powering some actuators on the active +particle.31 The use of polarizing filters in front of the photovoltaic components could +allow the polarization-dependent control of specific actuators to steer the particle’s +self-propulsion with the light polarization (Fig. 4d). +Transfer of momentum +The existence of light momentum is foundational to the whole field of optical trapping and +optical micromanipulation.7,8,130 When light interacts with matter, the change in linear and +angular optical momentum induces forces and torques. +In most cases, these forces and +15 + +a +E +500 μm +20 μm +C +入=2150 nm, +[a.u. +Extinction | +500 1000 1500 2000 2500 3000 +Polarisation Filter +Wavelength [nm]torques are used to hold particles in place or to generate some deterministic, controllable +motion. +There are nevertheless some cases where they have been employed to alter the +random motion of microscopic particles in interesting ways. For example, the forces produced +by random light fields have been employed to alter the diffusion of Brownian particles131–137 +leading also to superdiffusive behavior in the presence of time-varying patterns.132,138,139 +Within the field of active matter, recent work exploited the transfer of momentum us- +ing unfocused light to propel microscopic vehicles with incorporated plasmonic or dielectric +metasurfaces that generate lateral optical forces due to directional light scattering along the +side of the structure (Fig. 5a-c).20,140,141 For example, microvehicles with directional light- +scattering nanostructures arranged in parallel can propel forward upon illumination with +linearly polarized light due to transfer of linear momentum (Fig. 5a,b),20,140 and they can +be steered left or right by circularly polarized light thanks to transfer of angular momentum +(Fig. 5b).20 As an example of application, these metavehicles were also employed for the mi- +cromanipulation of colloidal particles and micro-organisms (Fig. 5b).20 Alternately, rotation +under plain linearly polarized light can be achieved by arranging the scatterers in a circle +(Fig. 5c).140,141 Moreover, microvehicles with four individually addressable chiral plasmonic +nanoantennas acting as nanomotor enable full motion control in two dimensions in all three +independent degrees of freedom (two translational and one rotational):141 Similar to macro- +scopic drones but in two dimensions, these microvehicles are maneuvered by adjusting the +optical power for each nanomotor using two overlapping unfocused light fields at λ = 830 nm +and λ = 980 nm, each with right- or left-handed polarization (Fig. 5c).141 +Further possible uses of optical forces and torques for the field of active matter are the +following: +• Complex optical fields. The use of complex light fields (e.g., random speckle light +fields) is a promising way to generate non-trivial optical potentials that can influence +the individual and collective motion behavior of active particles (Fig. 5d). For example, +the motion of non-light-driven (e.g., catalytic) active particles within random light +16 + +Figure 5: Active matter systems controlled by transfer of light momentum. a) +Plasmonic linear nanomotor driven by momentum transfer of light. Adapted with permis- +sion from ref.140 Copyright [2020] [American Association for the Advancement of Science]. +b) Microvehicles containing a directional scattering metasurface. The transfer of momentum +leads to a straight propulsion under linearly polarized light and circular motion under cir- +cularly polarized light. These metavehicles can be used also to move some microorganisms +present in the solution (bottom panel). Adapted with permission from ref.20 Copyright [2021] +[Springer Nature]. c) Light-driven microparticles containing four chiral plasmonic resonators +are maneuvered by adjusting the optical power for each resonator using two overlapping +unfocused light fields at 830 nm (orange arrow) and 980 nm (red arrow) with right- and left- +handed circular polarization, respectively. Adapted with permission from ref.141 Copyright +[2022] [Springer Nature]. d-g) Potential future uses of momentum transfer in active mat- +ter. d) Active particles either propelled by light or by chemical fuels can explore a speckle +light pattern according to some non-trivial random motion statistics (e.g., by a Fickian yet +non-Gaussian diffusion134). e) Active momentum-driven particles with different shapes can +generate emergent collective self-assembly behaviors, as already theoretically modeled.142,143 +f) Active birefringent Janus particles with a metal cap on one side144 can combine the propul- +sion of Janus particles with the orientation in polarized light of birefringent particles. Such +particles would move along the polarization direction of linearly polarized light or show cir- +cular motion in circularly polarized light. g) Solar sails5 propelled by the light of the sun +could self-assemble in space into complex devices, e.g., space telescopes. +17 + +0ms +50 +a +500n +0ms +50 +un +e +8888 +888 +口 +口口 +8888 +100 +8888fields may show a competition between their propulsion activity and the retardation +introduced by the speckle field as function of increasing light intensity, which has been +predicted in theoretical work.145 On the other hand, active particles that are driven +by light may be accelerated in speckle fields leading to the emergence of superdiffusive +patterns. +• Complex metavehicles. Metavehicles propelled by momentum transfer can be fab- +ricated in any shape without interfering with their propulsion mechanism (Fig. 5e). +This makes them a promising model system to study collective phase behaviors of +active particles as a function of particle shape, which has recently been theoretically +modeled.142,143 +• Birefringent active particles. Birefringent particles, e.g., metamaterial nanopar- +ticles,144 containing an absorbing cap on one side would combine the propulsion of +Janus particles with the re-orientation capabilities in polarized light of birefringent +particles144 (Fig. 5f). Such hybrid particles can be expected to propel parallel to the +polarization of light but also feature circular motion under circularly polarized light +due to transfer of angular momentum. +• Active matter in space. Self-organization of multiple active particles under the +action of optical forces can find fruitful applications in space exploration. For example, +several solar sails5 might interact through optical binding to generate complex collective +behaviors and to produce large self-organized devices: e.g., new space telescopes with +effective lenses made by self-organized active particles which might be much larger +than the Webb telescope (Fig. 5g). +18 + +Further actuation by structured light +A more holistic approach to the use of light for active matter will entail the full control +in space and time of its properties, including amplitude, phase, polarization and momen- +tum.146,147 Simple forms of structured light have already been employed in some active +matter experiments. For example, the fact that Janus particles in a critical mixture of wa- +ter–lutidine feature negative phototaxis in light gradients by drifting towards lower light in- +tensities because of diffusiophoretic torques (Fig. 6a) has been exploited in saw-tooth-shaped +static light profiles to make particles undergo directed motion over arbitrarily long distances +(Fig. 6a).68 Furthermore, structured light has been used to guide traveling motion waves +among photochemically-activated oscillating colloids.148 Silver chloride (AgCl) particles in +dilute hydrogen peroxide solutions under UV light illumination exhibit both single-particle +and collective oscillations in their motion, which arise due to an oscillatory, reversible con- +version of AgCl to silver metal at the particle’s surface.149 These motion waves can be guided +by spatial light patterns, thus enabling a precise and programmable control over the motion +waves’ origin, path and direction (Fig. 6b).148 In biological systems, structured UV light +enabled the creation of defined 3D spatio-temporal chemical landscapes by releasing caged +chemoattractants, which were used to investigate the chemotactic navigation mechanism of +sperm (Fig. 6c).150 Further, Euglena gracilis algae swim in polygonal trajectories when ex- +posed to a sudden increase in light intensity.151 In spatially structured light landscapes with +different light intensities, algae coming from low light to high light intensity start polygonal +swimming or localized spinning, making the cells turn around.151 +Structured light can also be used to induce body deformation in microswimmers lead- +ing to their motion. For example, spatiotemporally structured light based on interference +patterns was used to power and control intra-body shape changes in microrobots consist- +ing of photoactive liquid-crystal elastomers,101 which were able to self-propel by generating +a traveling-wave motion (Fig. 6d).101,153 Temporally structured light was further used to +actuate soft microrobots. These microrobots consist of a temperature-responsive hydrogel +19 + +Figure 6: Active matter systems interacting with structured light. a) Janus particles +in a critical mixture of water–lutidine align such that they move along the gradient of light +toward low light intensities. Directed particle transport over arbitrarily long distances can +then be achieved using periodic saw-tooth-like light profiles. Adapted with permission from +ref.68 Copyright [2016] [Springer Nature]. b) Structured light can guide the motion waves +among photochemically-activated colloids. Adapted with permission from ref.148 Copyright +[2022] [American Physical Society]. c) Structured UV light can create spatio-temporal chem- +ical landscapes by releasing caged chemo-attractants, which guide the movement of sperms. +Adapted with permission from ref.150 Copyright [2015] [Springer Nature]. d) Spatiotempo- +rally structured light induces intra-body shape changes in microrobots consisting of photoac- +tive liquid-crystal elastomers, which enables self-propulsion by generating a traveling-wave +motion. Adapted with permission from ref.101 Copyright [2016] [Springer Nature]. e) Tem- +porally structured light enables rapid dynamic switching between the configurations of he- +lical composite hydrogel microrobots, enabling translational movement near a solid surface. +Adapted with permission from ref.66 Copyright [2016] [John Wiley and Sons]. f-i) Potential +future uses of structured light to actuate and control active matter. f) Metavehicles are able +to change their motion direction (liner or circular) depending on the polarized polarization +of the illuminating light.20,141 Structured light landscapes with different local polarization +could guide the movement of such microvehicles. g) Microrobots could comprise temperature- +responsive bodies that shrink upon irradiation with IR light, thus reducing their drag force +and increasing their propulsion magnitude. Structured light with different local wavelengths +could spatiotemporally change their propulsion magnitude. h) Spatiotemporally structured +light could locally bend hydrogel nanoribbons66,101,152 and induce a snake-like motion, which +could be exploited to propel microparticles. i) Microwalkers could be driven by hydrogel-gold +nanoparticle composites, which serve as artificial muscles and joints in response to light. Us- +ing spatiotemporally structured light, each artificial element could be addressed individually +to enable microscale artificial walking. +20 + +Laseron +C +d +a +e +(milliseconds) +15μm +Laseroff +(milliseconds) +t=6s +t=3s +=09 +50μmfilled with gold nanorods that enable fast heating/cooling dynamics upon irradiation with +IR light combined with volumetric shape changes.66 Covering the hydrogel composite on +one side with a thin gold layer restricts the swelling and shrinking, leading to the formation +of helical configurations.66,152,154 Temporally structured light enables rapid dynamic switch- +ing between left-handed and right-handed helical configurations152 as well as translational +movement near a solid surface (Fig. 6e).66 +As the possibilities for structuring light increase with the advancement of light modulating +devices, control of active matter with structured light could include: +• Control of microvehicles by structured polarization. +Structured light with +different local polarizations or wavelengths could guide the movement of microvehi- +cles20,141 that can change their motion patters depending on the polarization of the +illuminating light (Fig. 6f). +• Shrinkable microrobots in structured intensity fields. Structured light could +spatiotemporally change the propulsion magnitude of microrobots comprising temperature- +responsive bodies that shrink upon illumination with infrared light,63,66,67 thus reduc- +ing their drag force and increasing their propulsion magnitude or propulsion direc- +tion.(Fig. 6g). +• Propulsion by deformable nanoribbons in spatiotemporally oscillating fields. +Spatiotemporally structured light could induce a snake-like motion in hydrogel nanorib- +bons66,101,152 functionalized with microparticles, which would then propel (Fig. 6h). +• 3D-printed articulated microbots actuated by spatiotemporally structured +light. Recent advances in 3D printing have enabled the precise programmable control +over the shape morphing and folding properties of soft stimuli-responsive composite +materials.65,155–157 For example, embedded gold nanorods have been employed to en- +hance light absorption in temperature-responsive hydrogel composites.66,152 Printing +composite hydrogels including gold nanoparticles of different sizes and aspect ratios +21 + +could then enable researchers to address such hydrogels individually by exploiting dif- +ferent plasmonic resonances to realize artificial muscles and joints for microscale walkers +and crawlers (Fig. 6i). +Conclusions +In this perspective, we have discussed the ongoing progress towards actuating and control- +ling soft active matter by exploiting the different properties of light. While changing the +light intensity provides a remote effortless means to adjust the speed and direction of light- +activated particles, the potential of other properties of light to control soft active matter +actuation has been mostly left untapped. For example, selectivity to light wavelength can +enable multiple propulsion mechanisms to coexist on a single particle, e.g., by triggering ex- +clusive light-matter interactions at different sites of the particle.18,109–111 Furthermore, light +polarization and the transfer of its linear and angular momentum can enable complex combi- +nations of translation and rotation in the propulsion of active particles without the need for +any additional fuel source.20,140,141 Finally, the use of spatiotemporally structured light can +combine all of light’s different degrees of freedom into one powerful tool to control the actu- +ation of active matter systems by light in a way that is flexible, selective and adaptive, yet +concomitantly easy to operate. Such level of control through light can enable active matter +researchers to test theory (e.g., phase transitions, optimal navigation strategies) as well as +to develop applications in energy conversion, catalysis, drug-delivery and tissue engineering +taking advantage of the fact that light is a broadly available and biocompatible source of +energy. Conversely, a higher degree of control of active matter with light could prove useful +to develop materials and devices based on active systems that can mold the flow of light +in non-conventional ways, e.g., to realize novel light sources, neuromorphic computers and, +even, displays. +22 + +Acknowledgements +M.R acknowledges Antonio Ciarlo, Giuseppe Pesce and Martin Wittmann for fruitful dis- +cussions. M.R. acknowledges funding from Marie Sklodowska-Curie Individual Fellowship +(Grant No.101064381). G.V. (Giorgio Volpe) acknowledges sponsorship for this work by +the US Office of Naval Research Global (Award No. N62909-18-1-2170). G.V. 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Commun. 2013, 4, 1712. +40 + +Graphical TOC Entry +41 + diff --git a/vtFPT4oBgHgl3EQfOjSp/content/tmp_files/load_file.txt b/vtFPT4oBgHgl3EQfOjSp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe63b7a2beadf4609af5964fc1ff8734bc1bec0f --- /dev/null +++ b/vtFPT4oBgHgl3EQfOjSp/content/tmp_files/load_file.txt @@ -0,0 +1,2628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf,len=2627 +page_content='Light, Matter, Action: Shining light on active matter Marcel Rey,∗,† Giovanni Volpe,∗,† and Giorgio Volpe∗,‡ †Physics Department, University of Gothenburg, 41296 Gothenburg, Sweden ‡Department of Chemistry, University College London, 20 Gordon Street, WC1H 0AJ London, United Kingdom E-mail: marcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='rey@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='se;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='volpe@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='se;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='volpe@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='uk Abstract Light carries energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' It can therefore alter the motion of objects from atomic to astronomical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Being widely available, readily controllable and broadly biocompatible, light is also an ideal tool to propel microscopic particles, drive them out of thermodynamic equilibrium and make them active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Thus, light-driven particles have become a recent focus of research in the field of soft active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In this perspective, we discuss recent advances in the control of soft active matter with light, which has mainly been achieved using light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' We also highlight some first attempts to utilize light’s additional degrees of freedom, such as its wavelength, polarization, and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' We then argue that fully exploiting light with all of its properties will play a critical role to increase the level of control over the actuation of active matter as well as the flow of light itself through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This enabling step will advance the design of soft active matter systems, their functionalities and their transfer towards technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='13034v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='soft] 30 Jan 2023 Introduction In the last half century, the possibility of transporting and actuating objects with light has left the realm of science fiction to impact several fields of science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nowadays, a tremendous number of disciplines and applications benefit from actuating objects with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' These include optical manipulation,1 microfluidics,2 nanomedicine,3 manufacturing4 and even space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='5,6 Light carries energy and momentum that can be transferred to materials via different types of light–matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='7 While these effects are usually too small to be appreciated in our everyday life, their magnitude is big enough to influence the motion of microscopic objects,8 whose energy fluctuations are comparable to the characteristic thermal energy kBT with kB the Boltzmann constant and T the absolute temperature (kBT ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='14 · 10−21 J at room temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This is the realm of soft matter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', the branch of science that studies systems and materials that can be deformed by relatively low energies on the order of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Since the characteristic energy of a visible light photon is comparable to kBT, light is particularly well suited to interact with soft materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', a green photon has energy Ephoton = hc/λ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='8 · 10−19 J ≈ 90 kBT, where h is the Planck constant, c is the speed of light, and λ = 532 nm is the wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Active matter is a term used to include all living and artificial systems that can au- tonomously perform work for different tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', move, transport cargoes, energy conver- sion) by utilizing energy available to them in their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' These systems can develop rich forms of self-organization and collective dynamics,9 leading to the emergence of com- plex properties, such as the possibility of interacting and evolving autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='10,11 At the macroscopic scale, examples of living active matter include animal groups and human crowds,9 while active granular matter10 and robotic swarms12 represent their artificial coun- terparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' At the microscale (the focus of this perspective), examples of living active matter include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', bacterial cells13 and sperm cells,14 while self-propelling colloids and micro- robots are their man-made analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='15 Systems at this scale are characterized by two main 2 features: (1) Brownian fluctuations can influence these systems’ motion and (2) inertia can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='16 Figure 1: Actuation of active matter by different properties of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Different properties of light (intensity, wavelength, polarization and momentum) can be employed to control active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This schematics represents the main properties of light (inner circle) and some prominent examples of actuation (outer circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (Top) The intensity of light typ- ically correlates with the magnitude of the respective particle’s propulsion mechanism,17 so that the speed of the active particles can be adjusted by intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (Right) Different wave- lengths can address different parts of heterogeneous active Janus particles, enabling control over propulsion direction and magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='18 (Bottom) Nanomotors consisting of nanowires with a high dichroic ratio preferentially absorb polarized light, enabling polarotactic active movement controlled by the polarization state of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='19 (Left) Light also carries momentum, which can propel matter: for example, microvehicles bearing metasurfaces that scatter light directionally can be accelerated via transfer of light momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20 Being widely available, readily controllable and broadly biocompatible, light is an ideal tool to control microscopic particles and drive them out of thermodynamic equilibrium, thus making them active (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' While the active matter community has enthusiastically adopted this tool to control microscopic active particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=", bacteria, active colloids and 3 OF 'S SPEED NSI E IN L E O 7 7 W Z G 米 TH NOI MO' E Eactive droplets), most studies have focused on exploiting the intensity of light, while neglect- ing the other degrees of freedom offered by light, such as its wavelength, its polarization and its linear and angular momenta (Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Higher control will enable more fundamental scientific discoveries about far-from-equilibrium phenomena,15 while being also useful for applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', in sensing, nanomedicine and materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='21 Nowadays the prospects for light actuation and advanced particle tracking are ever brighter thanks to the develop- ment of several new technologies, such as cheaper lasers at all wavelengths, more versatile spatial light modulators, and higher-speed cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In this perspective, we first discuss recent advances in the control of soft active matter actuation with light using its simplest degree of freedom (light intensity), with a focus on microscopic active systems such as microorganisms, active colloids and droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' We then highlight first attempts and potential future mechanisms to utilize light’s additional degrees of freedom, such as its wavelength, polarization and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Finally, we propose potential avenues to increase the level of control over the actuation of soft active matter based on fully exploiting light’s degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Active matter actuation by light intensity Thanks to the technological developments in the last decades, highly-controllable lasers and other light sources are nowadays easily available to most research labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light intensity is the most obvious light property that can be exploited to enable control over the actuation of microscopic active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This has been done at various scales, from molecular motors22,23 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2a), to microscopic colloids, bacteria and droplets15 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2b-d), to microrobots24,25 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2e) and macroscopic robots26 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In the following, we analyze how light intensity has been used to control the behavior of these systems, with an emphasis on systems at the micrometric scale (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2b-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4 Figure 2: Active matter systems controlled by light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light intensity has been employed to control active matter systems at all length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' a) On the molecular scale, light-responsive nanomotors can undergo a photochemical isomerization around the central double bond upon irradiation with UV light that results in helicity inversion (from right-handed to left-handed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='27 This motor is very effective at inducing helical organization in a liquid-crystal film, which can be harnessed to move microparticles placed on top of it ((i)-(iv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='27 Copyright [2006] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b- e) Microscopic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b) light-activated Janus colloids self-organize into clusters under blue light but dissolve when the light source is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='28 Copyright [2013] [American Association for the Advancement of Science].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c) Bacte- ria, genetically modified to swim smoothly with a light-controllable speed, can be arranged into complex and reconfigurable density patterns such as a portrait of Mona Lisa using a simple digital light projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='29 Copyright [2018] [eLife Sciences Publications Ltd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d) Self-propelling droplets of chiral nematic liquid crystals in surfactant-rich water propel in a screw-like motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Photo-invertible chiral dopants allow converting between right-handed and left-handed trajectories upon UV irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='30 Copyright [2019] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' e) Electronically-integrated micromotors consisting of a body containing standard silicon electronics and surface elec- trochemical actuator legs are able to walk by directing laser light to its photovoltaics that alternately bias the front and back legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='31 Copyright [2020] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' f) On the macroscale, phototactic robots can respond to light gra- dients, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', by adjusting their speed in response to the measured light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='26 Copyright [2016] [American Physical Society].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Micro-organisms Several micro-organisms, including archea, bacteria and protists, have evolved to sense and respond to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=" Phototaxis, whether towards (positive) or away from (negative) a light source, can be advantageous to optimize biological and physiological functions, such as pho- 5 Molecular motors Soft active matter Macroscopic active matter nm' μm mm m a Rotor &t, Yt Axle Stator 200μm C 50 μm iv 200 μm 20 μmtosynthesis, growth and the uptake of resources in competitive ecological contexts." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For exam- ple, positive phototaxis can be beneficial for phototropic microorganisms, helping them to po- sition and orient themselves to efficiently perform photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='32 In these microorganisms, the response to light is usually mediated by photoreceptor proteins (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', proteorhodopsin and rhodopsin pigments) that are sensitive to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='32 Most of these micro-organisms can measure light intensity gradients and move accordingly performing a biased random walk towards higher or lower light intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This is either achieved by probing changes in the signal over time33 or, in more complex micro-organisms, by directly measuring the gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='34,35 A beautiful example of the latter is represented by the unicellular cyanobac- terium Synechocystis that can accurately and directly sense the position of a light source as the cell itself acts as a spherical microlens, allowing it to see the source and move towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='35 The interaction among multiple phototactic micro-organisms can lead to the emergence of collective phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, bioconvective flows form in systems of phototac- tic algae due to the formation of uneven mass distributions of the cells moving towards a light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='36,37 Time and space variations of the source lead to the dynamic triggering and reconfiguration of these bioconvective plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='36,37 Beyond naturally photoresponsive microorganisms, the advent of optogenetics has en- abled researchers to introduce exogenous DNA into non-photosensitive cells to express the production of light-sensitive proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='38 For example, scientists have engineered E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' coli bac- terial cells to respond to red, green and blue light with the production of different pigments creating color photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='39 Light-sensitive proteins have been also expressed in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' coli to modulate their motility and consequently their population density by light,40 permitting the generation of dynamic bacterial patterns and images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='29,41 Micromotors Inspired by these phototactic micro-organisms, various man-made self-propelling microscopic particles that can move in response to light have been developed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', these recent 6 reviews21,42–48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In a homogeneous light-field, their motion can be described as a persis- tent random walk, similar to motile micro-organisms in homogeneous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In the presence of a light intensity gradient, their motion can become biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='15 A paradigmatic example of micromotors is constituted by Janus particles (named after the two-faced Roman god).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='49 These are colloidal particles whose surface presents two different physico-chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='50,51 This asymmetry induces a local gradient in some thermodynamic properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', concentration, interfacial energy or temperature) across the particle that leads to its self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='15,49 Light can be used to create such an asymmetry across an illuminated particle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', in its temperature profile or surface chemistry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' One approach relies on coat- ing one side of the particle with a photocatalytic material (such as platinum, palladium, hematite or titania) to locally decompose a chemical fuel (usually hydrogen peroxide) in water and create a local concentration to drive the particle’s self-diffusiophoresis28,52–54 or self-electrophoresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='55–57 Alternatively, light absorption in Janus particles half-coated with a light-absorbing material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', gold or carbon) can also lead to self-propulsion directly or indirectly due to the formation of a local temperature gradient across the particle because of selective heating at the absorbing side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='17,51,58–62 Differently from their biological counterparts which can move by body deformation, most of these synthetic micromotors have rigid shapes and only recently light-responsive reconfigurable microswimmers have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='63–67 As in the case of micro-organisms, artificial Janus particles can also orient in the light-field and feature a biased directional phototactic behavior in a light intensity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='17,68 While at the individual particle level these artificial micromotors have found interest as a promising route to develop novel applications in nanomedicine and environmental re- mediation,44,46 complex collective behaviors have also been reported when these individual units self-organize into larger light-activated clusters, including the formation of living crys- tals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2b),28 inverse crystallization,69 crystal annealing,70 different lattice structures,71 active colloidal molecules,72–74 dynamic pattern formation,75–77 metamachines78 and even functional photonic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='79 These emerging behaviors are interesting as model systems 7 to study self-organization in living matter but also as a novel route to develop next-generation materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Mostly, these complex collective behaviors are governed by physical forces such as steric, phoretic and hydrodynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Recently, however, light has been used to en- code more complex behaviors which also include feedback interactions among the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='80–82 Active Droplets Droplets are small volumes of liquid separated from their surroundings by at least one in- terface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='83 Because of their small size, they can be used as versatile transport vessels and reactors in microfluidics for applications in chemistry, biology, and nanomedicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='83 Active droplets are a particular class of droplets that can either self-propel in isolation or do so in response to other neighboring droplets in an emulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='84,85 The main physical effect that induces these droplets’ motion is the Marangoni effect, where a gradient of surface tension drives mass transport towards areas of higher surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In the case of active droplets, the Marangoni effect is (self-)induced by the droplet itself or by surrounding droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Such gradients of surface energy can be produced optically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', by illuminating the droplet sur- face and harnessing the thermal or photochemical effects of the light absorbed within the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='2,85–87 For example, lipophilic droplets stabilized by photoresponsive surfactants can move in light gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light irradiation induces the dissociation of photoresponsive sur- factants combined with a rapid pH change in the surrounding aqueous phase, which results in fast movement of the droplet away from the light source due to a change in surface ten- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='88 In self-propelling liquid crystal droplets containing photo-invertible chiral dopants, light irradiation allows converting between right-handed or left-handed screw-like trajecto- ries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='30 Recently, similar light-induced motility effects away or towards a light source were also reported in droplets of various materials with the addition of different photore- sponsive molecules, such as surfactants89,90 and inorganic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='91 8 Microrobots The possibility of fabricating active particles with complex functionalities has driven the development of novel microrobots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', robots with characteristc sizes below 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='24,92 While these systems are at the very edge of what is typically considered soft active mat- ter, they are a powerful reminder of what can be achieved by light actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Due to their relatively larger size, inertial effects, rather than viscous, are more prominent in determin- ing their motion, and Brownian fluctuations less relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='93 In particular, liquid crystalline elastomers (LCE) are a common material employed to realize biomimetic micromotors ca- pable of autonomous locomotion in response to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='94 Beyond the realization of devices at the millimeter scale,95–99 these materials have been successfully employed to realize mi- crorobots in the submillimeter range, such as walkers on solid surfaces100 and biomimetic swimmers in fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='101 Similar shape-changing walkers and grippers were realized based on photo-sensitive spiropyran-based hydrogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='102,103 Recently, a new class of microrobots has been demonstrated that integrated electronic components with light actuation, thus pro- viding a stepping stone towards mass-manufacturing silicon-based functional robots at the microscopic scale (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='31 Actuation by other properties of light Differently from intensity, other light properties have not been extensively exploited for the actuation of soft active matter yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In this section, we will briefly review how they have been used so far in this context (with a focus on wavelength, polarization and transfer of momentum) and what further possibilities they offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Wavelength The wavelength determines the color of the light as well as the energy carried by each photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Optimal light conditions crucial for microorganism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' cyanobacteria,104 that 9 grow by capturing energy from sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' They demonstrate positive phototaxis towards green light as it is their preferred energy source for oxygenic photosynthesis, while they show negative phototaxis away from strong light or UV light as it causes cell damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='104 Similarly, marine zooplankton Platynereis larvae exposed to UV light swim downwards, away from the light source, while cyan light makes the larvae swim in the reverse, upwards direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='105 In the ocean, UV light is most intense near the surface, while cyan light reaches greater depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Platynereis larvae may thus use the ratio between UV and cyan light as a “depth gauge” during vertical migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='105 In artificial active matter, wavelength is the second most important and exploited prop- erty of light after intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The wavelength is often a boundary condition imposed by the materials employed in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, Janus particles with metallic caps can be heated by light of a specific wavelength depending on the cap’s material, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', green matches the plasmon resonance of gold (λ ≈ 530 nm), blue that of silver (λ ≈ 400 nm), and UV that of platinum (λ ≈ 260 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Multiple wavelengths have been combined in a single experiment to control the propulsion of different types of active particles108 (or different parts of an active particle) to achieve more complex particle’s behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='18,109–111 For example, TiO2 Janus particles with either cobalt oxide caps111 or metal caps18,109,110 combine a complex interplay between adsorption of light at different wavelengths and the respective catalytic and photochemical processes occurring on each side of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adjusting the wavelength of light enables control over the propulsion direction (including its on-demand reversal)18,109 and magnitude18,109–111 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Similarly, hybrid active particles made from two different photocatalysts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' TiO2 and CuO2, catalyze hydrogen peroxide over differing ranges of wavelength,112 which can lead to a wavelength-dependent translational and rotational swimming behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='112 Alternatively, photoelectrochemically driven nanotree microswimmer loaded with photosensitizer dyes can be driven and steered by visible light using different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='108 Strategies to manipulate liquid interfaces and droplets typically employ azobenzene- 10 Figure 3: Active matter systems controlled by wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' a) Forward trajectory of Au-coated anatase TiO2 Janus particles in a H2O2 solution upon illumination with UV light (top, magenta trajectory) and reverse direction upon illumination with green light (bottom, green trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='18 Copyright [2020] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b) AzoTAB-stabilized Janus emulsions under bright-field blue light irradiation self-assemble towards a localized UV light spot (filled circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='106 Copyright [2020] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c-e) Potential future uses of wavelength as control strategy for active particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c) Elongated particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', ellipsoids) with two different metal patches can be steered using two wavelengths of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d) U-shaped particle with three metal patches can work as cargo carriers, with full control over their planar movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' e) In stimuli-responsive polymer brush coated Janus particles decorated with plasmonic nanoparticles, external stimuli such as temperature, pH or salt concentration can collapse the polymer and shift the absorbance spectrum,107 affecting the particle mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' derived surfactants such as AzoTAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='113 These molecules can be reversibly switched between two conformations of different polarity by subsequent illumination with UV (365 nm) and blue (475 nm) light, which affects the interfacial tension of interfaces stabilized by the surfac- tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, this principle enabled the omnidirectional manipulation of oil droplets114 and liquid marbles115 floating on the surface of an aqueous solution of AzoTAB surfactants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' By changing the wavelength of the light used to illuminate the edge of the droplets or liquid marbles, it was possible to reversibly repel them from the incident beam (UV illumination) 11 Os 35 s 110 s a 5 μm 100μm 入~260 nm Ag 入~400nm 入400 nm Ag D: Au 入530 nm Au 入~530 nm Absorbance [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='] [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='] e Temperature Hd Absorbance I Salt 400 500 600 700 800 400 500 600 700 800 Wavelength [nm] Wavelength [nm]or attract them towards it (blue illumination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='114,115 Similarly, oil droplets can be propelled in the proximity of azobenzene-stabilized micelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='116 Changing the wavelength of the light induces a change in micelle geometry, which impacts the movement pattern of the droplets, resulting in a run-and-halt behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='116 Further, Janus emulsions stabilized with AzoTAB under blue light irradiation move towards a UV light spot around which they self-assemble in an ordered fashion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='106 Finally, fatty acid droplets containing photo-sensitive spiropyran can move towards visible light sources and away from UV light sources, thus enabling their manipulation in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='117 Potential future uses of the wavelength as a control mechanism of an active system may include: Multiple resonant shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Complex active particle shapes can be designed to re- spond to different wavelengths, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', by exploiting characteristic plasmonic resonances (hence wavelength-selective enhanced light absorption) of different metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For ex- ample, ellipsoidal or rod-shaped anisotropic Janus particles with two different metal patches instead of one118 would enable addressing each individual patch or both si- multaneously by adjusting the wavelength of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The anisotropic nature of the particle should allow control over the direction and magnitude of propulsion, where particles can either move straight or rotate clockwise or counterclockwis depending on the light wavelength and intensity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The same underlying principle can be used to design even more complex units, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', U-shaped particles with multiple plas- monic patches, which could additionally allow the reversal of the direction of motion for loading/unloading cargoes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Stimuli-responsive plasmonic or photonic resonance shifts via elastic defor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The elastic deformation of stimuli-responsive polymer brushes decorated with plasmonic nanoparticles has been employed to tune the wavelength of the light absorbed by this composite structure: external stimuli such as changes in pH,119 tem- perature120 or salt concentration107 can collapse the polymer brush and bring the 12 plasmonic nanoparticles in closer contact, thus red-shifting the absorption spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' We suggest to employ the same concept for active particles, which will then be able to adapt their activity to their local environment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' A further, though slightly more futuristic, approach is inspired by the skin of chameleons and cephalopods: these animals can camouflage by actively tuning the photonic response of their skin through elastic deformation, thus changing the wavelength of the reflected and absorbed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Similarly, photonic active particles could be realized with stimuli-responsive hydrogel opal films,121 where the photonic band gap can be adjusted via external stimuli such as temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Polarization Polarization is the property of light waves which describes the oscillation of the electric field in the direction perpendicular to the wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, light from the sun is unpolarized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', there is no preferred orientation for this oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Unpolarized light can become polarized when it is scattered or passes through polarizing filters that select only certain orientations of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In linearly polarized light, the electric field oscillates in a single direction perpendicular to the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In circularly polarized light, the field rotates at a constant rate around the direction of propagation as the wave travels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Sensitivity to polarization is not uncommon in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, many insect species bear photoreceptors in a small dorsal rim area of the eye that detect polarized skylight to improve their navigation skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='122 Furthermore, polarization sensitivity helps squids detect transparent, yet polarization-active zooplankton under partly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='123 The vision of the mantis shrimp holds the world record for the most complex visual system: these marine crustaceans have up to 16 photoreceptors and can see UV, visible and polarized light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' they are also the only animal known to detect circularly polarized light, which may serve as a secret communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='124 At the microscale, Euglena gracilis cells (an alga) exhibit polarotaxis behavior, which aligns their motion direction perpendicular to the light 13 polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='125 Their polarotaxis can also be used to guide the collective movement of these algae (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='126 In the manmade world, polarized light has been widely exploited in optical manipulation to control the orientation and rotation of particles via transfer of linear and angular mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='7 However, only recently, nanomotors with a high dichroic ratio have been shown to respond to polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' These nanomotors are constituted of nanowires with a ZnO shell and a Sb2Se3 core, whose anisotropic crystal structure preferentially absorbs light po- larized along the wire, hence enhancing their self-propulsion speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='19 By connecting two cross-aligned dichroic nanowires, the authors of this work were able to realize artificial po- larotactic active particles, whose navigation can be controlled by the polarization state of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='19 However, when it comes to actuating artificial active matter, polarization is still under- explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Examples of possible future uses of polarization to control the motion of active particles are: Polarization-dependent absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The scope for active particles whose self- phoretic forces depend on the absorption of specific polarizations of light is broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This can be achieved employing dielectric structures (as in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='19) or metallic struc- tures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', the plasmonic nanocrescents in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4c, which feature multiple polarization- dependent resonances combined with near-field enhancement at their tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='128,129 The propulsion speed of active particles featuring similar nanostructures would therefore depend on their orientation relative to the polarization of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4c, this would lead to an enhanced propulsion of the particles in the direction parallel to the polarization of light, thus leadign to a polarotactic behavior as the algae cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='126 Polarized photovoltaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The integration of electronic components in microrobots31 and metavehicles20 could be exploited to generate polarization-dependent motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The 14 Figure 4: Active matter systems controlled by polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' a) Polarotaxis in pho- toresponsive algae (Euglena gracilis) leads to movement perpendicular to the polarization of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='126 Copyright [2021] [American Physical Society].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b) Nanomotors consisting of nanowires with a high dichroic ratio preferentially absorb polarized light, enabling polarotactic active movement and steering controlled by the polarization state of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='19 Copyright [2019] [John Wiley and Sons].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c,d) Potential future uses of polarization as control mechanism in active particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c) Plasmonic nanocrescents feature polarization-dependent resonances and near-field enhance- ment at their tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='127 For a fixed wavelength, the propulsion strength of active particles driven by similar nanostructures therefore would depend on their orientation relative to the polarization of light, leading to predominant motion in the direction where the absorption of a given polarization is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d) Electronically-integrated micromotors31 equipped with polarizing filters in front of photovoltaic components could allow the polarization-dependent control of specific actuators to steer the particle’s self-propulsion with the light polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' electronic components could be simple circuits made from standard inorganic or or- ganic photovoltaics and metal interconnects powering some actuators on the active particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='31 The use of polarizing filters in front of the photovoltaic components could allow the polarization-dependent control of specific actuators to steer the particle’s self-propulsion with the light polarization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Transfer of momentum The existence of light momentum is foundational to the whole field of optical trapping and optical micromanipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='7,8,130 When light interacts with matter, the change in linear and angular optical momentum induces forces and torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' In most cases, these forces and 15 a E 500 μm 20 μm C 入=2150 nm, [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Extinction | 500 1000 1500 2000 2500 3000 Polarisation Filter Wavelength [nm]torques are used to hold particles in place or to generate some deterministic, controllable motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' There are nevertheless some cases where they have been employed to alter the random motion of microscopic particles in interesting ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, the forces produced by random light fields have been employed to alter the diffusion of Brownian particles131–137 leading also to superdiffusive behavior in the presence of time-varying patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='132,138,139 Within the field of active matter, recent work exploited the transfer of momentum us- ing unfocused light to propel microscopic vehicles with incorporated plasmonic or dielectric metasurfaces that generate lateral optical forces due to directional light scattering along the side of the structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20,140,141 For example, microvehicles with directional light- scattering nanostructures arranged in parallel can propel forward upon illumination with linearly polarized light due to transfer of linear momentum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5a,b),20,140 and they can be steered left or right by circularly polarized light thanks to transfer of angular momentum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20 As an example of application, these metavehicles were also employed for the mi- cromanipulation of colloidal particles and micro-organisms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20 Alternately, rotation under plain linearly polarized light can be achieved by arranging the scatterers in a circle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='140,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='141 Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' microvehicles with four individually addressable chiral plasmonic nanoantennas acting as nanomotor enable full motion control in two dimensions in all three independent degrees of freedom (two translational and one rotational):141 Similar to macro- scopic drones but in two dimensions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' these microvehicles are maneuvered by adjusting the optical power for each nanomotor using two overlapping unfocused light fields at λ = 830 nm and λ = 980 nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' each with right- or left-handed polarization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='141 Further possible uses of optical forces and torques for the field of active matter are the following: Complex optical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The use of complex light fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', random speckle light fields) is a promising way to generate non-trivial optical potentials that can influence the individual and collective motion behavior of active particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, the motion of non-light-driven (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', catalytic) active particles within random light 16 Figure 5: Active matter systems controlled by transfer of light momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' a) Plasmonic linear nanomotor driven by momentum transfer of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permis- sion from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='140 Copyright [2020] [American Association for the Advancement of Science].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b) Microvehicles containing a directional scattering metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' The transfer of momentum leads to a straight propulsion under linearly polarized light and circular motion under cir- cularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' These metavehicles can be used also to move some microorganisms present in the solution (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20 Copyright [2021] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c) Light-driven microparticles containing four chiral plasmonic resonators are maneuvered by adjusting the optical power for each resonator using two overlapping unfocused light fields at 830 nm (orange arrow) and 980 nm (red arrow) with right- and left- handed circular polarization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='141 Copyright [2022] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d-g) Potential future uses of momentum transfer in active mat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d) Active particles either propelled by light or by chemical fuels can explore a speckle light pattern according to some non-trivial random motion statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', by a Fickian yet non-Gaussian diffusion134).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' e) Active momentum-driven particles with different shapes can generate emergent collective self-assembly behaviors, as already theoretically modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='142,143 f) Active birefringent Janus particles with a metal cap on one side144 can combine the propul- sion of Janus particles with the orientation in polarized light of birefringent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Such particles would move along the polarization direction of linearly polarized light or show cir- cular motion in circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' g) Solar sails5 propelled by the light of the sun could self-assemble in space into complex devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', space telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 17 0ms 50 a 500n 0ms 50 un e 8888 888 口 口口 8888 100 8888fields may show a competition between their propulsion activity and the retardation introduced by the speckle field as function of increasing light intensity, which has been predicted in theoretical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='145 On the other hand, active particles that are driven by light may be accelerated in speckle fields leading to the emergence of superdiffusive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Complex metavehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Metavehicles propelled by momentum transfer can be fab- ricated in any shape without interfering with their propulsion mechanism (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' This makes them a promising model system to study collective phase behaviors of active particles as a function of particle shape, which has recently been theoretically modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='142,143 Birefringent active particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Birefringent particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', metamaterial nanopar- ticles,144 containing an absorbing cap on one side would combine the propulsion of Janus particles with the re-orientation capabilities in polarized light of birefringent particles144 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Such hybrid particles can be expected to propel parallel to the polarization of light but also feature circular motion under circularly polarized light due to transfer of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Active matter in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Self-organization of multiple active particles under the action of optical forces can find fruitful applications in space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, several solar sails5 might interact through optical binding to generate complex collective behaviors and to produce large self-organized devices: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', new space telescopes with effective lenses made by self-organized active particles which might be much larger than the Webb telescope (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 18 Further actuation by structured light A more holistic approach to the use of light for active matter will entail the full control in space and time of its properties, including amplitude, phase, polarization and momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='146,147 Simple forms of structured light have already been employed in some active matter experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, the fact that Janus particles in a critical mixture of wa- ter–lutidine feature negative phototaxis in light gradients by drifting towards lower light in- tensities because of diffusiophoretic torques (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6a) has been exploited in saw-tooth-shaped static light profiles to make particles undergo directed motion over arbitrarily long distances (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='68 Furthermore, structured light has been used to guide traveling motion waves among photochemically-activated oscillating colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='148 Silver chloride (AgCl) particles in dilute hydrogen peroxide solutions under UV light illumination exhibit both single-particle and collective oscillations in their motion, which arise due to an oscillatory, reversible con- version of AgCl to silver metal at the particle’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='149 These motion waves can be guided by spatial light patterns, thus enabling a precise and programmable control over the motion waves’ origin, path and direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='148 In biological systems, structured UV light enabled the creation of defined 3D spatio-temporal chemical landscapes by releasing caged chemoattractants, which were used to investigate the chemotactic navigation mechanism of sperm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='150 Further, Euglena gracilis algae swim in polygonal trajectories when ex- posed to a sudden increase in light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='151 In spatially structured light landscapes with different light intensities, algae coming from low light to high light intensity start polygonal swimming or localized spinning, making the cells turn around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='151 Structured light can also be used to induce body deformation in microswimmers lead- ing to their motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, spatiotemporally structured light based on interference patterns was used to power and control intra-body shape changes in microrobots consist- ing of photoactive liquid-crystal elastomers,101 which were able to self-propel by generating a traveling-wave motion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='101,153 Temporally structured light was further used to actuate soft microrobots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' These microrobots consist of a temperature-responsive hydrogel 19 Figure 6: Active matter systems interacting with structured light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' a) Janus particles in a critical mixture of water–lutidine align such that they move along the gradient of light toward low light intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Directed particle transport over arbitrarily long distances can then be achieved using periodic saw-tooth-like light profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='68 Copyright [2016] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' b) Structured light can guide the motion waves among photochemically-activated colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='148 Copyright [2022] [American Physical Society].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' c) Structured UV light can create spatio-temporal chem- ical landscapes by releasing caged chemo-attractants, which guide the movement of sperms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='150 Copyright [2015] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' d) Spatiotempo- rally structured light induces intra-body shape changes in microrobots consisting of photoac- tive liquid-crystal elastomers, which enables self-propulsion by generating a traveling-wave motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='101 Copyright [2016] [Springer Nature].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' e) Tem- porally structured light enables rapid dynamic switching between the configurations of he- lical composite hydrogel microrobots, enabling translational movement near a solid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adapted with permission from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='66 Copyright [2016] [John Wiley and Sons].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' f-i) Potential future uses of structured light to actuate and control active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' f) Metavehicles are able to change their motion direction (liner or circular) depending on the polarized polarization of the illuminating light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20,141 Structured light landscapes with different local polarization could guide the movement of such microvehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' g) Microrobots could comprise temperature- responsive bodies that shrink upon irradiation with IR light, thus reducing their drag force and increasing their propulsion magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Structured light with different local wavelengths could spatiotemporally change their propulsion magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' h) Spatiotemporally structured light could locally bend hydrogel nanoribbons66,101,152 and induce a snake-like motion, which could be exploited to propel microparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' i) Microwalkers could be driven by hydrogel-gold nanoparticle composites, which serve as artificial muscles and joints in response to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Us- ing spatiotemporally structured light, each artificial element could be addressed individually to enable microscale artificial walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 20 Laseron C d a e (milliseconds) 15μm Laseroff (milliseconds) t=6s t=3s =09 50μmfilled with gold nanorods that enable fast heating/cooling dynamics upon irradiation with IR light combined with volumetric shape changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='66 Covering the hydrogel composite on one side with a thin gold layer restricts the swelling and shrinking, leading to the formation of helical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='66,152,154 Temporally structured light enables rapid dynamic switch- ing between left-handed and right-handed helical configurations152 as well as translational movement near a solid surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='66 As the possibilities for structuring light increase with the advancement of light modulating devices, control of active matter with structured light could include: Control of microvehicles by structured polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Structured light with different local polarizations or wavelengths could guide the movement of microvehi- cles20,141 that can change their motion patters depending on the polarization of the illuminating light (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Shrinkable microrobots in structured intensity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Structured light could spatiotemporally change the propulsion magnitude of microrobots comprising temperature- responsive bodies that shrink upon illumination with infrared light,63,66,67 thus reduc- ing their drag force and increasing their propulsion magnitude or propulsion direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Propulsion by deformable nanoribbons in spatiotemporally oscillating fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Spatiotemporally structured light could induce a snake-like motion in hydrogel nanorib- bons66,101,152 functionalized with microparticles, which would then propel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 3D-printed articulated microbots actuated by spatiotemporally structured light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Recent advances in 3D printing have enabled the precise programmable control over the shape morphing and folding properties of soft stimuli-responsive composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='65,155–157 For example, embedded gold nanorods have been employed to en- hance light absorption in temperature-responsive hydrogel composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='66,152 Printing composite hydrogels including gold nanoparticles of different sizes and aspect ratios 21 could then enable researchers to address such hydrogels individually by exploiting dif- ferent plasmonic resonances to realize artificial muscles and joints for microscale walkers and crawlers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 6i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Conclusions In this perspective, we have discussed the ongoing progress towards actuating and control- ling soft active matter by exploiting the different properties of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' While changing the light intensity provides a remote effortless means to adjust the speed and direction of light- activated particles, the potential of other properties of light to control soft active matter actuation has been mostly left untapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' For example, selectivity to light wavelength can enable multiple propulsion mechanisms to coexist on a single particle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', by triggering ex- clusive light-matter interactions at different sites of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='18,109–111 Furthermore, light polarization and the transfer of its linear and angular momentum can enable complex combi- nations of translation and rotation in the propulsion of active particles without the need for any additional fuel source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='20,140,141 Finally, the use of spatiotemporally structured light can combine all of light’s different degrees of freedom into one powerful tool to control the actu- ation of active matter systems by light in a way that is flexible, selective and adaptive, yet concomitantly easy to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Such level of control through light can enable active matter researchers to test theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', phase transitions, optimal navigation strategies) as well as to develop applications in energy conversion, catalysis, drug-delivery and tissue engineering taking advantage of the fact that light is a broadly available and biocompatible source of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Conversely, a higher degree of control of active matter with light could prove useful to develop materials and devices based on active systems that can mold the flow of light in non-conventional ways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=', to realize novel light sources, neuromorphic computers and, even, displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 22 Acknowledgements M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='R acknowledges Antonio Ciarlo, Giuseppe Pesce and Martin Wittmann for fruitful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' acknowledges funding from Marie Sklodowska-Curie Individual Fellowship (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='101064381).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (Giorgio Volpe) acknowledges sponsorship for this work by the US Office of Naval Research Global (Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' N62909-18-1-2170).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (Giovanni Volpe) acknowledges funding from the Horizon Europe ERC Consolidator Grant MAPEI (grant number 101001267) and the Knut and Alice Wallenberg Foundation (grant number 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content='0079).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 23 References (1) Volpe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Maragò, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (7) Zemánek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Volpe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Jonáš, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Brzobohat`y, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Perspective on light-induced trans- port of particles: From optical forces to phoretic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Photonics 2019, 11, 577–678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (8) Jones, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' H.' metadata={'source': 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University Press: Cambridge, UK, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (9) Vicsek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Zafeiris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2012, 517, 71–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (10) Ramaswamy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Theory Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2017, 2017, 054002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (11) Needleman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Dogic, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Active matter at the interface between materials science and cell biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Berg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Bacterial motility: Machinery and mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Microbiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2022, 20, 161—-173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (14) Gaffney, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Gadêlha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Smith, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (35) Schuergers, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Lenn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Kampmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Meissner, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Wilde, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Cyanobacteria use micro-optics to sense light direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' eLife 2016, 5, e12620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (36) Dervaux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Capellazzi Resta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Brunet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-controlled flows in active fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Tuval, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light control of localized photo- bioconvection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2019, 123, 158101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Song, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Voigt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Engineering RGB color vision into Escherichia coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Poon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Painting with light- powered bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2018, 9, 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (43) Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Mou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Gong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Guan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-driven micro/nanomotors: From fundamentals to applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2017, 46, 6905–6926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (44) Safdar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Simmchen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Jänis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-driven micro-and nanomotors for environ- mental remediation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nano 2017, 4, 1602–1616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (45) Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Du, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-powered micro/nanomotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Micromachines 2018, 9, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (46) Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Zhan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-driven micro/nanomotor for promising biomedical tools: Principle, challenge, and prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2018, 51, 1957–1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (47) Villa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Pumera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Fuel-free light-driven micro/nanomachines: Artificial active matter mimicking nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2019, 48, 4966–4978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (48) Palagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Singh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Light-controlled micromotors and soft microrobots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Materials 2019, 7, 1900370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (49) Howse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Vafabakhsh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Golestanian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Self-motile colloidal particles: From directed propulsion to random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Engineering shapes of active colloids for tunable dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Current Opinion in Colloid and Interface Science 2022, 61, 101608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 28 (52) Ibele, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Mallouk, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Volpe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Isa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Microscale Marangoni surfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rev.' 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S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Alegria, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Isa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Self-propelling col- loidal finite state machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2022;' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Löwen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Bechinger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phototaxis of synthetic mi- croswimmers in optical landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2016, 7, 12828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (69) Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Misko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Gobeil, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Fassbender, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Cuniberti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Makarov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Baraban, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Inverse solidification induced by active Janus particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Materials 2020, 30, 2003851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (70) Ramananarivo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Palacci, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Targeted assembly and synchroniza- tion of self-spinning microgears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2018, 14, 1114–1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (79) Trivedi, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2020, 152, 040901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (94) Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Actuators based on liquid crystalline elastomer materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nanoscale 2013, 5, 5225–5240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (95) Rogóż, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Zeng, H.' metadata={'source': 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Light-driven, caterpillar-inspired miniature inching robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Macromol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Rapid Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2018, 39, 1700224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (98) Shahsavan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Aghakhani, A.' metadata={'source': 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of phototaxis by a c-di-GMP-dependent process with high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' eLife 2022, 11, e73405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (105) Verasztó, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Gühmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Jia, H.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Ferrari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Optical trapping and manipulation of nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nanotech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2013, 8, 807–819.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Marco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Palagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Gait learning for soft microrobots controlled by light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Möller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Microgel that swims to the beat of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' E 2021, 44, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Shape-morphing materials from stimuli- responsive hydrogel hybrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Accounts Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' 2017, 50, 161–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (156) Jeon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' (157) Erb, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Sander, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Grisch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Studart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Self-shaping composites with programmable bioinspired microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFPT4oBgHgl3EQfOjSp/content/2301.13034v1.pdf'} 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